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| − | '''Author's Note.''' The text that follows is a collection of notes that will eventually be developed into a paper on [[Charles Sanders Peirce]]'s [[Logic of Relatives (1870)|1870 memoir on the logic of relative terms]]. [[User:Jon Awbrey|Jon Awbrey]] 06:06, 8 October 2007 (PDT) | + | {{DISPLAYTITLE:Peirce's 1870 Logic Of Relatives}} |
| | + | '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_1870_Logic_Of_Relatives InterSciWiki version]. |
| | | | |
| − | ----
| + | '''Author: [[User:Jon Awbrey|Jon Awbrey]]''' |
| | | | |
| − | ==Selection 1==
| + | Peirce's text employs lower case letters for logical terms of general reference and upper case letters for logical terms of individual reference. General terms fall into types — absolute terms, dyadic relative terms, higher adic relative terms — and Peirce employs different typefaces to distinguish these. The following Tables indicate the typefaces that are used in the text below for Peirce's examples of general terms. |
| | + | |
| | + | <br> |
| | | | |
| − | <pre>
| + | {| align="center" cellspacing="6" width="90%" |
| − | | The letters of the alphabet will denote logical signs. | + | |+ <math>\text{Absolute Terms (Monadic Relatives)}\!</math> |
| − | | Now logical terms are of three grand classes. | |
| | | | | | |
| − | | The first embraces those whose logical form involves only the
| + | <math>\begin{array}{ll} |
| − | | conception of quality, and which therefore represent a thing
| + | \mathrm{a}. & \text{animal} |
| − | | simply as "a ---". These discriminate objects in the most
| + | \\ |
| − | | rudimentary way, which does not involve any consciousness | + | \mathrm{b}. & \text{black} |
| − | | of discrimination. They regard an object as it is in | + | \\ |
| − | | itself as 'such' ('quale'); for example, as horse, | + | \mathrm{f}. & \text{Frenchman} |
| − | | tree, or man. These are 'absolute terms'.
| + | \\ |
| | + | \mathrm{h}. & \text{horse} |
| | + | \\ |
| | + | \mathrm{m}. & \text{man} |
| | + | \\ |
| | + | \mathrm{p}. & \text{President of the United States Senate} |
| | + | \\ |
| | + | \mathrm{r}. & \text{rich person} |
| | + | \\ |
| | + | \mathrm{u}. & \text{violinist} |
| | + | \\ |
| | + | \mathrm{v}. & \text{Vice-President of the United States} |
| | + | \\ |
| | + | \mathrm{w}. & \text{woman} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | <br> |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | |+ <math>\text{Simple Relative Terms (Dyadic Relatives)}\!</math> |
| | | | | | |
| − | | The second class embraces terms whose logical form involves the
| + | <math>\begin{array}{ll} |
| − | | conception of relation, and which require the addition of another
| + | \mathit{a}. & \text{enemy} |
| − | | term to complete the denotation. These discriminate objects with a
| + | \\ |
| − | | distinct consciousness of discrimination. They regard an object as
| + | \mathit{b}. & \text{benefactor} |
| − | | over against another, that is as relative; as father of, lover of,
| + | \\ |
| − | | or servant of. These are 'simple relative terms'.
| + | \mathit{c}. & \text{conqueror} |
| | + | \\ |
| | + | \mathit{e}. & \text{emperor} |
| | + | \\ |
| | + | \mathit{h}. & \text{husband} |
| | + | \\ |
| | + | \mathit{l}. & \text{lover} |
| | + | \\ |
| | + | \mathit{m}. & \text{mother} |
| | + | \\ |
| | + | \mathit{n}. & \text{not} |
| | + | \\ |
| | + | \mathit{o}. & \text{owner} |
| | + | \\ |
| | + | \mathit{s}. & \text{servant} |
| | + | \\ |
| | + | \mathit{w}. & \text{wife} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | <br> |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | |+ <math>\text{Conjugative Terms (Higher Adic Relatives)}\!</math> |
| | | | | | |
| − | | The third class embraces terms whose logical form involves the
| + | <math>\begin{array}{ll} |
| − | | conception of bringing things into relation, and which require
| + | \mathfrak{b}. & \text{betrayer to ------ of ------} |
| − | | the addition of more than one term to complete the denotation.
| + | \\ |
| − | | They discriminate not only with consciousness of discrimination,
| + | \mathfrak{g}. & \text{giver to ------ of ------} |
| − | | but with consciousness of its origin. They regard an object
| + | \\ |
| − | | as medium or third between two others, that is as conjugative;
| + | \mathfrak{t}. & \text{transferrer from ------ to ------} |
| − | | as giver of --- to ---, or buyer of --- for --- from ---.
| + | \\ |
| − | | These may be termed 'conjugative terms'.
| + | \mathfrak{w}. & \text{winner over of ------ to ------ from ------} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | <br> |
| | + | |
| | + | Individual terms are taken to denote individual entities falling under a general term. Peirce uses upper case Roman letters for individual terms, for example, the individual horses <math>\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}</math> falling under the general term <math>\mathrm{h}\!</math> for ''horse''. |
| | + | |
| | + | The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context. Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual <math>\mathrm{X}\!</math> of the genus <math>\mathrm{x}\!</math> and the element <math>x\!</math> of the set <math>X\!</math> as we pass between the two styles of text. |
| | + | |
| | + | __TOC__ |
| | + | |
| | + | ==Selection 1== |
| | + | |
| | + | ===Use of the Letters=== |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | The conjugative term involves the conception of 'third', the relative that of
| + | <p>The letters of the alphabet will denote logical signs.</p> |
| − | | second or 'other', the absolute term simply considers 'an' object. No fourth
| + | |
| − | | class of terms exists involving the conception of 'fourth', because when that
| + | <p>Now logical terms are of three grand classes.</p> |
| − | | of 'third' is introduced, since it involves the conception of bringing objects
| + | |
| − | | into relation, all higher numbers are given at once, inasmuch as the conception
| + | <p>The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as “a ——”. These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as ''such'' (''quale''); for example, as horse, tree, or man. These are ''absolute terms''.</p> |
| − | | of bringing objects into relation is independent of the number of members of the
| |
| − | | relationship. Whether this 'reason' for the fact that there is no fourth class
| |
| − | | of terms fundamentally different from the third is satisfactory of not, the fact
| |
| − | | itself is made perfectly evident by the study of the logic of relatives.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.63
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre> | |
| | | | |
| − | ==Commentary Note 1==
| + | <p>The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are ''simple relative terms''.</p> |
| | | | |
| − | <pre> | + | <p>The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of —— to ——, or buyer of —— for —— from ——. These may be termed ''conjugative terms''.</p> |
| − | I am going to experiment with an interlacing commentary
| |
| − | on Peirce's 1870 "Logic of Relatives" paper, revisiting
| |
| − | some critical transitions from several different angles
| |
| − | and calling attention to a variety of puzzles, problems, | |
| − | and potentials that are not so often remarked or tapped.
| |
| | | | |
| − | What strikes me about the initial installment this time around is its
| + | <p>The conjugative term involves the conception of ''third'', the relative that of second or ''other'', the absolute term simply considers ''an'' object. No fourth class of terms exists involving the conception of ''fourth'', because when that of ''third'' is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this ''reason'' for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives.</p> |
| − | use of a certain pattern of argument that I can recognize as invoking
| |
| − | a "closure principle", and this is a figure of reasoning that Peirce
| |
| − | uses in three other places: his discussion of "continuous relations",
| |
| − | his definition of sign relations, and in the pragmatic maxim itself.
| |
| | | | |
| − | One might also call attention to the following two statements:
| + | <p>(Peirce, CP 3.63).</p> |
| | + | |} |
| | | | |
| − | | Now logical terms are of three grand classes.
| + | I am going to experiment with an interlacing commentary on Peirce's 1870 “Logic of Relatives” paper, revisiting some critical transitions from several different angles and calling attention to a variety of puzzles, problems, and potentials that are not so often remarked or tapped. |
| | | | |
| − | | No fourth class of terms exists involving the conception of 'fourth',
| + | What strikes me about the initial installment this time around is its use of a certain pattern of argument that I can recognize as invoking a ''closure principle'', and this is a figure of reasoning that Peirce uses in three other places: his discussion of [[continuous predicates]], his definition of [[sign relations]], and in the [[pragmatic maxim]] itself. |
| − | | because when that of 'third' is introduced, since it involves the
| |
| − | | conception of bringing objects into relation, all higher numbers are
| |
| − | | given at once, inasmuch as the conception of bringing objects into
| |
| − | | relation is independent of the number of members of the relationship.
| |
| − | </pre>
| |
| | | | |
| − | ==Selection 2==
| + | One might also call attention to the following two statements: |
| | | | |
| − | <pre> | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | | Numbers Corresponding to Letters
| |
| | | | | | |
| − | | I propose to use the term "universe" to denote that class of individuals
| + | <p>Now logical terms are of three grand classes.</p> |
| − | | 'about' which alone the whole discourse is understood to run. The universe,
| + | |- |
| − | | therefore, in this sense, as in Mr. De Morgan's, is different on different
| |
| − | | occasions. In this sense, moreover, discourse may run upon something which
| |
| − | | is not a subjective part of the universe; for instance, upon the qualities | |
| − | | or collections of the individuals it contains.
| |
| | | | | | |
| − | | I propose to assign to all logical terms, numbers; to an absolute term,
| + | <p>No fourth class of terms exists involving the conception of ''fourth'', because when that of ''third'' is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.</p> |
| − | | the number of individuals it denotes; to a relative term, the average
| + | |} |
| − | | number of things so related to one individual. Thus in a universe of
| |
| − | | perfect men ('men'), the number of "tooth of" would be 32. The number
| |
| − | | of a relative with two correlates would be the average number of things
| |
| − | | so related to a pair of individuals; and so on for relatives of higher
| |
| − | | numbers of correlates. I propose to denote the number of a logical term
| |
| − | | by enclosing the term in square brackets, thus ['t'].
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.65
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre> | |
| | | | |
| − | ==Commentary Note 2== | + | ==Selection 2== |
| | | | |
| − | <pre>
| + | ===Numbers Corresponding to Letters=== |
| − | Peirce's remarks at CP 3.65 are so replete with remarkable ideas,
| |
| − | some of them so taken for granted in mathematical discourse that
| |
| − | they usually escape explicit mention, and others so suggestive
| |
| − | of things to come in a future remote from his time of writing,
| |
| − | and yet so smoothly introduced in passing that it's all too
| |
| − | easy to overlook their consequential significance, that I
| |
| − | can do no better here than to highlight these ideas in
| |
| − | other words, whose main advantage is to be a little
| |
| − | more jarring to the mind's sensibilities.
| |
| | | | |
| − | | Numbers Corresponding to Letters | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | |
| |
| − | | I propose to use the term "universe" to denote that class of individuals
| |
| − | | 'about' which alone the whole discourse is understood to run. The universe,
| |
| − | | therefore, in this sense, as in Mr. De Morgan's, is different on different
| |
| − | | occasions. In this sense, moreover, discourse may run upon something which
| |
| − | | is not a subjective part of the universe; for instance, upon the qualities
| |
| − | | or collections of the individuals it contains.
| |
| | | | | | |
| − | | I propose to assign to all logical terms, numbers; to an absolute term,
| + | <p>I propose to use the term “universe” to denote that class of individuals ''about'' which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.</p> |
| − | | the number of individuals it denotes; to a relative term, the average
| |
| − | | number of things so related to one individual. Thus in a universe of
| |
| − | | perfect men ('men'), the number of "tooth of" would be 32. The number
| |
| − | | of a relative with two correlates would be the average number of things
| |
| − | | so related to a pair of individuals; and so on for relatives of higher
| |
| − | | numbers of correlates. I propose to denote the number of a logical term
| |
| − | | by enclosing the term in square brackets, thus ['t'].
| |
| − | |
| |
| − | | C.S. Peirce, 'Collected Papers', CP 3.65
| |
| | | | |
| − | 1. This mapping of letters to numbers, or logical terms to mathematical quantities,
| + | <p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (''men''), the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus <math>[t].\!</math></p> |
| − | is the very core of what "quantification theory" is all about, and definitely
| |
| − | more to the point than the mere "innovation" of using distinctive symbols
| |
| − | for the so-called "quantifiers". We will speak of this more later on.
| |
| | | | |
| − | 2. The mapping of logical terms to numerical measures,
| + | <p>(Peirce, CP 3.65).</p> |
| − | to express it in current language, would probably be
| + | |} |
| − | recognizable as some kind of "morphism" or "functor"
| |
| − | from a logical domain to a quantitative co-domain.
| |
| | | | |
| − | 3. Notice that Peirce follows the mathematician's usual practice, | + | Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse that they usually escape explicit mention, and others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly introduced in passing that it's all too easy to overlook their consequential significance, that I can do no better here than to highlight these ideas in other words, whose main advantage is to be a little more jarring to the mind's sensibilities. |
| − | then and now, of making the status of being an "individual" or
| |
| − | a "universal" relative to a discourse in progress. I have come
| |
| − | to appreciate more and more of late how radically different this
| |
| − | "patchwork" or "piecewise" approach to things is from the way of
| |
| − | some philosophers who seem to be content with nothing less than
| |
| − | many worlds domination, which means that they are never content
| |
| − | and rarely get started toward the solution of any real problem.
| |
| − | Just my observation, I hope you understand.
| |
| | | | |
| − | 4. It is worth noting that Peirce takes the "plural denotation"
| + | :* This mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what "quantification theory" is all about, and definitely more to the point than the mere "innovation" of using distinctive symbols for the so-called "quantifiers". We will speak of this more later on. |
| − | of terms for granted, or what's the number of a term for,
| |
| − | if it could not vary apart from being one or nil?
| |
| | | | |
| − | 5. I also observe that Peirce takes the individual objects of a particular
| + | :* The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as some kind of "morphism" or "functor" from a logical domain to a quantitative co-domain. |
| − | universe of discourse in a "generative" way, not a "totalizing" way,
| |
| − | and thus they afford us with the basis for talking freely about
| |
| − | collections, constructions, properties, qualities, subsets,
| |
| − | and "higher types", as the phrase is mint.
| |
| − | </pre>
| |
| | | | |
| − | ==Selection 3==
| + | :* Notice that Peirce follows the mathematician's usual practice, then and now, of making the status of being an "individual" or a "universal" relative to a discourse in progress. I have come to appreciate more and more of late how radically different this "patchwork" or "piecewise" approach to things is from the way of some philosophers who seem to be content with nothing less than many worlds domination, which means that they are never content and rarely get started toward the solution of any real problem. Just my observation, I hope you understand. |
| | | | |
| − | <pre>
| + | :* It is worth noting that Peirce takes the "plural denotation" of terms for granted, or what's the number of a term for, if it could not vary apart from being one or nil? |
| − | | The Signs of Inclusion, Equality, Etc.
| + | |
| | + | :* I also observe that Peirce takes the individual objects of a particular universe of discourse in a "generative" way, not a "totalizing" way, and thus they afford us with the basis for talking freely about collections, constructions, properties, qualities, subsets, and "higher types", as the phrase is mint. |
| | + | |
| | + | ==Selection 3== |
| | + | |
| | + | ===The Signs of Inclusion, Equality, Etc.=== |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | I shall follow Boole in taking the sign of equality to signify identity.
| + | <p>I shall follow Boole in taking the sign of equality to signify identity. Thus, if <math>\mathrm{v}\!</math> denotes the Vice-President of the United States, and <math>\mathrm{p}~\!</math> the President of the Senate of the United States,</p> |
| − | | Thus, if v denotes the Vice-President of the United States, and p the
| + | |- |
| − | | President of the Senate of the United States,
| + | | align="center" | <math>\mathrm{v} = \mathrm{p}\!</math> |
| | + | |- |
| | | | | | |
| − | | v = p | + | <p>means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President.</p> |
| | + | |
| | + | <p>The sign “less than” is to be so taken that</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{f} < \mathrm{m}~\!</math> |
| | + | |- |
| | | | | | |
| − | | means that every Vice-President of the United States is President of the
| + | <p>means that every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. It will follow from these significations of <math>=\!</math> and <math><\!</math> that the sign <math>-\!\!\!<\!</math> (or <math>\leqq</math>, “as small as”) will mean “is”. Thus,</p> |
| − | | Senate, and every President of the United States Senate is Vice-President. | + | |- |
| − | | The sign "less than" is to be so taken that | + | | align="center" | <math>\mathrm{f} ~-\!\!\!< \mathrm{m}</math> |
| | + | |- |
| | | | | | |
| − | | f < m | + | <p>means “every Frenchman is a man”, without saying whether there are any other men or not. So,</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{m} ~-\!\!\!< \mathit{l}</math> |
| | + | |- |
| | | | | | |
| − | | means that every Frenchman is a man, but there are men besides Frenchmen.
| + | <p>will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of <math>=\!</math> and <math><\!</math> plainly conform to the indispensable conditions. Upon the transitive character of these relations the syllogism depends, for by virtue of it, from</p> |
| − | | Drobisch has used this sign in the same sense. It will follow from these
| + | |- |
| − | | significations of '=' and '<' that the sign '-<' (or '=<', "as small as")
| |
| − | | will mean "is". Thus, | |
| | | | | | |
| − | | f -< m | + | {| width="100%" |
| | + | | width="25%" | |
| | + | | align="center" | <math>\mathrm{f} ~-\!\!\!< \mathrm{m}</math> |
| | + | | width="25%" | |
| | + | |- |
| | + | | <p>and</p> |
| | + | | align="center" | <math>\mathrm{m} ~-\!\!\!< \mathrm{a}</math> |
| | + | | |
| | + | |- |
| | + | | <p>we can infer that</p> |
| | + | | align="center" | <math>\mathrm{f} ~-\!\!\!< \mathrm{a}</math> |
| | + | | |
| | + | |} |
| | + | |- |
| | | | | | |
| − | | means "every Frenchman is a man", without saying whether there are any
| + | <p>that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.</p> |
| − | | other men or not. So,
| + | |
| − | |
| + | <p>But not only do the significations of <math>=\!</math> and <math><\!</math> here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write <math>5 < 7\!</math> is to say that <math>5\!</math> is part of <math>7\!</math>, just as to write <math>\mathrm{f} < \mathrm{m}~\!</math> is to say that Frenchmen are part of men. Indeed, if <math>\mathrm{f} < \mathrm{m}~\!</math>, then the number of Frenchmen is less than the number of men, and if <math>\mathrm{v} = \mathrm{p}\!</math>, then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.</p> |
| − | | 'm' -< 'l'
| + | |
| − | |
| + | <p>(Peirce, CP 3.66).</p> |
| − | | will mean that every mother of anything is a lover of the same thing;
| + | |} |
| − | | although this interpretation in some degree anticipates a convention to
| |
| − | | be made further on. These significations of '=' and '<' plainly conform
| |
| − | | to the indispensable conditions. Upon the transitive character of these
| |
| − | | relations the syllogism depends, for by virtue of it, from
| |
| − | |
| |
| − | | f -< m
| |
| − | |
| |
| − | | and
| |
| − | |
| |
| − | | m -< a,
| |
| − | |
| |
| − | | we can infer that
| |
| − | |
| |
| − | | f -< a;
| |
| − | |
| |
| − | | that is, from every Frenchman being a man and every
| |
| − | | man being an animal, that every Frenchman is an animal.
| |
| − | |
| |
| − | | But not only do the significations of '=' and '<' here adopted fulfill all
| |
| − | | absolute requirements, but they have the supererogatory virtue of being very
| |
| − | | nearly the same as the common significations. Equality is, in fact, nothing
| |
| − | | but the identity of two numbers; numbers that are equal are those which are
| |
| − | | predicable of the same collections, just as terms that are identical are those
| |
| − | | which are predicable of the same classes. So, to write 5 < 7 is to say that 5
| |
| − | | is part of 7, just as to write f < m is to say that Frenchmen are part of men.
| |
| − | | Indeed, if f < m, then the number of Frenchmen is less than the number of men,
| |
| − | | and if v = p, then the number of Vice-Presidents is equal to the number of
| |
| − | | Presidents of the Senate; so that the numbers may always be substituted
| |
| − | | for the terms themselves, in case no signs of operation occur in the
| |
| − | | equations or inequalities.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.66
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre> | |
| − | ==Commentary Note 3==
| |
| | | | |
| − | <pre>
| + | The quantifier mapping from terms to their numbers that Peirce signifies by means of the square bracket notation <math>[t]\!</math> has one of its principal uses in providing a basis for the computation of frequencies, probabilities, and all of the other statistical measures that can be constructed from these, and thus in affording what may be called a ''principle of correspondence'' between probability theory and its limiting case in the forms of logic. |
| − | The quantifier mapping from terms to their numbers that Peirce signifies | |
| − | by means of the square bracket notation has one of its principal uses in | |
| − | providing a basis for the computation of frequencies, probabilities, and | |
| − | all of the other statistical measures that can be constructed from these, | |
| − | and thus in affording what may be called a "principle of correspondence" | |
| − | between probability theory and its limiting case in the forms of logic. | |
| | | | |
| − | This brings us once again to the relativity of contingency and necessity, | + | This brings us once again to the relativity of contingency and necessity, as one way of approaching necessity is through the avenue of probability, describing necessity as a probability of 1, but the whole apparatus of probability theory only figures in if it is cast against the backdrop of probability space axioms, the reference class of distributions, and the sample space that we cannot help but to abduce upon the scene of observations. Aye, there's the snake eyes. And with them we can see that there is always an irreducible quantum of facticity to all our necessities. More plainly spoken, it takes a fairly complex conceptual infrastructure just to begin speaking of probabilities, and this setting can only be set up by means of abductive, fallible, hypothetical, and inherently risky mental acts. |
| − | as one way of approaching necessity is through the avenue of probability, | |
| − | describing necessity as a probability of 1, but the whole apparatus of | |
| − | probability theory only figures in if it is cast against the backdrop | |
| − | of probability space axioms, the reference class of distributions, | |
| − | and the sample space that we cannot help but to abdeuce upon the | |
| − | scene of observations. Aye, there's the snake eyes. And with | |
| − | them we can see that there is always an irreducible quantum | |
| − | of facticity to all our necessities. More plainly spoken, | |
| − | it takes a fairly complex conceptual infrastructure just | |
| − | to begin speaking of probabilities, and this setting | |
| − | can only be set up by means of abductive, fallible, | |
| − | hypothetical, and inherently risky mental acts. | |
| | | | |
| − | Pragmatic thinking is the logic of abduction, which is just another | + | Pragmatic thinking is the logic of abduction, which is just another way of saying that it addresses the question: “What may be hoped?” We have to face the possibility that it may be just as impossible to speak of “absolute identity” with any hope of making practical philosophical sense as it is to speak of “absolute simultaneity” with any hope of making operational physical sense. |
| − | way of saying that it addresses the question: "What may be hoped?" | |
| − | We have to face the possibility that it may be just as impossible | |
| − | to speak of "absolute identity" with any hope of making practical | |
| − | philosophical sense as it is to speak of "absolute simultaneity" | |
| − | with any hope of making operational physical sense. | |
| − | </pre>
| |
| | | | |
| | ==Selection 4== | | ==Selection 4== |
| | | | |
| − | <pre>
| + | ===The Signs for Addition=== |
| − | | The Signs for Addition
| + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | The sign of addition is taken by Boole so that
| + | <p>The sign of addition is taken by Boole so that</p> |
| | + | |- |
| | + | | align="center" | <math>x + y\!</math> |
| | + | |- |
| | | | | | |
| − | | x + y | + | <p>denotes everything denoted by <math>x\!</math>, and, ''besides'', everything denoted by <math>y\!</math>.</p> |
| | + | |
| | + | <p>Thus</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m} + \mathrm{w}~\!</math> |
| | + | |- |
| | | | | | |
| − | | denotes everything denoted by x, and, 'besides',
| + | <p>denotes all men, and, besides, all women.</p> |
| − | | everything denoted by y. | + | |
| | + | <p>This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken ''besides'' the objects denoted by the other.</p> |
| | + | |
| | + | <p>For example,</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{f} + \mathrm{u}\!</math> |
| | + | |- |
| | | | | | |
| − | | Thus | + | <p>means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are ''besides themselves''.</p> |
| | + | |
| | + | <p>For this reason alone, in a paper which is published in the Proceedings of the Academy for March 17, 1867, I preferred to take as the regular addition of logic a non-invertible process, such that</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m} ~+\!\!,~ \mathrm{b}</math> |
| | + | |- |
| | | | | | |
| − | | m + w
| + | <p>stands for all men and black things, without any implication that the black things are to be taken besides the men; and the study of the logic of relatives has supplied me with other weighty reasons for the same determination.</p> |
| − | |
| + | |
| − | | denotes all men, and, besides, all women.
| + | <p>Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in a tract called ''Pure Logic, or the Logic of Quality'' [1864], had anticipated me in substituting the same operation for Boole's addition, although he rejects Boole's operation entirely and writes the new one with a <math>+\!</math> sign while withholding from it the name of addition.</p> |
| − | |
| + | |
| − | | This signification for this sign is needed for
| + | <p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of ''taking together'' involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.</p> |
| − | | connecting the notation of logic with that of the
| + | |
| − | | theory of probabilities. But if there is anything
| + | <p>Addition being taken in this sense, ''nothing'' is to be denoted by ''zero'', for then</p> |
| − | | which is denoted by both terms of the sum, the latter
| + | |- |
| − | | no longer stands for any logical term on account of
| + | | align="center" | <math>x ~+\!\!,~ 0 ~=~ x</math> |
| − | | its implying that the objects denoted by one term
| + | |- |
| − | | are to be taken 'besides' the objects denoted by
| |
| − | | the other.
| |
| − | |
| |
| − | | For example,
| |
| − | |
| |
| − | | f + u
| |
| − | |
| |
| − | | means all Frenchmen besides all violinists, and,
| |
| − | | therefore, considered as a logical term, implies
| |
| − | | that all French violinists are 'besides themselves'.
| |
| − | |
| |
| − | | For this reason alone, in a paper which is published
| |
| − | | in the Proceedings of the Academy for March 17, 1867,
| |
| − | | I preferred to take as the regular addition of logic
| |
| − | | a non-invertible process, such that
| |
| − | |
| |
| − | | m +, b
| |
| − | |
| |
| − | | stands for all men and black things, without any implication that
| |
| − | | the black things are to be taken besides the men; and the study of
| |
| − | | the logic of relatives has supplied me with other weighty reasons for
| |
| − | | the same determination.
| |
| − | |
| |
| − | | Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in
| |
| − | | a tract called 'Pure Logic, or the Logic of Quality' [1864], had anticipated me in
| |
| − | | substituting the same operation for Boole's addition, although he rejects Boole's
| |
| − | | operation entirely and writes the new one with a '+' sign while withholding from
| |
| − | | it the name of addition.
| |
| | | | | | |
| − | | It is plain that both the regular non-invertible addition
| + | <p>whatever is denoted by <math>x\!</math>; and this is the definition of ''zero''. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of ''zero'' and that of nothing, and because we shall thus have</p> |
| − | | and the invertible addition satisfy the absolute conditions.
| + | |- |
| − | | But the notation has other recommendations. The conception
| + | | align="center" | <math>[0] ~=~ 0.</math> |
| − | | of 'taking together' involved in these processes is strongly
| + | |- |
| − | | analogous to that of summation, the sum of 2 and 5, for example,
| |
| − | | being the number of a collection which consists of a collection of | |
| − | | two and a collection of five. Any logical equation or inequality | |
| − | | in which no operation but addition is involved may be converted
| |
| − | | into a numerical equation or inequality by substituting the
| |
| − | | numbers of the several terms for the terms themselves -- | |
| − | | provided all the terms summed are mutually exclusive.
| |
| | | | | | |
| − | | Addition being taken in this sense,
| + | <p>(Peirce, CP 3.67).</p> |
| − | | 'nothing' is to be denoted by 'zero',
| + | |} |
| − | | for then
| |
| − | |
| |
| − | | x +, 0 = x,
| |
| − | |
| |
| − | | whatever is denoted by x; and this is the definition
| |
| − | | of 'zero'. This interpretation is given by Boole, and
| |
| − | | is very neat, on account of the resemblance between the
| |
| − | | ordinary conception of 'zero' and that of nothing, and
| |
| − | | because we shall thus have
| |
| − | |
| |
| − | | [0] = 0.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.67
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre> | |
| | | | |
| − | ==Commentary Note 4==
| + | A wealth of issues arises here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing. |
| | | | |
| − | <pre>
| + | The two papers that precede this one in CP 3 are Peirce's papers of March and September 1867 in the ''Proceedings of the American Academy of Arts and Sciences'', titled “On an Improvement in Boole's Calculus of Logic” and “Upon the Logic of Mathematics”, respectively. Among other things, these two papers provide us with further clues about the motivating considerations that brought Peirce to introduce the “number of a term” function, signified here by square brackets. I have already quoted from the “Logic of Mathematics” paper in a related connection. Here are the links to those excerpts: |
| − | A wealth of issues arise here that I hope
| |
| − | to take up in depth at a later point, but | |
| − | for the moment I shall be able to mention
| |
| − | only the barest sample of them in passing.
| |
| | | | |
| − | The two papers that precede this one in CP 3 are Peirce's papers of
| + | <dl style="margin-left:30px;"> |
| − | March and September 1867 in the 'Proceedings of the American Academy
| + | <dt>Limited Mark Universes |
| − | of Arts and Sciences', titled "On an Improvement in Boole's Calculus
| + | <dd>[http://web.archive.org/web/20140429004255/http://suo.ieee.org/ontology/msg04349.html (1)] |
| − | of Logic" and "Upon the Logic of Mathematics", respectively. Among
| + | <dd>[http://web.archive.org/web/20140429004359/http://suo.ieee.org/ontology/msg04350.html (2)] |
| − | other things, these two papers provide us with further clues about
| + | <dd>[http://web.archive.org/web/20140429004130/http://suo.ieee.org/ontology/msg04351.html (3)] |
| − | the motivating considerations that brought Peirce to introduce the
| + | </dl> |
| − | "number of a term" function, signified here by square brackets.
| |
| − | I have already quoted from the "Logic of Mathematics" paper in
| |
| − | a related connection. Here are the links to those excerpts:
| |
| | | | |
| − | http://suo.ieee.org/ontology/msg04350.html
| + | In setting up a correspondence between “letters” and “numbers”, Peirce constructs a structure-preserving map from a logical domain to a numerical domain. That he does this deliberately is evidenced by the care that he takes with the conditions under which the chosen aspects of structure are preserved, along with his recognition of the critical fact that zeroes are preserved by the mapping. |
| − | http://suo.ieee.org/ontology/msg04351.html
| |
| | | | |
| − | In setting up a correspondence between "letters" and "numbers",
| + | Incidentally, Peirce appears to have an inkling of the problems that would later be caused by using the plus sign for inclusive disjunction, but his advice was overridden by the dialects of applied logic that developed in various communities, retarding the exchange of information among engineering, mathematical, and philosophical specialties all throughout the subsequent century. |
| − | my sense is that Peirce is "nocking an arrow", or constructing
| |
| − | some kind of structure-preserving map from a logical domain to
| |
| − | a numerical domain, and this interpretation is here reinforced
| |
| − | by the careful attention that he gives to the conditions under
| |
| − | which precisely which aspects of structure are preserved, plus
| |
| − | his telling recognition of the criterial fact that zeroes are | |
| − | preserved by the mapping. But here's the catch, the arrow is
| |
| − | from the qualitative domain to the quantitative domain, which
| |
| − | is just the opposite of what I tend to expect, since I think
| |
| − | of quantitative measures as preserving more information than
| |
| − | qualitative measures. To curtail the story, it is possible
| |
| − | to sort this all out, but that is a story for another day.
| |
| | | | |
| − | Other than that, I just want to red flag the beginnings
| + | ==Selection 5== |
| − | of another one of those "failures to communicate" that
| |
| − | so dogged the disciplines in the 20th Century, namely,
| |
| − | the fact that Peirce seemed to have an inkling about
| |
| − | the problems that would be caused by using the plus
| |
| − | sign for inclusive disjunction, but, as it happens,
| |
| − | his advice was overridden by the usages in various
| |
| − | different communities, rendering the exchange of
| |
| − | information among engineering, mathematical, and
| |
| − | philosophical specialties a minefield in place
| |
| − | of mindfield to this very day.
| |
| − | </pre>
| |
| | | | |
| − | ==Selection 5== | + | ===The Signs for Multiplication=== |
| | | | |
| − | <pre> | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | | The Signs for Multiplication
| |
| | | | | | |
| − | | I shall adopt for the conception of multiplication
| + | <p>I shall adopt for the conception of multiplication ''the application of a relation'', in such a way that, for example, <math>\mathit{l}\mathrm{w}~\!</math> shall denote whatever is lover of a woman. This notation is the same as that used by Mr. De Morgan, although he appears not to have had multiplication in his mind.</p> |
| − | | 'the application of a relation', in such a way that,
| + | |
| − | | for example, 'l'w shall denote whatever is lover of
| + | <p><math>\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w})</math> will, then, denote whatever is servant of anything of the class composed of men and women taken together. So that:</p> |
| − | | a woman. This notation is the same as that used by
| + | |- |
| − | | Mr. De Morgan, although he apears not to have had
| + | | align="center" | <math>\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w}) ~=~ \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}.</math> |
| − | | multiplication in his mind.
| + | |- |
| | | | | | |
| − | | 's'(m +, w) will, then, denote whatever is
| + | <p><math>(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w}</math> will denote whatever is lover or servant to a woman, and:</p> |
| − | | servant of anything of the class composed | + | |- |
| − | | of men and women taken together. So that: | + | | align="center" | <math>(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w} ~=~ \mathit{l}\mathrm{w} ~+\!\!,~ \mathit{s}\mathrm{w}.</math> |
| | + | |- |
| | | | | | |
| − | | 's'(m +, w) = 's'm +, 's'w. | + | <p><math>(\mathit{s}\mathit{l})\mathrm{w}\!</math> will denote whatever stands to a woman in the relation of servant of a lover, and:</p> |
| | + | |- |
| | + | | align="center" | <math>(\mathit{s}\mathit{l})\mathrm{w} ~=~ \mathit{s}(\mathit{l}\mathrm{w}).</math> |
| | + | |- |
| | | | | | |
| − | | ('l' +, 's')w will denote whatever is
| + | <p>Thus all the absolute conditions of multiplication are satisfied.</p> |
| − | | lover or servant to a woman, and: | + | |
| | + | <p>The term “identical with ——” is a unity for this multiplication. That is to say, if we denote “identical with ——” by <math>\mathit{1}\!</math> we have:</p> |
| | + | |- |
| | + | | align="center" | <math>x \mathit{1} ~=~ x ~ ,</math> |
| | + | |- |
| | | | | | |
| − | | ('l' +, 's')w = 'l'w +, 's'w.
| + | <p>whatever relative term <math>x\!</math> may be. For what is a lover of something identical with anything, is the same as a lover of that thing.</p> |
| − | |
| |
| − | | ('sl')w will denote whatever stands to
| |
| − | | a woman in the relation of servant of
| |
| − | | a lover, and:
| |
| − | |
| |
| − | | ('sl')w = 's'('l'w).
| |
| − | |
| |
| − | | Thus all the absolute conditions
| |
| − | | of multiplication are satisfied.
| |
| − | |
| |
| − | | The term "identical with ---" is a unity
| |
| − | | for this multiplication. That is to say,
| |
| − | | if we denote "identical with ---" by !1!
| |
| − | | we have:
| |
| − | |
| |
| − | | 'x'!1! = 'x',
| |
| − | |
| |
| − | | whatever relative term 'x' may be.
| |
| − | | For what is a lover of something
| |
| − | | identical with anything, is the
| |
| − | | same as a lover of that thing.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.68
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre> | |
| | | | |
| − | ==Commentary Note 5==
| + | <p>(Peirce, CP 3.68).</p> |
| | + | |} |
| | | | |
| − | <pre>
| + | Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of “information” to explain what it is that signs convey in the process. By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn. So let's take a moment to draw out a few of these characters. |
| − | Peirce in 1870 is five years down the road from the Peirce of 1865-1866 | |
| − | who lectured extensively on the role of sign relations in the logic of | |
| − | scientific inquiry, articulating their involvement in the three types | |
| − | of inference, and inventing the concept of "information" to explain | |
| − | what it is that signs convey in the process. By this time, then, | |
| − | the semiotic or sign relational approach to logic is so implicit | |
| − | in his way of working that he does not always take the trouble | |
| − | to point out its distinctive features at each and every turn. | |
| − | So let's take a moment to draw out a few of these characters. | |
| | | | |
| − | Sign relations, like any non-trivial brand of 3-adic relations, | + | [[Sign relations]], like any brand of non-trivial [[3-adic relations]], can become overwhelming to think about once the cardinality of the object, sign, and interpretant domains or the complexity of the relation itself ascends beyond the simplest examples. Furthermore, most of the strategies that we would normally use to control the complexity, like neglecting one of the domains, in effect, projecting the 3-adic sign relation onto one of its 2-adic faces, or focusing on a single ordered triple of the form <math>(o, s, i)\!</math> at a time, can result in our receiving a distorted impression of the sign relation's true nature and structure. |
| − | can become overwhelming to think about once the cardinality of | |
| − | the object, sign, and interpretant domains or the complexity | |
| − | of the relation itself ascends beyond the simplest examples. | |
| − | Furthermore, most of the strategies that we would normally | |
| − | use to control the complexity, like neglecting one of the | |
| − | domains, in effect, projecting the 3-adic sign relation | |
| − | onto one of its 2-adic faces, or focusing on a single | |
| − | ordered triple of the form <o, s, i> at a time, can | |
| − | result in our receiving a distorted impression of | |
| − | the sign relation's true nature and structure. | |
| | | | |
| − | I find that it helps me to draw, or at least to imagine drawing, | + | I find that it helps me to draw, or at least to imagine drawing, diagrams of the following form, where I can keep tabs on what's an object, what's a sign, and what's an interpretant sign, for a selected set of sign-relational triples. |
| − | diagrams of the following form, where I can keep tabs on what's | |
| − | an object, what's a sign, and what's an interpretant sign, for | |
| − | a selected set of sign-relational triples. | |
| | | | |
| − | Here is how I would picture Peirce's example of equivalent terms: | + | Here is how I would picture Peirce's example of equivalent terms, <math>\mathrm{v} = \mathrm{p},\!</math> where <math>{}^{\backprime\backprime} \mathrm{v} {}^{\prime\prime}\!</math> denotes the Vice-President of the United States, and <math>{}^{\backprime\backprime} \mathrm{p} {}^{\prime\prime}\!</math> denotes the President of the Senate of the United States. |
| − | v = p, where "v" denotes the Vice-President of the United States, | |
| − | and "p" denotes the President of the Senate of the United States. | |
| | | | |
| − | o-----------------------------o-----------------------------o
| + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" |
| − | | Objective Framework (OF) | Interpretive Framework (IF) | | + | | [[Image:LOR 1870 Figure 1.jpg]] |
| − | o-----------------------------o-----------------------------o
| + | |- |
| − | | Objects | Signs |
| + | | height="20px" valign="top" | <math>\text{Figure 1}~\!</math> |
| − | o-----------------------------o-----------------------------o
| + | |} |
| − | | | | |
| − | | o "v" |
| |
| − | | / |
| |
| − | | / |
| |
| − | | / |
| |
| − | | o ... o-----------@ |
| |
| − | | \ |
| |
| − | | \ |
| |
| − | | \ | | |
| − | | o "p" |
| |
| − | | |
| |
| − | o-----------------------------o-----------------------------o
| |
| | | | |
| − | Depending on whether we interpret the terms "v" and "p" as applying to | + | Depending on whether we interpret the terms <math>{}^{\backprime\backprime} \mathrm{v} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{p} {}^{\prime\prime}\!</math> as applying to persons who hold these offices at one particular time or as applying to all those persons who have held these offices over an extended period of history, their denotations may be either singular of plural, respectively. |
| − | persons who hold these offices at one particular time or as applying to | |
| − | all those persons who have held these offices over an extended period of | |
| − | history, their denotations may be either singular of plural, respectively. | |
| | | | |
| − | As a shortcut technique for indicating general denotations or plural referents, | + | As a shortcut technique for indicating general denotations or plural referents, I will use the ''elliptic convention'' that represents these by means of figures like “o o o” or “o … o”, placed at the object ends of sign relational triads. |
| − | I will use the "elliptic convention" that represents these by means of figures | |
| − | like "o o o" or "o ... o", placed at the object ends of sign relational triads. | |
| | | | |
| − | For a more complex example, here is how I would picture Peirce's example | + | For a more complex example, here is how I would picture Peirce's example of an equivalence between terms that comes about by applying one of the distributive laws, for relative multiplication over absolute summation. |
| − | of an equivalence between terms that comes about by applying one of the | |
| − | distributive laws, for relative multiplication over absolute summation. | |
| | | | |
| − | o-----------------------------o-----------------------------o
| + | {| align="center" border="0" cellspacing="10" style="text-align:center; width:100%" |
| − | | Objective Framework (OF) | Interpretive Framework (IF) | | + | | [[Image:LOR 1870 Figure 2.jpg]] |
| − | o-----------------------------o-----------------------------o
| + | |- |
| − | | Objects | Signs | | + | | height="20px" valign="top" | <math>\text{Figure 2}\!</math> |
| − | o-----------------------------o-----------------------------o
| + | |} |
| − | | | | |
| − | | o "'s'(m +, w)" |
| |
| − | | / |
| |
| − | | / |
| |
| − | | / |
| |
| − | | o ... o-----------@ |
| |
| − | | \ |
| |
| − | | \ |
| |
| − | | \ |
| |
| − | | o "'s'm +, 's'w" |
| |
| − | | |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | </pre> | |
| | | | |
| | ==Selection 6== | | ==Selection 6== |
| | | | |
| − | <pre>
| + | ===The Signs for Multiplication (cont.)=== |
| − | | The Signs for Multiplication (cont.)
| + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | A conjugative term like 'giver' naturally requires two correlates,
| + | <p>A conjugative term like ''giver'' naturally requires two correlates, one denoting the thing given, the other the recipient of the gift.</p> |
| − | | one denoting the thing given, the other the recipient of the gift.
| + | |
| | + | <p>We must be able to distinguish, in our notation, the giver of <math>\mathrm{A}\!</math> to <math>\mathrm{B}\!</math> from the giver to <math>\mathrm{A}\!</math> of <math>\mathrm{B}\!</math>, and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that “giver of —— to ——” and “giver to —— of ——” will be expressed by different letters.</p> |
| | + | |
| | + | <p>Let <math>\mathfrak{g}</math> denote the latter of these conjugative terms. Then, the correlates or multiplicands of this multiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}\mathit{x}\mathit{y}</math> |
| | + | |- |
| | | | | | |
| − | | We must be able to distinguish, in our notation, the
| + | <p>will denote a giver to <math>\mathit{x}\!</math> of <math>\mathit{y}\!</math>.</p> |
| − | | giver of A to B from the giver to A of B, and, therefore,
| + | |
| − | | I suppose the signification of the letter equivalent to such
| + | <p>But according to the notation, <math>\mathit{x}\!</math> here multiplies <math>\mathit{y}\!</math>, so that if we put for <math>\mathit{x}\!</math> owner (<math>\mathit{o}\!</math>), and for <math>\mathit{y}\!</math> horse (<math>\mathrm{h}\!</math>),</p> |
| − | | a relative to distinguish the correlates as first, second, third,
| + | |- |
| − | | etc., so that "giver of --- to ---" and "giver to --- of ---" will | + | | align="center" | <math>\mathfrak{g}\mathit{o}\mathrm{h}</math> |
| − | | be expressed by different letters. | + | |- |
| | | | | | |
| − | | Let `g` denote the latter of these conjugative terms. Then, the correlates
| + | <p>appears to denote the giver of a horse to an owner of a horse. But let the individual horses be <math>\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}</math>, etc.</p> |
| − | | or multiplicands of this multiplier cannot all stand directly after it, as is | + | |
| − | | usual in multiplication, but may be ranged after it in regular order, so that: | + | <p>Then:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{h} ~=~ \mathrm{H} ~+\!\!,~ \mathrm{H}^{\prime} ~+\!\!,~ \mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}</math> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}\mathit{o}\mathrm{h} ~=~ \mathfrak{g}\mathit{o}(\mathrm{H} ~+\!\!,~ \mathrm{H}^{\prime} ~+\!\!,~ \mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}) ~=~ \mathfrak{g}\mathit{o}\mathrm{H} ~+\!\!,~ \mathfrak{g}\mathit{o}\mathrm{H}^{\prime} ~+\!\!,~ \mathfrak{g}\mathit{o}\mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}</math> |
| | + | |- |
| | | | | | |
| − | | `g`xy | + | <p>Now this last member must be interpreted as a giver of a horse to the owner of ''that'' horse, and this, therefore must be the interpretation of <math>\mathfrak{g}\mathit{o}\mathrm{h}</math>. This is always very important. ''A term multiplied by two relatives shows that the same individual is in the two relations.''</p> |
| | + | |
| | + | <p>If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}\mathit{l}\mathrm{w}\mathrm{h}</math>, |
| | + | |- |
| | | | | | |
| − | | will denote a giver to x of y.
| + | <p>we have written giver of a woman to a lover of her, and if we add brackets, thus,</p> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}(\mathit{l}\mathrm{w})\mathrm{h}</math>, |
| | + | |- |
| | | | | | |
| − | | But according to the notation,
| + | <p>we abandon the associative principle of multiplication.</p> |
| − | | x here multiplies y, so that | + | |
| − | | if we put for x owner ('o'), | + | <p>A little reflection will show that the associative principle must in some form or other be abandoned at this point. But while this principle is sometimes falsified, it oftener holds, and a notation must be adopted which will show of itself when it holds. We already see that we cannot express multiplication by writing the multiplicand directly after the multiplier; let us then affix subjacent numbers after letters to show where their correlates are to be found. The first number shall denote how many factors must be counted from left to right to reach the first correlate, the second how many ''more'' must be counted to reach the second, and so on.</p> |
| − | | and for y horse (h), | + | |
| | + | <p>Then, the giver of a horse to a lover of a woman may be written:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}_{12} \mathit{l}_1 \mathrm{w} \mathrm{h} ~=~ \mathfrak{g}_{11} \mathit{l}_2 \mathrm{h} \mathrm{w} ~=~ \mathfrak{g}_{2(-1)} \mathrm{h} \mathit{l}_1 \mathrm{w}</math>. |
| | + | |- |
| | | | | | |
| − | | `g`'o'h
| + | <p>Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number.</p> |
| | + | |
| | + | <p>A subjacent ''zero'' makes the term itself the correlate.</p> |
| | + | |
| | + | <p>Thus,</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{l}_0\!</math> |
| | + | |- |
| | | | | | |
| − | | appears to denote the giver of a horse
| + | <p>denotes the lover of ''that'' lover or the lover of himself, just as <math>\mathfrak{g}\mathit{o}\mathrm{h}</math> denotes that the horse is given to the owner of itself, for to make a term doubly a correlate is, by the distributive principle, to make each individual doubly a correlate, so that:</p> |
| − | | to an owner of a horse. But let the
| + | |- |
| − | | individual horses be H, H', H", etc. | + | | align="center" | <math>\mathit{l}_0 ~=~ \mathit{L}_0 ~+\!\!,~ \mathit{L}_0^{\prime} ~+\!\!,~ \mathit{L}_0^{\prime\prime} ~+\!\!,~ \text{etc.}</math> |
| | + | |- |
| | | | | | |
| − | | Then: | + | <p>A subjacent sign of infinity may indicate that the correlate is indeterminate, so that:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{l}_\infty</math> |
| | + | |- |
| | | | | | |
| − | | h = H +, H' +, H" +, etc. | + | <p>will denote a lover of something. We shall have some confirmation of this presently.</p> |
| | + | |
| | + | <p>If the last subjacent number is a ''one'' it may be omitted. Thus we shall have:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{l}_1 ~=~ \mathit{l}</math>, |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}_{11} ~=~ \mathfrak{g}_1 ~=~ \mathfrak{g}</math>. |
| | + | |- |
| | | | | | |
| − | | `g`'o'h = `g`'o'(H +, H' +, H" +, etc.)
| + | <p>This enables us to retain our former expressions <math>\mathit{l}\mathrm{w}~\!</math>, <math>\mathfrak{g}\mathit{o}\mathrm{h}</math>, etc.</p> |
| | + | |
| | + | <p>(Peirce, CP 3.69–70).</p> |
| | + | |} |
| | + | |
| | + | ===Comment : Sets as Logical Sums=== |
| | + | |
| | + | Peirce's way of representing sets as logical sums may seem archaic, but it is quite often used, and is actually the tool of choice in many branches of algebra, combinatorics, computing, and statistics to this very day. |
| | + | |
| | + | Peirce's application to logic is fairly novel, and the degree of his elaboration of the logic of relative terms is certainly original with him, but this particular genre of representation, commonly going under the handle of ''generating functions'', goes way back, well before anyone thought to stick a flag in set theory as a separate territory or to try to fence off our native possessions of it with expressly decreed axioms. And back in the days when a computer was just a person who computed, before we had the sorts of ''electronic register machines'' that we take so much for granted today, mathematicians were constantly using generating functions as a rough and ready type of addressable memory to sort, store, and keep track of their accounts of a wide variety of formal objects of thought. |
| | + | |
| | + | Let us look at a few simple examples of generating functions, much as I encountered them during my own first adventures in the Fair Land Of Combinatoria. |
| | + | |
| | + | Suppose that we are given a set of three elements, say, <math>\{ a, b, c \},\!</math> and we are asked to find all the ways of choosing a subset from this collection. |
| | + | |
| | + | We can represent this problem setup as the problem of computing the following product: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>(1 + a)(1 + b)(1 + c).\!</math> |
| | + | |} |
| | + | |
| | + | The factor <math>(1 + a)\!</math> represents the option that we have, in choosing a subset of <math>\{ a, b, c \},\!</math> to leave the element <math>a\!</math> out (signified by the <math>1\!</math>), or else to include it (signified by the <math>a\!</math>), and likewise for the other elements <math>b\!</math> and <math>c\!</math> in their turns. |
| | + | |
| | + | Probably on account of all those years I flippered away playing the oldtime pinball machines, I tend to imagine a product like this being displayed in a vertical array: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | = `g`'o'H +, `g`'o'H' +, `g`'o'H" +, etc. | + | <math>\begin{matrix} |
| | + | (1 ~+~ a) |
| | + | \\ |
| | + | (1 ~+~ b) |
| | + | \\ |
| | + | (1 ~+~ c) |
| | + | \end{matrix}</math> |
| | + | |} |
| | + | |
| | + | I picture this as a playboard with six bumpers, the ball chuting down the board in such a career that it strikes exactly one of the two bumpers on each and every one of the three levels. |
| | + | |
| | + | So a trajectory of the ball where it hits the <math>a\!</math> bumper on the 1st level, hits the <math>1\!</math> bumper on the 2nd level, hits the <math>c\!</math> bumper on the 3rd level, and then exits the board, represents a single term in the desired product and corresponds to the subset <math>\{ a, c \}.\!</math> |
| | + | |
| | + | Multiplying out the product <math>(1 + a)(1 + b)(1 + c),\!</math> one obtains: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Now this last member must be interpreted as a giver
| + | <math>\begin{array}{*{15}{c}} |
| − | | of a horse to the owner of 'that' horse, and this, | + | 1 & + & a & + & b & + & c & + & ab & + & ac & + & bc & + & abc. |
| − | | therefore must be the interpretation of `g`'o'h. | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | And this informs us that the subsets of choice are: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | This is always very important. | + | <math>\begin{matrix} |
| | + | \varnothing, & \{ a \}, & \{ b \}, & \{ c \}, & \{ a, b \}, & \{ a, c \}, & \{ b, c \}, & \{ a, b, c \}. |
| | + | \end{matrix}</math> |
| | + | |} |
| | + | |
| | + | ==Selection 7== |
| | + | |
| | + | ===The Signs for Multiplication (cont.)=== |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | 'A term multiplied by two relatives shows that
| + | <p>The associative principle does not hold in this counting of factors. Because it does not hold, these subjacent numbers are frequently inconvenient in practice, and I therefore use also another mode of showing where the correlate of a term is to be found. This is by means of the marks of reference, <math>\dagger ~ \ddagger ~ \parallel ~ \S ~ \P</math>, which are placed subjacent to the relative term and before and above the correlate. Thus, giver of a horse to a lover of a woman may be written:</p> |
| − | | the same individual is in the two relations.'
| + | |- |
| | + | | align="center" | <math>\mathfrak{g}_{\dagger\ddagger} \, ^\dagger\mathit{l}_\parallel \, ^\parallel\mathrm{w} \, ^\ddagger\mathrm{h}</math> |
| | + | |- |
| | | | | | |
| − | | If we attempt to express the giver of a horse to | + | <p>The asterisk I use exclusively to refer to the last correlate of the last relative of the algebraic term.</p> |
| − | | a lover of a woman, and for that purpose write:
| + | |
| | + | <p>Now, considering the order of multiplication to be: — a term, a correlate of it, a correlate of that correlate, etc. — there is no violation of the associative principle. The only violations of it in this mode of notation are that in thus passing from relative to correlate, we skip about among the factors in an irregular manner, and that we cannot substitute in such an expression as <math>\mathfrak{g}\mathit{o}\mathrm{h}</math> a single letter for <math>\mathit{o}\mathrm{h}.\!</math></p> |
| | + | |
| | + | <p>I would suggest that such a notation may be found useful in treating other cases of non-associative multiplication. By comparing this with what was said above [in CP 3.55] concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation. I am therefore using two alphabets, the Greek and Kennerly, where only one was necessary. But it is convenient to use both.</p> |
| | + | |
| | + | <p>(Peirce, CP 3.71–72).</p> |
| | + | |} |
| | + | |
| | + | ===Comment : Proto-Graphical Syntax=== |
| | + | |
| | + | It is clear from our last excerpt that Peirce is already on the verge of a graphical syntax for the logic of relatives. Indeed, it seems likely that he had already reached this point in his own thinking. |
| | + | |
| | + | For instance, it seems quite impossible to read his last variation on the theme of a “giver of a horse to a lover of a woman” without drawing lines of identity to connect up the corresponding marks of reference, like this: |
| | + | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 3.jpg]] || (3) |
| | + | |} |
| | + | |
| | + | ==Selection 8== |
| | + | |
| | + | ===The Signs for Multiplication (cont.)=== |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | `g`'l'wh, | + | <p>Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p> |
| | + | |
| | + | <p>Now the absolute term “man” is really exactly equivalent to the relative term “man that is ——”, and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.</p> |
| | + | |
| | + | <p>Then “man that is black” will be written:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m},\!\mathrm{b}\!</math> |
| | + | |- |
| | | | | | |
| − | | we have written giver of a woman to a lover of her,
| + | <p>But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one.</p> |
| − | | and if we add brackets, thus, | + | |
| | + | <p>Then:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> |
| | + | |- |
| | | | | | |
| − | | `g`('l'w)h, | + | <p>will denote a lover of a woman that is a servant of that woman.</p> |
| | + | |
| | + | <p>The comma here after <math>\mathit{l}\!</math> should not be considered as altering at all the meaning of <math>\mathit{l}\!</math>, but as only a subjacent sign, serving to alter the arrangement of the correlates.</p> |
| | + | |
| | + | <p>In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates.</p> |
| | + | |
| | + | <p>So:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m},\!,\!\mathrm{b},\!\mathrm{r}</math> |
| | + | |- |
| | | | | | |
| − | | we abandon the associative principle of multiplication. | + | <p>interpreted like</p> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}\mathit{o}\mathrm{h}</math> |
| | + | |- |
| | | | | | |
| − | | A little reflection will show that the associative principle must
| + | <p>means a man that is a rich individual and is a black that is that rich individual.</p> |
| − | | in some form or other be abandoned at this point. But while this
| + | |
| − | | principle is sometimes falsified, it oftener holds, and a notation
| + | <p>But this has no other meaning than:</p> |
| − | | must be adopted which will show of itself when it holds. We already
| + | |- |
| − | | see that we cannot express multiplication by writing the multiplicand
| + | | align="center" | <math>\mathrm{m},\!\mathrm{b},\!\mathrm{r}</math> |
| − | | directly after the multiplier; let us then affix subjacent numbers after
| + | |- |
| − | | letters to show where their correlates are to be found. The first number | |
| − | | shall denote how many factors must be counted from left to right to reach | |
| − | | the first correlate, the second how many 'more' must be counted to reach | |
| − | | the second, and so on. | |
| | | | | | |
| − | | Then, the giver of a horse to a lover of a woman may be written:
| + | <p>or a man that is a black that is rich.</p> |
| − | | | + | |
| − | | `g`_12 'l'_1 w h = `g`_11 'l'_2 h w = `g`_2(-1) h 'l'_1 w. | + | <p>Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number.</p> |
| − | | | + | |
| − | | Of course a negative number indicates that
| + | <p>If, therefore, <math>\mathit{l},\!,\!\mathit{s}\mathrm{w}</math> is not the same as <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates.</p> |
| − | | the former correlate follows the latter
| + | |
| − | | by the corresponding positive number.
| + | <p>And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should}, we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series “that is —— and is —— and is —— etc.”</p> |
| | + | |
| | + | <p>Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates?</p> |
| | + | |
| | + | <p>Any term may be regarded as having an infinite number of factors, those at the end being ''ones'', thus:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{l},\!\mathit{s}\mathrm{w} ~=~ \mathit{l},\!\mathit{s}\mathit{w},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1}, ~\text{etc.}</math> |
| | + | |- |
| | | | | | |
| − | | A subjacent 'zero' makes the term itself the correlate.
| + | <p>A subjacent number may therefore be as great as we please.</p> |
| | + | |
| | + | <p>But all these ''ones'' denote the same identical individual denoted by <math>\mathrm{w}\!</math>; what then can be the subjacent numbers to be applied to <math>\mathit{s}\!</math>, for instance, on account of its infinite “''that is''”'s? What numbers can separate it from being identical with <math>\mathrm{w}\!</math>? There are only two. The first is ''zero'', which plainly neutralizes a comma completely, since</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}</math> |
| | + | |- |
| | | | | | |
| − | | Thus, | + | <p>and the other is infinity; for as <math>1^\infty</math> is indeterminate in ordinary algbra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of ''ones'' is to leave it indeterminate.</p> |
| | + | |
| | + | <p>Accordingly,</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m},_\infty</math> |
| | + | |- |
| | | | | | |
| − | | 'l'_0
| + | <p>should be regarded as expressing ''some'' man.</p> |
| | + | |
| | + | <p>Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros.</p> |
| | + | |
| | + | <p>“Something” may then be expressed by:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{1}_\infty\!</math> |
| | + | |- |
| | | | | | |
| − | | denotes the lover of 'that' lover or the lover of himself, just as | + | <p>I shall for brevity frequently express this by an antique figure one <math>(\mathfrak{1}).</math></p> |
| − | | `g`'o'h denotes that the horse is given to the owner of itself, for | + | |
| − | | to make a term doubly a correlate is, by the distributive principle, | + | <p>“Anything” by:</p> |
| − | | to make each individual doubly a correlate, so that: | + | |- |
| | + | | align="center" | <math>\mathit{1}_0\!</math> |
| | + | |- |
| | | | | | |
| − | | 'l'_0 = L_0 +, L_0' +, L_0" +, etc.
| + | <p>I shall often also write a straight <math>1\!</math> for ''anything''.</p> |
| − | |
| + | |
| − | | A subjacent sign of infinity may
| + | <p>(Peirce, CP 3.73).</p> |
| − | | indicate that the correlate is
| + | |} |
| − | | indeterminate, so that:
| + | |
| − | |
| + | ===Commentary Note 8.1=== |
| − | | 'l'_oo
| + | |
| − | |
| + | To my way of thinking, CP 3.73 is one of the most remarkable passages in the history of logic. In this first pass over its deeper contents I won't be able to accord it much more than a superficial dusting off. |
| − | | will denote a lover of something.
| |
| − | | We shall have some confirmation
| |
| − | | of this presently.
| |
| − | | | |
| − | | If the last subjacent number is a 'one'
| |
| − | | it may be omitted. Thus we shall have:
| |
| − | |
| |
| − | | 'l'_1 = 'l',
| |
| − | |
| |
| − | | `g`_11 = `g`_1 = `g`.
| |
| − | |
| |
| − | | This enables us to retain our former expressions 'l'w, `g`'o'h, etc.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.69-70
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre>
| |
| | | | |
| − | ==Commentary Note 6==
| + | Let us imagine a concrete example that will serve in developing the uses of Peirce's notation. Entertain a discourse whose universe <math>X\!</math> will remind us a little of the cast of characters in Shakespeare's ''Othello''. |
| | | | |
| − | <pre> | + | {| align="center" cellspacing="6" width="90%" |
| − | Peirce's way of representing sets as sums may seem archaic, but it is
| + | | <math>X ~=~ \{ \mathrm{Bianca}, \mathrm{Cassio}, \mathrm{Clown}, \mathrm{Desdemona}, \mathrm{Emilia}, \mathrm{Iago}, \mathrm{Othello} \}.</math> |
| − | quite often used, and is actually the tool of choice in many branches
| + | |} |
| − | of algebra, combinatorics, computing, and statistics to this very day.
| |
| | | | |
| − | Peirce's application to logic is fairly novel, and the degree of his
| + | The universe <math>X\!</math> is “that class of individuals ''about'' which alone the whole discourse is understood to run” but its marking out for special recognition as a universe of discourse in no way rules out the possibility that “discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains” (CP 3.65). |
| − | elaboration of the logic of relative terms is certainly original with
| |
| − | him, but this particular genre of representation, commonly going under
| |
| − | the handle of "generating functions", goes way back, well before anyone
| |
| − | thought to stick a flag in set theory as a separate territory or to try
| |
| − | to fence off our native possessions of it with expressly decreed axioms.
| |
| − | And back in the days when computers were people, before we had the sorts
| |
| − | of "electronic register machines" that we take so much for granted today,
| |
| − | mathematicians were constantly using generating functions as a rough and
| |
| − | ready type of addressable memory to sort, store, and keep track of their
| |
| − | accounts of a wide variety of formal objects of thought.
| |
| | | | |
| − | Let us look at a few simple examples of generating functions,
| + | In order to provide ourselves with the convenience of abbreviated terms, while preserving Peirce's conventions about capitalization, we may use the alternate names <math>^{\backprime\backprime}\mathrm{u}^{\prime\prime}</math> for the universe <math>X\!</math> and <math>^{\backprime\backprime}\mathrm{Jeste}^{\prime\prime}</math> for the character <math>\mathrm{Clown}.~\!</math> This permits the above description of the universe of discourse to be rewritten in the following fashion: |
| − | much as I encountered them during my own first adventures in
| |
| − | the Fair Land Of Combinatoria. | |
| | | | |
| − | Suppose that we are given a set of three elements,
| + | {| align="center" cellspacing="6" width="90%" |
| − | say, {a, b, c}, and we are asked to find all the
| + | | <math>\mathrm{u} ~=~ \{ \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{I}, \mathrm{J}, \mathrm{O} \}</math> |
| − | ways of choosing a subset from this collection.
| + | |} |
| | | | |
| − | We can represent this problem setup as the
| + | This specification of the universe of discourse could be summed up in Peirce's notation by the following equation: |
| − | problem of computing the following product:
| |
| | | | |
| − | (1 + a)(1 + b)(1 + c).
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{*{15}{c}} |
| | + | \mathbf{1} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | The factor (1 + a) represents the option that we have, in choosing
| + | Within this discussion, then, the ''individual terms'' are as follows: |
| − | a subset of {a, b, c}, to leave the 'a' out (signified by the "1"),
| |
| − | or else to include it (signified by the "a"), and likewise for the
| |
| − | other elements 'b' and 'c' in their turns.
| |
| | | | |
| − | Probably on account of all those years I flippered away
| + | {| align="center" cellspacing="6" width="90%" |
| − | playing the oldtime pinball machines, I tend to imagine
| + | | |
| − | a product like this being displayed in a vertical array:
| + | <math>\begin{matrix} |
| | + | ^{\backprime\backprime}\mathrm{B}^{\prime\prime}, & |
| | + | ^{\backprime\backprime}\mathrm{C}^{\prime\prime}, & |
| | + | ^{\backprime\backprime}\mathrm{D}^{\prime\prime}, & |
| | + | ^{\backprime\backprime}\mathrm{E}^{\prime\prime}, & |
| | + | ^{\backprime\backprime}\mathrm{I}^{\prime\prime}, & |
| | + | ^{\backprime\backprime}\mathrm{J}^{\prime\prime}, & |
| | + | ^{\backprime\backprime}\mathrm{O}^{\prime\prime} |
| | + | \end{matrix}</math> |
| | + | |} |
| | | | |
| − | (1 + a)
| + | Each of these terms denotes in a singular fashion the corresponding individual in <math>X.\!</math> |
| − | (1 + b)
| |
| − | (1 + c)
| |
| | | | |
| − | I picture this as a playboard with six "bumpers",
| + | By way of ''general terms'' in this discussion, we may begin with the following set: |
| − | the ball chuting down the board in such a career
| |
| − | that it strikes exactly one of the two bumpers
| |
| − | on each and every one of the three levels.
| |
| | | | |
| − | So a trajectory of the ball where it
| + | {| align="center" cellspacing="6" width="90%" |
| − | hits the "a" bumper on the 1st level,
| + | | |
| − | hits the "1" bumper on the 2nd level,
| + | <math>\begin{array}{ccl} |
| − | hits the "c" bumper on the 3rd level,
| + | ^{\backprime\backprime}\mathrm{b}^{\prime\prime} |
| − | and then exits the board, represents
| + | & = & |
| − | a single term in the desired product
| + | ^{\backprime\backprime}\mathrm{black}^{\prime\prime} |
| − | and corresponds to the subset {a, c}.
| + | \\[6pt] |
| | + | ^{\backprime\backprime}\mathrm{m}^{\prime\prime} |
| | + | & = & |
| | + | ^{\backprime\backprime}\mathrm{man}^{\prime\prime} |
| | + | \\[6pt] |
| | + | ^{\backprime\backprime}\mathrm{w}^{\prime\prime} |
| | + | & = & |
| | + | ^{\backprime\backprime}\mathrm{woman}^{\prime\prime} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | Multiplying out (1 + a)(1 + b)(1 + c), one obtains:
| + | The denotation of a general term may be given by means of an equation between terms: |
| | | | |
| − | 1 + a + b + c + ab + ac + bc + abc.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{*{15}{c}} |
| | + | \mathrm{b} |
| | + | & = & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | And this informs us that the subsets of choice are:
| + | ===Commentary Note 8.2=== |
| | | | |
| − | {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.
| + | I continue with my commentary on CP 3.73, developing the ''Othello'' example as a way of illustrating Peirce's concepts. |
| − | </pre>
| |
| | | | |
| − | ==Selection 7==
| + | In the development of the story so far, we have a universe of discourse that can be characterized by means of the following system of equations: |
| | | | |
| − | <pre>
| + | {| align="center" cellspacing="6" width="90%" |
| − | | The Signs for Multiplication (cont.) | |
| | | | | | |
| − | | The associative principle does not hold in this counting
| + | <math>\begin{array}{*{15}{c}} |
| − | | of factors. Because it does not hold, these subjacent
| + | \mathbf{1} |
| − | | numbers are frequently inconvenient in practice, and
| + | & = & \mathrm{B} |
| − | | I therefore use also another mode of showing where
| + | & +\!\!, & \mathrm{C} |
| − | | the correlate of a term is to be found. This is
| + | & +\!\!, & \mathrm{D} |
| − | | by means of the marks of reference, † ‡ || § ¶,
| + | & +\!\!, & \mathrm{E} |
| − | | which are placed subjacent to the relative | + | & +\!\!, & \mathrm{I} |
| − | | term and before and above the correlate.
| + | & +\!\!, & \mathrm{J} |
| − | | Thus, giver of a horse to a lover of
| + | & +\!\!, & \mathrm{O} |
| − | | a woman may be written: | + | \\[6pt] |
| | + | \mathrm{b} |
| | + | & = & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | This much provides a basis for collection of absolute terms that I plan to use in this example. Let us now consider how we might represent a sufficiently exemplary collection of relative terms. |
| | + | |
| | + | Consider the genesis of relative terms, for example: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | `g`_†‡ †'l'_|| ||w ‡h. | + | <math>\begin{array}{l} |
| | + | ^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime} |
| | + | \\[6pt] |
| | + | ^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~}\, ^{\prime\prime} |
| | + | \\[6pt] |
| | + | ^{\backprime\backprime}\, \text{winner over of}\, \underline{~~ ~~}\, \text{to}\, \underline{~~ ~~}\, \text{from}\, \underline{~~ ~~}\, ^{\prime\prime} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | We may regard these fill-in-the-blank forms as being derived by a kind of ''rhematic abstraction'' from the corresponding instances of absolute terms. |
| | + | |
| | + | In other words: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | The asterisk I use exclusively to refer to the last
| + | <p>The relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}</math></p> |
| − | | correlate of the last relative of the algebraic term.
| + | |
| | + | <p>can be reached by removing the absolute term <math>^{\backprime\backprime}\, \text{Emilia}\, ^{\prime\prime}</math></p> |
| | + | |
| | + | <p>from the absolute term <math>^{\backprime\backprime}\, \text{lover of Emilia}\, ^{\prime\prime}.</math></p> |
| | + | |
| | + | <p><math>\text{Iago}</math> is a lover of <math>\text{Emilia},</math> so the relate-correlate pair <math>\mathrm{I}:\mathrm{E}</math></p> |
| | + | |
| | + | <p>lies in the 2-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}.</math></p> |
| | + | |- |
| | | | | | |
| − | | Now, considering the order of multiplication to be: --
| + | <p>The relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~}\, ^{\prime\prime}</math></p> |
| − | | a term, a correlate of it, a correlate of that correlate,
| + | |
| − | | etc. -- there is no violation of the associative principle.
| + | <p>can be reached by removing the absolute terms <math>^{\backprime\backprime}\, \text{Othello}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{Desdemona}\, ^{\prime\prime}</math></p> |
| − | | The only violations of it in this mode of notation are that
| + | |
| − | | in thus passing from relative to correlate, we skip about
| + | <p>from the absolute term <math>^{\backprime\backprime}\, \text{betrayer to Othello of Desdemona}\, ^{\prime\prime}.</math></p> |
| − | | among the factors in an irregular manner, and that we
| + | |
| − | | cannot substitute in such an expression as `g`'o'h | + | <p><math>\text{Iago}</math> is a betrayer to <math>\text{Othello}</math> of <math>\text{Desdemona},</math> so the relate-correlate-correlate triple <math>\mathrm{I}:\mathrm{O}:\mathrm{D}</math></p> |
| − | | a single letter for 'o'h.
| + | |
| | + | <p>lies in the 3-adic relation assciated with the relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~}\, ^{\prime\prime}.\!</math></p> |
| | + | |- |
| | | | | | |
| − | | I would suggest that such a notation may be found useful in treating other
| + | <p>The relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{~~ ~~}\, \text{to}\, \underline{~~ ~~}\, \text{from}\, \underline{~~ ~~}\, ^{\prime\prime}</math></p> |
| − | | cases of non-associative multiplication. By comparing this with what was
| + | |
| − | | said above [in CP 3.55] concerning functional multiplication, it appears
| + | <p>can be reached by removing the absolute terms <math>^{\backprime\backprime}\, \text{Othello}\, ^{\prime\prime},</math> <math>^{\backprime\backprime}\, \text{Iago}\, ^{\prime\prime},</math> and <math>^{\backprime\backprime}\, \text{Cassio}\, ^{\prime\prime}</math></p> |
| − | | that multiplication by a conjugative term is functional, and that the
| + | |
| − | | letter denoting such a term is a symbol of operation. I am therefore
| + | <p>from the absolute term <math>^{\backprime\backprime}\, \text{winner over of Othello to Iago from Cassio}\, ^{\prime\prime}.</math></p> |
| − | | using two alphabets, the Greek and Kennerly, where only one was
| + | |
| − | | necessary. But it is convenient to use both. | + | <p><math>\text{Iago}</math> is a winner over of <math>\text{Othello}</math> to <math>\text{Iago}</math> from <math>\text{Cassio},\!</math> so the elementary relative term <math>\mathrm{I}:\mathrm{O}:\mathrm{I}:\mathrm{C}</math></p> |
| | + | |
| | + | <p>lies in the 4-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{~~ ~~}\, \text{to}\, \underline{~~ ~~}\, \text{from}\, \underline{~~ ~~}\, ^{\prime\prime}.</math></p> |
| | + | |} |
| | + | |
| | + | ===Commentary Note 8.3=== |
| | + | |
| | + | Speaking very strictly, we need to be careful to distinguish a ''relation'' from a ''relative term''. |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | C.S. Peirce, CP 3.71-72 | + | <p>The relation is an ''object'' of thought that may be regarded ''in extension'' as a set of ordered tuples that are known as its ''elementary relations''.</p> |
| | + | |- |
| | | | | | |
| − | | Charles Sanders Peirce,
| + | <p>The relative term is a ''sign'' that denotes certain objects, called its ''relates'', as these are determined in relation to certain other objects, called its ''correlates''. Under most circumstances, one may also regard the relative term as denoting the corresponding relation.</p> |
| − | |"Description of a Notation for the Logic of Relatives,
| + | |} |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | </pre> | |
| | | | |
| − | ==Commentary Note 7==
| + | Returning to the Othello example, let us take up the 2-adic relatives <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{servant of}\, \underline{~~ ~~}\, ^{\prime\prime}.</math> |
| | | | |
| − | <pre>
| + | Ignoring the many splendored nuances appurtenant to the idea of love, we may regard the relative term <math>\mathit{l}\!</math> for <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}</math> to be given by the following equation: |
| − | NB. On account of the invideous circumstance that various
| |
| − | listservers balk at Peirce's "marks of reference" -- or is
| |
| − | it only the Microsoft Cryptkeeper's kryptonizing of them? --
| |
| − | I will make the following substitutions in Peirce's text:
| |
| | | | |
| − | @ = dagger symbol
| + | {| align="center" cellspacing="6" width="90%" |
| − | # = double dagger
| + | | |
| − | || = parallel sign | + | <math>\begin{array}{*{13}{c}} |
| − | $ = section symbol
| + | \mathit{l} |
| − | % = paragraph mark
| + | & = & |
| | + | \mathrm{B}:\mathrm{C} |
| | + | & +\!\!, & |
| | + | \mathrm{C}:\mathrm{B} |
| | + | & +\!\!, & |
| | + | \mathrm{D}:\mathrm{O} |
| | + | & +\!\!, & |
| | + | \mathrm{E}:\mathrm{I} |
| | + | & +\!\!, & |
| | + | \mathrm{I}:\mathrm{E} |
| | + | & +\!\!, & |
| | + | \mathrm{O}:\mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | It is clear from our last excerpt that Peirce is already on the verge
| + | If for no better reason than to make the example more interesting, let us put aside all distinctions of rank and fealty, collapsing the motley crews of attendant, servant, subordinate, and so on, under the heading of a single service, denoted by the relative term <math>\mathit{s}\!</math> for <math>^{\backprime\backprime}\, \text{servant of}\, \underline{~~ ~~}\, ^{\prime\prime}.</math> The terms of this service are: |
| − | of a graphical syntax for the logic of relatives. Indeed, it seems | |
| − | likely that he had already reached this point in his own thinking.
| |
| | | | |
| − | For instance, it seems quite impossible to read his last variation on the
| + | {| align="center" cellspacing="6" width="90%" |
| − | theme of a "giver of a horse to a lover of a woman" without drawing lines
| + | | |
| − | of identity to connect up the corresponding marks of reference, like this:
| + | <math>\begin{array}{*{11}{c}} |
| | + | \mathit{s} |
| | + | & = & |
| | + | \mathrm{C}:\mathrm{O} |
| | + | & +\!\!, & |
| | + | \mathrm{E}:\mathrm{D} |
| | + | & +\!\!, & |
| | + | \mathrm{I}:\mathrm{O} |
| | + | & +\!\!, & |
| | + | \mathrm{J}:\mathrm{D} |
| | + | & +\!\!, & |
| | + | \mathrm{J}:\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | o---------------------------------------o
| + | The term <math>\mathrm{I}:\mathrm{C}\!</math> may also be implied, but, since it is so hotly arguable, I will leave it out of the toll. |
| − | | |
| |
| − | | @ || |
| |
| − | | / \ / \ |
| |
| − | | o o o o |
| |
| − | | `g`_@# @'l'_|| ||w #h |
| |
| − | | o o |
| |
| − | | \______________/ |
| |
| − | | # |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Giver of a Horse to a Lover of a Woman
| |
| − | </pre> | |
| | | | |
| − | <pre>
| + | One more thing that we need to be duly wary about: There are many different conventions in the field as to the ordering of terms in their applications, and it happens that different conventions will be more convenient under different circumstances, so there does not appear to be much of a chance that any one of them can be canonized once and for all. |
| | | | |
| | + | In the current reading, we are applying relative terms from right to left, and so our conception of relative multiplication, or relational composition, will need to be adjusted accordingly. |
| | | | |
| − | LOR. Note 8
| + | ===Commentary Note 8.4=== |
| | | | |
| | + | To familiarize ourselves with the forms of calculation that are available in Peirce's notation, let us compute a few of the simplest products that we find at hand in the Othello case. |
| | | | |
| | + | Here are the absolute terms: |
| | | | |
| − | | The Signs for Multiplication (cont.) | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Thus far, we have considered the multiplication of relative terms only. | + | <math>\begin{array}{*{15}{c}} |
| − | | Since our conception of multiplication is the application of a relation,
| + | \mathbf{1} |
| − | | we can only multiply absolute terms by considering them as relatives. | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{b} |
| | + | & = & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | \end{array}\!</math> |
| | + | |} |
| | + | |
| | + | Here are the 2-adic relative terms: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Now the absolute term "man" is really exactly equivalent to
| + | <math>\begin{array}{*{13}{c}} |
| − | | the relative term "man that is ---", and so with any other.
| + | \mathit{l} |
| − | | I shall write a comma after any absolute term to show that | + | & = & \mathrm{B}:\mathrm{C} |
| − | | it is so regarded as a relative term. | + | & +\!\!, & \mathrm{C}:\mathrm{B} |
| | + | & +\!\!, & \mathrm{D}:\mathrm{O} |
| | + | & +\!\!, & \mathrm{E}:\mathrm{I} |
| | + | & +\!\!, & \mathrm{I}:\mathrm{E} |
| | + | & +\!\!, & \mathrm{O}:\mathrm{D} |
| | + | \\[6pt] |
| | + | \mathit{s} |
| | + | & = & \mathrm{C}:\mathrm{O} |
| | + | & +\!\!, & \mathrm{E}:\mathrm{D} |
| | + | & +\!\!, & \mathrm{I}:\mathrm{O} |
| | + | & +\!\!, & \mathrm{J}:\mathrm{D} |
| | + | & +\!\!, & \mathrm{J}:\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Here are a few of the simplest products among these terms: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Then: | + | <math>\begin{array}{lll} |
| | + | \mathit{l}\mathbf{1} |
| | + | & = & |
| | + | \text{lover of anything} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{D}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{anything except}~\mathrm{J} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | "man that is black" | + | <math>\begin{array}{lll} |
| | + | \mathit{l}\mathrm{b} |
| | + | & = & |
| | + | \text{lover of a black} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{D}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | \mathrm{O} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | will be written | + | <math>\begin{array}{lll} |
| | + | \mathit{l}\mathrm{m} |
| | + | & = & |
| | + | \text{lover of a man} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{D}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | m,b. | + | <math>\begin{array}{lll} |
| | + | \mathit{l}\mathrm{w} |
| | + | & = & |
| | + | \text{lover of a woman} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{D}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | But not only may any absolute term be thus regarded as a relative term,
| + | <math>\begin{array}{lll} |
| − | | but any relative term may in the same way be regarded as a relative with | + | \mathit{s}\mathbf{1} |
| − | | one correlate more. It is convenient to take this additional correlate
| + | & = & |
| − | | as the first one. | + | \text{servant of anything} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Then: | + | <math>\begin{array}{lll} |
| | + | \mathit{s}\mathrm{b} |
| | + | & = & |
| | + | \text{servant of a black} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | \mathrm{O} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | 'l','s'w | + | <math>\begin{array}{lll} |
| | + | \mathit{s}\mathrm{m} |
| | + | & = & |
| | + | \text{servant of a man} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | will denote a lover of a woman that is a servant of that woman.
| + | <math>\begin{array}{lll} |
| | + | \mathit{s}\mathrm{w} |
| | + | & = & |
| | + | \text{servant of a woman} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{E} ~+\!\!,~ \mathrm{J} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | The comma here after 'l' should not be considered as altering at
| + | <math>\begin{array}{lll} |
| − | | all the meaning of 'l', but as only a subjacent sign, serving to
| + | \mathit{l}\mathit{s} |
| − | | alter the arrangement of the correlates. | + | & = & |
| | + | \text{lover of a servant of}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{D}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{B}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | In point of fact, since a comma may be added in this way to any
| + | <math>\begin{array}{lll} |
| − | | relative term, it may be added to one of these very relatives | + | \mathit{s}\mathit{l} |
| − | | formed by a comma, and thus by the addition of two commas
| + | & = & |
| − | | an absolute term becomes a relative of two correlates.
| + | \text{servant of a lover of}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{D}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Among other things, one observes that the relative terms <math>\mathit{l}\!</math> and <math>\mathit{s}\!</math> do not commute, that is, <math>\mathit{l}\mathit{s}\!</math> is not equal to <math>\mathit{s}\mathit{l}.~\!</math> |
| | + | |
| | + | ===Commentary Note 8.5=== |
| | + | |
| | + | Since multiplication by a 2-adic relative term is a logical analogue of matrix multiplication in linear algebra, all of the products that we computed above can be represented in terms of logical matrices and logical vectors. |
| | + | |
| | + | Here are the absolute terms again, followed by their representation as ''coefficient tuples'', otherwise thought of as ''coordinate vectors''. |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | So: | + | <math>\begin{array}{ccrcccccccccccl} |
| | + | \mathbf{1} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[10pt] |
| | + | & = & (1 & , & 1 & , & 1 & , & 1 & , & 1 & , & 1 & , & 1) |
| | + | \\[20pt] |
| | + | \mathrm{b} |
| | + | & = & |
| | + | & & |
| | + | & & |
| | + | & & |
| | + | & & |
| | + | & & |
| | + | & & |
| | + | \mathrm{O} |
| | + | \\[10pt] |
| | + | & = & (0 & , & 0 & , & 0 & , & 0 & , & 0 & , & 0 & , & 1) |
| | + | \\[20pt] |
| | + | \mathrm{m} |
| | + | & = & |
| | + | & & \mathrm{C} |
| | + | & & |
| | + | & & |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[10pt] |
| | + | & = & (0 & , & 1 & , & 0 & , & 0 & , & 1 & , & 1 & , & 1) |
| | + | \\[20pt] |
| | + | \mathrm{w} |
| | + | & = & \mathrm{B} |
| | + | & & |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | & & |
| | + | & & |
| | + | & & |
| | + | \\[10pt] |
| | + | & = & (1 & , & 0 & , & 1 & , & 1 & , & 0 & , & 0 & , & 0) |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Since we are going to be regarding these tuples as ''column vectors'', it is convenient to arrange them into a table of the following form: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | m,,b,r | + | <math>\begin{array}{c|cccc} |
| | + | \text{ } & \mathbf{1} & \mathrm{b} & \mathrm{m} & \mathrm{w} |
| | + | \\ |
| | + | \text{---} & \text{---} & \text{---} & \text{---} & \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 1 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{C} & 1 & 0 & 1 & 0 |
| | + | \\ |
| | + | \mathrm{D} & 1 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{E} & 1 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{I} & 1 & 0 & 1 & 0 |
| | + | \\ |
| | + | \mathrm{J} & 1 & 0 & 1 & 0 |
| | + | \\ |
| | + | \mathrm{O} & 1 & 1 & 1 & 0 |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Here are the 2-adic relative terms again, followed by their representation as coefficient matrices, in this case bordered by row and column labels to remind us what the coefficient values are meant to signify. |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | interpreted like | + | <math>\begin{array}{*{13}{c}} |
| | + | \mathit{l} |
| | + | & = & \mathrm{B}:\mathrm{C} |
| | + | & +\!\!, & \mathrm{C}:\mathrm{B} |
| | + | & +\!\!, & \mathrm{D}:\mathrm{O} |
| | + | & +\!\!, & \mathrm{E}:\mathrm{I} |
| | + | & +\!\!, & \mathrm{I}:\mathrm{E} |
| | + | & +\!\!, & \mathrm{O}:\mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | `g`'o'h | + | <math>\begin{array}{c|ccccccc} |
| | + | \mathit{l} & |
| | + | \mathrm{B} & |
| | + | \mathrm{C} & |
| | + | \mathrm{D} & |
| | + | \mathrm{E} & |
| | + | \mathrm{I} & |
| | + | \mathrm{J} & |
| | + | \mathrm{O} |
| | + | \\ |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{C} & 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{D} & 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{E} & 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{I} & 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{J} & 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{O} & 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | means a man that is a rich individual and | + | <math>\begin{array}{*{13}{c}} |
| − | | is a black that is that rich individual. | + | \mathit{s} |
| | + | & = & \mathrm{C}:\mathrm{O} |
| | + | & +\!\!, & \mathrm{E}:\mathrm{D} |
| | + | & +\!\!, & \mathrm{I}:\mathrm{O} |
| | + | & +\!\!, & \mathrm{J}:\mathrm{D} |
| | + | & +\!\!, & \mathrm{J}:\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | But this has no other meaning than: | + | <math>\begin{array}{c|ccccccc} |
| | + | \mathit{s} & |
| | + | \mathrm{B} & |
| | + | \mathrm{C} & |
| | + | \mathrm{D} & |
| | + | \mathrm{E} & |
| | + | \mathrm{I} & |
| | + | \mathrm{J} & |
| | + | \mathrm{O} |
| | + | \\ |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{C} & 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{D} & 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{E} & 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | \mathrm{I} & 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{J} & 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | \mathrm{O} & 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Here are the matrix representations of the products that we calculated before: |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | m,b,r | + | <math>\begin{matrix} |
| | + | \mathit{l}\mathbf{1} & = & \text{lover of anything} & = |
| | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | or a man that is a black that is rich. | + | <math> |
| | + | \begin{bmatrix} |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 0 \\ 1 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | Thus we see that, after one comma is added, the
| + | <math>\begin{matrix} |
| − | | addition of another does not change the meaning
| + | \mathit{l}\mathrm{b} & = & \text{lover of a black} & = |
| − | | at all, so that whatever has one comma after it
| + | \end{matrix}</math> |
| − | | must be regarded as having an infinite number. | + | |- |
| | | | | | |
| − | | If, therefore, 'l',,'s'w is not the same as 'l','s'w (as it plainly is not,
| + | <math> |
| − | | because the latter means a lover and servant of a woman, and the former a | + | \begin{bmatrix} |
| − | | lover of and servant of and same as a woman), this is simply because the
| + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| − | | writing of the comma alters the arrangement of the correlates. | + | \\ |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | And if we are to suppose that absolute terms are multipliers
| + | <math>\begin{matrix} |
| − | | at all (as mathematical generality demands that we should},
| + | \mathit{l}\mathrm{m} & = & \text{lover of a man} & = |
| − | | we must regard every term as being a relative requiring
| + | \end{matrix}</math> |
| − | | an infinite number of correlates to its virtual infinite
| + | |- |
| − | | series "that is --- and is --- and is --- etc." | |
| | | | | | |
| − | | Now a relative formed by a comma of course receives its
| + | <math> |
| − | | subjacent numbers like any relative, but the question is, | + | \begin{bmatrix} |
| − | | What are to be the implied subjacent numbers for these
| + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| − | | implied correlates? | + | \\ |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | Any term may be regarded as having an
| + | <math>\begin{matrix} |
| − | | infinite number of factors, those
| + | \mathit{l}\mathrm{w} & = & \text{lover of a woman} & = |
| − | | at the end being 'ones', thus: | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | 'l','s'w = 'l','s'w,!1!,!1!,!1!,!1!,!1!,!1!,!1!, etc.
| + | <math> |
| | + | \begin{bmatrix} |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 |
| | + | \end{bmatrix}\! |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | A subjacent number may therefore be as great as we please. | + | <math>\begin{matrix} |
| | + | \mathit{s}\mathbf{1} & = & \text{servant of anything} & = |
| | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | But all these 'ones' denote the same identical individual denoted
| + | <math> |
| − | | by w; what then can be the subjacent numbers to be applied to 's',
| + | \begin{bmatrix} |
| − | | for instance, on account of its infinite "that is"'s? What numbers
| + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| − | | can separate it from being identical with w? There are only two. | + | \\ |
| − | | The first is 'zero', which plainly neutralizes a comma completely,
| + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| − | | since | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 1 \\ 1 \\ 1 \\ 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | 's',_0 w = 's'w
| + | <math>\begin{matrix} |
| | + | \mathit{s}\mathrm{b} & = & \text{servant of a black} & = |
| | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | and the other is infinity; for as 1^oo is indeterminate
| + | <math> |
| − | | in ordinary algbra, so it will be shown hereafter to be | + | \begin{bmatrix} |
| − | | here, so that to remove the correlate by the product of
| + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| − | | an infinite series of 'ones' is to leave it indeterminate. | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | Accordingly, | + | <math>\begin{matrix} |
| | + | \mathit{s}\mathrm{m} & = & \text{servant of a man} & = |
| | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | m,_oo | + | <math> |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | should be regarded as expressing 'some' man. | + | <math>\begin{matrix} |
| | + | \mathit{s}\mathrm{w} & = & \text{servant of a woman} & = |
| | + | \end{matrix}\!</math> |
| | + | |- |
| | | | | | |
| − | | Any term, then, is properly to be regarded as having an infinite | + | <math> |
| − | | number of commas, all or some of which are neutralized by zeros. | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1 \\ 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | "Something" may then be expressed by: | + | <math>\begin{matrix} |
| | + | \mathit{l}\mathit{s} & = & \text{lover of a servant of ---} & = |
| | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | !1!_oo.
| + | <math> |
| − | |
| + | \begin{bmatrix} |
| − | | I shall for brevity frequently express this by an antique figure one (`1`).
| + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| − | | | + | \\ |
| − | | "Anything" by: | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | + | |
| | + | {| align="center" cellpadding="6" width="90%" |
| | | | | | |
| − | | !1!_0. | + | <math>\begin{matrix} |
| | + | \mathit{s}\mathit{l} & = & \text{servant of a lover of ---} & = |
| | + | \end{matrix}</math> |
| | + | |- |
| | | | | | |
| − | | I shall often also write a straight 1 for 'anything'.
| + | <math> |
| − | |
| + | \begin{bmatrix} |
| − | | C.S. Peirce, CP 3.73
| + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| − | |
| + | \\ |
| − | | Charles Sanders Peirce,
| + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| − | |"Description of a Notation for the Logic of Relatives,
| + | \\ |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| + | \\ |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429). | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 1 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | | | |
| | + | ===Commentary Note 8.6=== |
| | | | |
| | + | The foregoing has hopefully filled in enough background that we can begin to make sense of the more mysterious parts of CP 3.73. |
| | | | |
| − | LOR. Commentary Note 8.1
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p> |
| | | | |
| | + | <p>Now the absolute term “man” is really exactly equivalent to the relative term “man that is ——”, and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.</p> |
| | | | |
| | + | <p>Then “man that is black” will be written:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m},\!\mathrm{b}</math> |
| | + | |- |
| | + | | |
| | + | <p>(Peirce, CP 3.73).</p> |
| | + | |} |
| | | | |
| − | To my way of thinking, CP 3.73 is one of the most remarkable passages
| + | In any system where elements are organized according to types, there tend to be any number of ways in which elements of one type are naturally associated with elements of another type. If the association is anything like a logical equivalence, but with the first type being lower and the second type being higher in some sense, then one may speak of a ''semantic ascent'' from the lower to the higher type. |
| − | in the history of logic. In this first pass over its deeper contents | |
| − | I won't be able to accord it much more than a superficial dusting off.
| |
| | | | |
| − | As always, it is probably best to begin with a concrete example.
| + | For example, it is common in mathematics to associate an element <math>a\!</math> of a set <math>A\!</math> with the constant function <math>f_a : X \to A</math> that has <math>f_a (x) = a\!</math> for all <math>x\!</math> in <math>X,\!</math> where <math>X\!</math> is an arbitrary set. Indeed, the correspondence is so close that one often uses the same name <math>{}^{\backprime\backprime} a {}^{\prime\prime}</math> to denote both the element <math>a\!</math> in <math>A\!</math> and the function <math>a = f_a : X \to A,</math> relying on the context or an explicit type indication to tell them apart. |
| − | So let us initiate a discourse, whose universe X may remind us
| |
| − | a little of the cast of characters in Shakespeare's 'Othello'. | |
| | | | |
| − | X = {Bianca, Cassio, Clown, Desdemona, Emilia, Iago, Othello}.
| + | For another example, we have the ''tacit extension'' of a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> to a <math>(k+1)\!</math>-place relation <math>L^\prime \subseteq X_1 \times \ldots \times X_{k+1}\!</math> that we get by letting <math>L^\prime = L \times X_{k+1},</math> that is, by maintaining the constraints of <math>L\!</math> on the first <math>k\!</math> variables and letting the last variable wander freely. |
| | | | |
| − | The universe X is "that class of individuals 'about' which alone
| + | What we have here, if I understand Peirce correctly, is another such type of natural extension, sometimes called the ''diagonal extension''. This extension associates a <math>k\!</math>-adic relative or a <math>k\!</math>-adic relation, counting the absolute term and the set whose elements it denotes as the cases for <math>k = 0,\!</math> with a series of relatives and relations of higher adicities. |
| − | the whole discourse is understood to run" but its marking out for | |
| − | special recognition as a universe of discourse in no way rules out
| |
| − | the possibility that "discourse may run upon something which is not | |
| − | a subjective part of the universe; for instance, upon the qualities | |
| − | or collections of the individuals it contains" (CP 3.65).
| |
| | | | |
| − | In order to provide ourselves with the convenience of abbreviated terms,
| + | A few examples will suffice to anchor these ideas. |
| − | while staying a bit closer to Peirce's conventions about capitalization,
| |
| − | let us rename the universe "u", the Clown "Jeste", and then rewrite the
| |
| − | above description of the universe of discourse in the following fashion:
| |
| | | | |
| − | u = {B, C, D, E, I, J, O}.
| + | Absolute terms: |
| | | | |
| − | This specification of the universe of discourse could be
| + | {| align="center" cellspacing="6" width="90%" |
| − | summed up in Peirce's notation by the following equation:
| + | | |
| | + | <math>\begin{array}{*{11}{c}} |
| | + | \mathrm{m} |
| | + | & = & \text{man} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{n} |
| | + | & = & \text{noble} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w} |
| | + | & = & \text{woman} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 1 = B +, C +, D +, E +, I +, J +, O.
| + | Diagonal extensions: |
| | | | |
| − | Within this discussion, then, the "individual terms" are
| + | {| align="center" cellspacing="6" width="90%" |
| − | "B", "C", "D", "E", "I", "J", "O", each of which denotes | + | | |
| − | in a singular fashion the corresponding individual in X.
| + | <math>\begin{array}{*{11}{c}} |
| | + | \mathrm{m,} |
| | + | & = & \text{man that is}\, \underline{~~ ~~} |
| | + | & = & \mathrm{C}:\mathrm{C} |
| | + | & +\!\!, & \mathrm{I}:\mathrm{I} |
| | + | & +\!\!, & \mathrm{J}:\mathrm{J} |
| | + | & +\!\!, & \mathrm{O}:\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{n,} |
| | + | & = & \text{noble that is}\, \underline{~~ ~~} |
| | + | & = & \mathrm{C}:\mathrm{C} |
| | + | & +\!\!, & \mathrm{D}:\mathrm{D} |
| | + | & +\!\!, & \mathrm{O}:\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w,} |
| | + | & = & \text{woman that is}\, \underline{~~ ~~} |
| | + | & = & \mathrm{B}:\mathrm{B} |
| | + | & +\!\!, & \mathrm{D}:\mathrm{D} |
| | + | & +\!\!, & \mathrm{E}:\mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | As "general terms" of this discussion,
| + | Sample products: |
| − | we might begin with the following set:
| |
| | | | |
| − | "b" = "black" | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathrm{m},\!\mathrm{n} |
| | + | & = & |
| | + | \text{man that is noble} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C} ~+\!\!,~ \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | "m" = "man" | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathrm{n},\!\mathrm{m} |
| | + | & = & |
| | + | \text{noble that is a man} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{C} ~+\!\!,~ \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | "w" = "woman" | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathrm{w},\!\mathrm{n} |
| | + | & = & |
| | + | \text{woman that is noble} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | In Peirce's notation, the denotation of a general term
| + | {| align="center" cellspacing="6" width="90%" |
| − | can be expressed by means of an equation between terms:
| + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathrm{n},\!\mathrm{w} |
| | + | & = & |
| | + | \text{noble that is a woman} |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | & & |
| | + | \times |
| | + | \\ |
| | + | & & |
| | + | (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | b = O
| + | ==Selection 9== |
| | | | |
| − | m = C +, I +, J +, O
| + | ===The Signs for Multiplication (cont.)=== |
| | | | |
| − | w = B +, D +, E
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>It is obvious that multiplication into a multiplicand indicated by a comma is commutative<sup>1</sup>, that is,</p> |
| | + | |- |
| | + | | align="center" | <math>\mathit{s},\!\mathit{l} ~=~ \mathit{l},\!\mathit{s}</math> |
| | + | |- |
| | + | | |
| | + | <p>This multiplication is effectively the same as that of Boole in his logical calculus. Boole's unity is my <math>\mathbf{1},</math> that is, it denotes whatever is.</p> |
| | | | |
| | + | #<p>It will often be convenient to speak of the whole operation of affixing a comma and then multiplying as a commutative multiplication, the sign for which is the comma. But though this is allowable, we shall fall into confusion at once if we ever forget that in point of fact it is not a different multiplication, only it is multiplication by a relative whose meaning — or rather whose syntax — has been slightly altered; and that the comma is really the sign of this modification of the foregoing term.</p> |
| | | | |
| | + | <p>(Peirce, CP 3.74).</p> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 8.2
| + | ===Commentary Note 9.1=== |
| | | | |
| | + | Let us backtrack a few years, and consider how George Boole explained his twin conceptions of ''selective operations'' and ''selective symbols''. |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>Let us then suppose that the universe of our discourse is the actual universe, so that words are to be used in the full extent of their meaning, and let us consider the two mental operations implied by the words “white” and “men”. The word “men” implies the operation of selecting in thought from its subject, the universe, all men; and the resulting conception, ''men'', becomes the subject of the next operation. The operation implied by the word “white” is that of selecting from its subject, “men”, all of that class which are white. The final resulting conception is that of “white men”.</p> |
| | | | |
| − | I will continue with my commentary on CP 3.73, developing
| + | <p>Now it is perfectly apparent that if the operations above described had been performed in a converse order, the result would have been the same. Whether we begin by forming the conception of “''men''”, and then by a second intellectual act limit that conception to “white men”, or whether we begin by forming the conception of “white objects”, and then limit it to such of that class as are “men”, is perfectly indifferent so far as the result is concerned. It is obvious that the order of the mental processes would be equally indifferent if for the words “white” and “men” we substituted any other descriptive or appellative terms whatever, provided only that their meaning was fixed and absolute. And thus the indifference of the order of two successive acts of the faculty of Conception, the one of which furnishes the subject upon which the other is supposed to operate, is a general condition of the exercise of that faculty. It is a law of the mind, and it is the real origin of that law of the literal symbols of Logic which constitutes its formal expression (1) Chap. II, [ namely, <math>xy = yx~\!</math> ].</p> |
| − | the Othello example as a way of illustrating its concepts. | |
| | | | |
| − | In the development of the story so far, we have a universe of discourse
| + | <p>It is equally clear that the mental operation above described is of such a nature that its effect is not altered by repetition. Suppose that by a definite act of conception the attention has been fixed upon men, and that by another exercise of the same faculty we limit it to those of the race who are white. Then any further repetition of the latter mental act, by which the attention is limited to white objects, does not in any way modify the conception arrived at, viz., that of white men. This is also an example of a general law of the mind, and it has its formal expression in the law ((2) Chap. II) of the literal symbols [ namely, <math>x^2 = x\!</math> ].</p> |
| − | that can be characterized by means of the following system of equations: | |
| | | | |
| − | 1 = B +, C +, D +, E +, I +, J +, O
| + | <p>(Boole, ''Laws of Thought'', 44–45).</p> |
| | + | |} |
| | | | |
| − | b = O
| + | ===Commentary Note 9.2=== |
| | | | |
| − | m = C +, I +, J +, O
| + | In setting up his discussion of selective operations and their corresponding selective symbols, Boole writes this: |
| | | | |
| − | w = B +, D +, E
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>The operation which we really perform is one of ''selection according to a prescribed principle or idea''. To what faculties of the mind such an operation would be referred, according to the received classification of its powers, it is not important to inquire, but I suppose that it would be considered as dependent upon the two faculties of Conception or Imagination, and Attention. To the one of these faculties might be referred the formation of the general conception; to the other the fixing of the mental regard upon those individuals within the prescribed universe of discourse which answer to the conception. If, however, as seems not improbable, the power of Attention is nothing more than the power of continuing the exercise of any other faculty of the mind, we might properly regard the whole of the mental process above described as referrible to the mental faculty of Imagination or Conception, the first step of the process being the conception of the Universe itself, and each succeeding step limiting in a definite manner the conception thus formed. Adopting this view, I shall describe each such step, or any definite combination of such steps, as a ''definite act of conception''.</p> |
| | | | |
| − | This much provides a basis for collection of absolute terms that
| + | <p>(Boole, ''Laws of Thought'', 43).</p> |
| − | I plan to use in this example. Let us now consider how we might
| + | |} |
| − | represent a sufficiently exemplary collection of relative terms.
| |
| | | | |
| − | If we consider the genesis of relative terms, for example, "lover of ---",
| + | ===Commentary Note 9.3=== |
| − | "betrayer to --- of ---", or "winner over of --- to --- from ---", we may
| |
| − | regard these fill-in-the-blank forms as being derived by way of a kind of
| |
| − | "rhematic abstraction" from the corresponding instances of absolute terms.
| |
| | | | |
| − | In other words: | + | In algebra, an ''idempotent element'' <math>x\!</math> is one that obeys the ''idempotent law'', that is, it satisfies the equation <math>xx = x.\!</math> Under most circumstances, it is usual to write this as <math>x^2 = x.\!</math> |
| | | | |
| − | 1. The relative term "lover of ---" can be constructed by abstracting
| + | If the algebraic system in question falls under the additional laws that are necessary to carry out the requisite transformations, then <math>x^2 = x\!</math> is convertible into <math>x - x^2 = 0,\!</math> and this into <math>x(1 - x) = 0.\!</math> |
| − | the absolute term "Emilia" from the absolute term "lover of Emilia".
| |
| − | Since Iago is a lover of Emilia, the relate-correlate pair denoted
| |
| − | by "Iago:Emilia" is a summand of the relative term "lover of ---".
| |
| | | | |
| − | 2. The relative term "betrayer to --- of ---" can be constructed
| + | If the algebraic system in question happens to be a boolean algebra, then the equation <math>x(1 - x) = 0\!</math> says that <math>x \land \lnot x</math> is identically false, in effect, a statement of the classical principle of non-contradiction. |
| − | by abstracting the absolute terms "Othello" and "Desdemona"
| |
| − | from the absolute term "betrayer to Othello of Desdemona".
| |
| − | In as much as Iago is a betrayer to Othello of Desdemona,
| |
| − | the relate-correlate-correlate triple denoted by "I:O:D"
| |
| − | belongs to the relative term "betrayer to --- of ---".
| |
| | | | |
| − | 3. The relative term "winner over of --- to --- from ---" can be constructed
| + | We have already seen how Boole found rationales for the commutative law and the idempotent law by contemplating the properties of ''selective operations''. |
| − | by abstracting the absolute terms "Othello", "Iago", and "Cassio" from the
| |
| − | absolute term "winner over of Othello to Iago from Cassio". Since Iago is
| |
| − | a winner over of Othello to Iago from Cassio, the elementary relative term
| |
| − | "I:O:I:C" belongs to the relative term "winner over of --- to --- from ---".
| |
| | | | |
| | + | It is time to bring these threads together, which we can do by considering the so-called ''idempotent representation'' of sets. This will give us one of the best ways to understand the significance that Boole attached to selective operations. It will also link up with the statements that Peirce makes about his adicity-augmenting comma operation. |
| | | | |
| | + | ===Commentary Note 9.4=== |
| | | | |
| − | LOR. Commentary Note 8.3
| + | Boole rationalized the properties of what we now call ''boolean multiplication'', roughly equivalent to logical conjunction, in terms of the laws that apply to selective operations. Peirce, in his turn, taking a very significant step of analysis that has seldom been recognized for what it would lead to, does not consider this multiplication to be a fundamental operation, but derives it as a by-product of relative multiplication by a comma relative. Thus, Peirce makes logical conjunction a special case of relative composition. |
| | | | |
| | + | This opens up a very wide field of investigation, ''the operational significance of logical terms'', one might say, but it will be best to advance bit by bit, and to lean on simple examples. |
| | | | |
| | + | Back to Venice, and the close-knit party of absolutes and relatives that we were entertaining when last we were there. |
| | | | |
| − | Speaking very strictly, we need to be careful to
| + | Here is the list of absolute terms that we were considering before, to which I have thrown in <math>\mathbf{1},</math> the universe of ''anything'', just for good measure: |
| − | distinguish a "relation" from a "relative term".
| |
| | | | |
| − | 1. The relation is an 'object' of thought | + | {| align="center" cellspacing="6" width="90%" |
| − | that may be regarded "in extension" as
| + | | |
| − | a set of ordered tuples that are known
| + | <math>\begin{array}{*{17}{l}} |
| − | as its "elementary relations".
| + | \mathbf{1} |
| | + | & = & \text{anything} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m} |
| | + | & = & \text{man} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{n} |
| | + | & = & \text{noble} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w} |
| | + | & = & \text{woman} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 2. The relative term is a 'sign' that denotes certain objects,
| + | Here is the list of ''comma inflexions'' or ''diagonal extensions'' of these terms: |
| − | called its "relates", as these are determined in relation
| |
| − | to certain other objects, called its "correlates". Under
| |
| − | most circumstances, one may also regard the relative term
| |
| − | as denoting the corresponding relation.
| |
| | | | |
| − | Returning to the Othello example, let us take up the
| + | {| align="center" cellspacing="6" width="90%" |
| − | 2-adic relatives "lover of ---" and "servant of ---".
| + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathbf{1,} |
| | + | & = & \text{anything that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[9pt] |
| | + | \mathrm{m,} |
| | + | & = & \text{man that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[9pt] |
| | + | \mathrm{n,} |
| | + | & = & \text{noble that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[9pt] |
| | + | \mathrm{w,} |
| | + | & = & \text{woman that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | Ignoring the many splendored nuances appurtenant to the idea of love,
| + | One observes that the diagonal extension of <math>\mathbf{1}</math> is the same thing as the identity relation <math>\mathit{1}.\!</math> |
| − | we may regard the relative term 'l' for "lover of ---" to be given by
| |
| − | the following equation: | |
| | | | |
| − | 'l' = B:C +, C:B +, D:O +, E:I +, I:E +, O:D.
| + | Working within our smaller sample of absolute terms, we have already computed the sorts of products that apply the diagonal extension of an absolute term to another absolute term, for instance, these products: |
| | | | |
| − | If for no better reason than to make the example more interesting,
| + | {| align="center" cellspacing="6" width="90%" |
| − | let us put aside all distinctions of rank and fealty, collapsing
| + | | |
| − | the motley crews of attendant, servant, subordinate, and so on,
| + | <math>\begin{array}{lllll} |
| − | under the heading of a single service, denoted by the relative
| + | \mathrm{m},\!\mathrm{n} |
| − | term 's' for "servant of ---". The terms of this service are:
| + | & = & \text{man that is noble} |
| | + | & = & \mathrm{C} ~+\!\!,~ \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{n},\!\mathrm{m} |
| | + | & = & \text{noble that is a man} |
| | + | & = & \mathrm{C} ~+\!\!,~ \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w},\!\mathrm{n} |
| | + | & = & \text{woman that is noble} |
| | + | & = & \mathrm{D} |
| | + | \\[6pt] |
| | + | \mathrm{n},\!\mathrm{w} |
| | + | & = & \text{noble that is a woman} |
| | + | & = & \mathrm{D} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 's' = C:O +, E:D +, I:O +, J:D +, J:O.
| + | This exercise gave us a bit of practical insight into why the commutative law holds for logical conjunction. |
| | | | |
| − | The term I:C may also be implied, but, since it is
| + | Further insight into the laws that govern this realm of logic, and the underlying reasons why they apply, might be gained by systematically working through the whole variety of different products that are generated by the operational means in sight, namely, the products indicated by <math>\{\mathbf{1}, \mathrm{m}, \mathrm{n}, \mathrm{w} \} , \{\mathbf{1}, \mathrm{m}, \mathrm{n}, \mathrm{w} \}.</math> |
| − | so hotly arguable, I will leave it out of the toll.
| |
| | | | |
| − | One more thing that we need to be duly wary about:
| + | But before we try to explore this territory more systematically, let us equip our intuitions with the forms of graphical and matrical representation that served us so well in our previous adventures. |
| − | There are many different conventions in the field
| |
| − | as to the ordering of terms in their applications,
| |
| − | and it happens that different conventions will be | |
| − | more convenient under different circumstances, so
| |
| − | there does not appear to be much of a chance that
| |
| − | any one of them can be canonized once and for all.
| |
| | | | |
| − | In the current reading, we are applying relative terms
| + | ===Commentary Note 9.5=== |
| − | from right to left, and so our conception of relative
| |
| − | multiplication, or relational composition, will need
| |
| − | to be adjusted accordingly.
| |
| | | | |
| | + | Peirce's comma operation, in its application to an absolute term, is tantamount to the representation of that term's denotation as an idempotent transformation, which is commonly represented as a diagonal matrix. Hence the alternate name, ''diagonal extension''. |
| | | | |
| | + | An idempotent element <math>x\!</math> is given by the abstract condition that <math>xx = x,\!</math> but elements like these are commonly encountered in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients. |
| | | | |
| − | LOR. Commentary Note 8.4
| + | Let's see how this looks in the matrix and graph pictures of absolute and relative terms: |
| | | | |
| | + | ====Absolute Terms==== |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{*{17}{l}} |
| | + | \mathbf{1} & = & \text{anything} & = & |
| | + | \mathrm{B} & +\!\!, & |
| | + | \mathrm{C} & +\!\!, & |
| | + | \mathrm{D} & +\!\!, & |
| | + | \mathrm{E} & +\!\!, & |
| | + | \mathrm{I} & +\!\!, & |
| | + | \mathrm{J} & +\!\!, & |
| | + | \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m} & = & \text{man} & = & |
| | + | \mathrm{C} & +\!\!, & |
| | + | \mathrm{I} & +\!\!, & |
| | + | \mathrm{J} & +\!\!, & |
| | + | \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{n} & = & \text{noble} & = & |
| | + | \mathrm{C} & +\!\!, & |
| | + | \mathrm{D} & +\!\!, & |
| | + | \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{w} & = & \text{woman} & = & |
| | + | \mathrm{B} & +\!\!, & |
| | + | \mathrm{D} & +\!\!, & |
| | + | \mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | To familiarize ourselves with the forms of calculation
| + | Previously, we represented absolute terms as column arrays. The above four terms are given by the columns of the following table: |
| − | that are available in Peirce's notation, let us compute
| |
| − | a few of the simplest products that we find at hand in
| |
| − | the Othello case.
| |
| | | | |
| − | Here are the absolute terms:
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{c|cccc} |
| | + | \text{ } & \mathbf{1} & \mathrm{m} & \mathrm{n} & \mathrm{w} \\ |
| | + | \text{---} & \text{---} & \text{---} & \text{---} & \text{---} \\ |
| | + | \mathrm{B} & 1 & 0 & 0 & 1 \\ |
| | + | \mathrm{C} & 1 & 1 & 1 & 0 \\ |
| | + | \mathrm{D} & 1 & 0 & 1 & 1 \\ |
| | + | \mathrm{E} & 1 & 0 & 0 & 1 \\ |
| | + | \mathrm{I} & 1 & 1 & 0 & 0 \\ |
| | + | \mathrm{J} & 1 & 1 & 0 & 0 \\ |
| | + | \mathrm{O} & 1 & 1 & 1 & 0 |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 1 = B +, C +, D +, E +, I +, J +, O
| + | The types of graphs known as ''bigraphs'' or ''bipartite graphs'' can be used to picture simple relative terms, dyadic relations, and their corresponding logical matrices. One way to bring absolute terms and their corresponding sets of individuals into the bigraph picture is to mark the nodes in some way, for example, hollow nodes for non-members and filled nodes for members of the indicated set, as shown below: |
| | | | |
| − | b = O
| + | {| align="center" cellpadding="10" width="90%" |
| | + | | [[Image:LOR 1870 Figure 4.1.jpg]] || (4.1) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 4.2.jpg]] || (4.2) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 4.3.jpg]] || (4.3) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 4.4.jpg]] || (4.4) |
| | + | |} |
| | | | |
| − | m = C +, I +, J +, O
| + | ====Diagonal Extensions==== |
| | | | |
| − | w = B +, D +, E | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathbf{1,} |
| | + | & = & \text{anything that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[9pt] |
| | + | \mathrm{m,} |
| | + | & = & \text{man that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[9pt] |
| | + | \mathrm{n,} |
| | + | & = & \text{noble that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[9pt] |
| | + | \mathrm{w,} |
| | + | & = & \text{woman that is}\, \underline{~~ ~~} |
| | + | \\[6pt] |
| | + | & = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | Here are the 2-adic relative terms:
| + | Naturally enough, the diagonal extensions are represented by diagonal matrices: |
| | | | |
| − | 'l' = B:C +, C:B +, D:O +, E:I +, I:E +, O:D
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | |- |
| | + | | |
| | + | <math>\begin{array}{c|ccccccc} |
| | + | \mathbf{1,} & |
| | + | \mathrm{B} & |
| | + | \mathrm{C} & |
| | + | \mathrm{D} & |
| | + | \mathrm{E} & |
| | + | \mathrm{I} & |
| | + | \mathrm{J} & |
| | + | \mathrm{O} |
| | + | \\ |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 1 & & & & & & |
| | + | \\ |
| | + | \mathrm{C} & & 1 & & & & & |
| | + | \\ |
| | + | \mathrm{D} & & & 1 & & & & |
| | + | \\ |
| | + | \mathrm{E} & & & & 1 & & & |
| | + | \\ |
| | + | \mathrm{I} & & & & & 1 & & |
| | + | \\ |
| | + | \mathrm{J} & & & & & & 1 & |
| | + | \\ |
| | + | \mathrm{O} & & & & & & & 1 |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 's' = C:O +, E:D +, I:O +, J:D +, J:O
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | |- |
| | + | | |
| | + | <math>\begin{array}{c|ccccccc} |
| | + | \mathrm{m,} & |
| | + | \mathrm{B} & |
| | + | \mathrm{C} & |
| | + | \mathrm{D} & |
| | + | \mathrm{E} & |
| | + | \mathrm{I} & |
| | + | \mathrm{J} & |
| | + | \mathrm{O} |
| | + | \\ |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 0 & & & & & & |
| | + | \\ |
| | + | \mathrm{C} & & 1 & & & & & |
| | + | \\ |
| | + | \mathrm{D} & & & 0 & & & & |
| | + | \\ |
| | + | \mathrm{E} & & & & 0 & & & |
| | + | \\ |
| | + | \mathrm{I} & & & & & 1 & & |
| | + | \\ |
| | + | \mathrm{J} & & & & & & 1 & |
| | + | \\ |
| | + | \mathrm{O} & & & & & & & 1 |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | Here are a few of the simplest products among these terms:
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | |- |
| | + | | |
| | + | <math>\begin{array}{c|ccccccc} |
| | + | \mathrm{n,} & |
| | + | \mathrm{B} & |
| | + | \mathrm{C} & |
| | + | \mathrm{D} & |
| | + | \mathrm{E} & |
| | + | \mathrm{I} & |
| | + | \mathrm{J} & |
| | + | \mathrm{O} |
| | + | \\ |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 0 & & & & & & |
| | + | \\ |
| | + | \mathrm{C} & & 1 & & & & & |
| | + | \\ |
| | + | \mathrm{D} & & & 1 & & & & |
| | + | \\ |
| | + | \mathrm{E} & & & & 0 & & & |
| | + | \\ |
| | + | \mathrm{I} & & & & & 0 & & |
| | + | \\ |
| | + | \mathrm{J} & & & & & & 0 & |
| | + | \\ |
| | + | \mathrm{O} & & & & & & & 1 |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 'l'1 = "lover of anybody"
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | |- |
| | + | | |
| | + | <math>\begin{array}{c|ccccccc} |
| | + | \mathrm{w,} & |
| | + | \mathrm{B} & |
| | + | \mathrm{C} & |
| | + | \mathrm{D} & |
| | + | \mathrm{E} & |
| | + | \mathrm{I} & |
| | + | \mathrm{J} & |
| | + | \mathrm{O} |
| | + | \\ |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} & |
| | + | \text{---} |
| | + | \\ |
| | + | \mathrm{B} & 1 & & & & & & |
| | + | \\ |
| | + | \mathrm{C} & & 0 & & & & & |
| | + | \\ |
| | + | \mathrm{D} & & & 1 & & & & |
| | + | \\ |
| | + | \mathrm{E} & & & & 1 & & & |
| | + | \\ |
| | + | \mathrm{I} & & & & & 0 & & |
| | + | \\ |
| | + | \mathrm{J} & & & & & & 0 & |
| | + | \\ |
| | + | \mathrm{O} & & & & & & & 0 |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(B +, C +, D +, E +, I +, J +, O)
| + | Cast into the bigraph picture of dyadic relations, the diagonal extension of an absolute term takes on a very distinctive sort of “straight-laced” character: |
| | | | |
| − | = B +, C +, D +, E +, I +, O
| + | {| align="center" cellpadding="10" width="90%" |
| | + | | [[Image:LOR 1870 Figure 5.1.jpg]] || (5.1) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 5.2.jpg]] || (5.2) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 5.3.jpg]] || (5.3) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 5.4.jpg]] || (5.4) |
| | + | |} |
| | | | |
| − | = "anybody except J"
| + | ===Commentary Note 9.6=== |
| | | | |
| − | 'l'b = "lover of a black"
| + | Just to be doggedly persistent about it, here is what ought to be a sufficient sample of products involving the multiplication of a comma relative onto an absolute term, presented in both matrix and bigraph pictures. |
| | | | |
| − | = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)O
| + | ====Example 1==== |
| | | | |
| − | = D
| + | {| align="center" cellpadding="6" width="90%" |
| | + | | <math>\mathbf{1,}\mathbf{1} ~=~ \mathbf{1}\!</math> |
| | + | |- |
| | + | | <math>\text{anything that is anything} ~=~ \text{anything}</math> |
| | + | |- |
| | + | | |
| | + | <math> |
| | + | \begin{bmatrix} |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 1 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | | | |
| − | 'l'm = "lover of a man"
| + | {| align="center" cellpadding="10" width="100%" |
| | + | | width="2%" | |
| | + | | width="48%" | [[Image:LOR 1870 Figure 6.1.jpg]] |
| | + | | width="50%" | (6.1) |
| | + | |} |
| | | | |
| − | = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(C +, I +, J +, O)
| + | ====Example 2==== |
| | | | |
| − | = B +, D +, E
| + | {| align="center" cellpadding="6" width="90%" |
| | + | | <math>\mathbf{1,}\mathrm{m} ~=~ \mathrm{m}</math> |
| | + | |- |
| | + | | <math>\text{anything that is a man} ~=~ \text{man}</math> |
| | + | |- |
| | + | | |
| | + | <math> |
| | + | \begin{bmatrix} |
| | + | 1 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 1 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 1 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | | | |
| − | 'l'w = "lover of a woman"
| + | {| align="center" cellpadding="10" width="100%" |
| | + | | width="2%" | |
| | + | | width="48%" | [[Image:LOR 1870 Figure 6.2.jpg]] |
| | + | | width="50%" | (6.2) |
| | + | |} |
| | | | |
| − | = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(B +, D +, E)
| + | ====Example 3==== |
| | | | |
| − | = C +, I +, O
| + | {| align="center" cellpadding="6" width="90%" |
| | + | | <math>\mathrm{m,}\mathbf{1} ~=~ \mathrm{m}</math> |
| | + | |- |
| | + | | <math>\text{man that is anything} ~=~ \text{man}</math> |
| | + | |- |
| | + | | |
| | + | <math> |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 1 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | | | |
| − | 's'1 = "servant of anybody"
| + | {| align="center" cellpadding="10" width="100%" |
| | + | | width="2%" | |
| | + | | width="48%" | [[Image:LOR 1870 Figure 6.3.jpg]] |
| | + | | width="50%" | (6.3) |
| | + | |} |
| | | | |
| − | = (C:O +, E:D +, I:O +, J:D +, J:O)(B +, C +, D +, E +, I +, J +, O)
| + | ====Example 4==== |
| | | | |
| − | = C +, E +, I +, J
| + | {| align="center" cellpadding="6" width="90%" |
| | + | | <math>\mathrm{m,}\mathrm{n} ~=~ \text{man that is noble}</math> |
| | + | |- |
| | + | | |
| | + | <math> |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 1 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 1 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | | | |
| − | 's'b = "servant of a black"
| + | {| align="center" cellpadding="10" width="100%" |
| | + | | width="2%" | |
| | + | | width="48%" | [[Image:LOR 1870 Figure 6.4.jpg]] |
| | + | | width="50%" | (6.4) |
| | + | |} |
| | | | |
| − | = (C:O +, E:D +, I:O +, J:D +, J:O)O
| + | ====Example 5==== |
| | | | |
| − | = C +, I +, J
| + | {| align="center" cellpadding="6" width="90%" |
| | + | | <math>\mathrm{n,}\mathrm{m} ~=~ \text{noble that is a man}</math> |
| | + | |- |
| | + | | |
| | + | <math> |
| | + | \begin{bmatrix} |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 1 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 1 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 0 |
| | + | \\ |
| | + | 0 & 0 & 0 & 0 & 0 & 0 & 1 |
| | + | \end{bmatrix} |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \\ 1 |
| | + | \end{bmatrix} |
| | + | = |
| | + | \begin{bmatrix} |
| | + | 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 |
| | + | \end{bmatrix} |
| | + | </math> |
| | + | |} |
| | | | |
| − | 's'm = "servant of a man"
| + | {| align="center" cellpadding="10" width="100%" |
| | + | | width="2%" | |
| | + | | width="48%" | [[Image:LOR 1870 Figure 6.5.jpg]] |
| | + | | width="50%" | (6.5) |
| | + | |} |
| | | | |
| − | = (C:O +, E:D +, I:O +, J:D +, J:O)(C +, I +, J +, O)
| + | ===Commentary Note 9.7=== |
| | | | |
| − | = C +, I +, J
| + | From this point forward we may think of idempotents, selectives, and zero-one diagonal matrices as being roughly equivalent notions. The only reason that I say ''roughly'' is that we are comparing ideas at different levels of abstraction in proposing these connections. |
| | | | |
| − | 's'w = "servant of a woman" | + | We have covered the way that Peirce uses his invention of the comma modifier to assimilate boolean multiplication, logical conjunction, and what we may think of as ''serial selection'' under his more general account of relative multiplication. |
| | | | |
| − | = (C:O +, E:D +, I:O +, J:D +, J:O)(B +, D +, E)
| + | But the comma functor has its application to relative terms of any arity, not just the zeroth arity of absolute terms, and so there will be a lot more to explore on this point. But now I must return to the anchorage of Peirce's text and hopefully get a chance to revisit this topic later. |
| | | | |
| − | = E +, J
| + | ==Selection 10== |
| | | | |
| − | 'ls' = "lover of a servant of ---"
| + | ===The Signs for Multiplication (cont.)=== |
| | | | |
| − | = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(C:O +, E:D +, I:O +, J:D +, J:O)
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>The sum <math>x + x\!</math> generally denotes no logical term. But <math>{x,}_\infty + \, {x,}_\infty</math> may be considered as denoting some two <math>x\!</math>'s.</p> |
| | | | |
| − | = B:O +, E:O +, I:D
| + | <p>It is natural to write:</p> |
| | | | |
| − | 'sl' = "servant of a lover of ---"
| + | {| align="center" width="100%" |
| | + | | width="20%" | |
| | + | | width="25%" align="right" | <math>x ~+~ x</math> |
| | + | | width="10%" align="center"| <math>=\!</math> |
| | + | | width="25%" align="left" | <math>\mathit{2}.x\!</math> |
| | + | | width="20%" | |
| | + | |- |
| | + | | |
| | + | |- |
| | + | | and |
| | + | | align="right" | <math>{x,}_\infty + \, {x,}_\infty</math> |
| | + | | align="center" | <math>=\!</math> |
| | + | | align="left" | <math>\mathit{2}.{x,}_\infty</math> |
| | + | | |
| | + | |} |
| | | | |
| − | = (C:O +, E:D +, I:O +, J:D +, J:O)(B:C +, C:B +, D:O +, E:I +, I:E +, O:D)
| + | <p>where the dot shows that this multiplication is invertible.</p> |
| | | | |
| − | = C:D +, E:O +, I:D +, J:D +, J:O
| + | <p>We may also use the antique figures so that:</p> |
| | | | |
| − | Among other things, one observes that the
| + | {| align="center" width="100%" |
| − | relative terms 'l' and 's' do not commute,
| + | | width="20%" | |
| − | that is to say, 'ls' is not equal to 'sl'.
| + | | width="25%" align="right" | <math>\mathit{2}.{x,}_\infty</math> |
| | + | | width="10%" align="center"| <math>=\!</math> |
| | + | | width="25%" align="left" | <math>\mathfrak{2}x</math> |
| | + | | width="20%" | |
| | + | |- |
| | + | | |
| | + | |- |
| | + | | just as |
| | + | | align="right" | <math>\mathit{1}_\infty</math> |
| | + | | align="center" | <math>=\!</math> |
| | + | | align="left" | <math>\mathfrak{1}</math> |
| | + | | |
| | + | |} |
| | | | |
| | + | <p>Then <math>\mathfrak{2}</math> alone will denote some two things.</p> |
| | | | |
| | + | <p>But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to ''one'' thing, and one thing ''alone'' bears it to a thing.</p> |
| | | | |
| − | LOR. Commentary Note 8.5
| + | <p>For instance, the lovers of two women are not the same as two lovers of women, that is:</p> |
| | | | |
| | + | {| align="center" width="100%" |
| | + | | width="20%" | |
| | + | | width="25%" align="right" | <math>\mathit{l}\mathfrak{2}.\mathrm{w}</math> |
| | + | | width="10%" align="center"| and |
| | + | | width="25%" align="left" | <math>\mathfrak{2}.\mathit{l}\mathrm{w}</math> |
| | + | | width="20%" | |
| | + | |} |
| | | | |
| | + | <p>are unequal; but the husbands of two women are the same as two husbands of women, that is:</p> |
| | | | |
| − | Since multiplication by a 2-adic relative term
| + | {| align="center" width="100%" |
| − | is a logical analogue of matrix multiplication
| + | | width="20%" | |
| − | in linear algebra, all of the products that we | + | | width="25%" align="right" | <math>\mathit{h}\mathfrak{2}.\mathrm{w}</math> |
| − | computed above can be represented in terms of
| + | | width="10%" align="center"| <math>=\!</math> |
| − | logical matrices and logical vectors.
| + | | width="25%" align="left" | <math>\mathfrak{2}.\mathit{h}\mathrm{w}</math> |
| | + | | width="20%" | |
| | + | |- |
| | + | | |
| | + | |- |
| | + | | and in general; |
| | + | | align="right" | <math>x,\!\mathfrak{2}.y</math> |
| | + | | align="center" | <math>=\!</math> |
| | + | | align="left" | <math>\mathfrak{2}.x,\!y</math> |
| | + | | |
| | + | |} |
| | | | |
| − | Here are the absolute terms again, followed by
| + | <p>(Peirce, CP 3.75).</p> |
| − | their representation as "coefficient tuples",
| + | |} |
| − | otherwise thought of as "coordinate vectors".
| |
| | | | |
| − | 1 = B +, C +, D +, E +, I +, J +, O | + | ===Commentary Note 10.1=== |
| | | | |
| − | = <1, 1, 1, 1, 1, 1, 1>
| + | What Peirce is attempting to do in CP 3.75 is absolutely amazing and I personally did not see anything on par with it again until I began to study the application of mathematical category theory to computation and logic, back in the mid 1980's. To completely evaluate the success of this attempt we would have to return to Peirce's earlier paper “Upon the Logic of Mathematics” (1867) to pick up some of the ideas about arithmetic that he set out there. |
| | | | |
| − | b = O
| + | Another branch of the investigation would require that we examine more carefully the entire syntactic mechanics of ''subjacent signs'' that Peirce uses to establish linkages among relational domains. It is important to note that these types of indices constitute a diacritical, interpretive, syntactic category under which Peirce also places the comma functor. |
| | | | |
| − | = <0, 0, 0, 0, 0, 0, 1>
| + | The way that I would currently approach both of these branches of the investigation would be to open up a wider context for the study of relational compositions, attempting to get at the essence of what is going on when we relate relations, possibly complex, to other relations, possibly simple. |
| | | | |
| − | m = C +, I +, J +, O
| + | ===Commentary Note 10.2=== |
| | | | |
| − | = <0, 1, 0, 0, 1, 1, 1>
| + | To say that a relative term “imparts a relation” is to say that it conveys information about the space of tuples in a cartesian product, that is, it determines a particular subset of that space. When we study the combinations of relative terms, from the most elementary forms of composition to the most complex patterns of correlation, we are considering the ways that these constraints, determinations, and informations, as imparted by relative terms, can be compounded in the formation of syntax. |
| | | | |
| − | w = B +, D +, E
| + | Let us go back and look more carefully at just how it happens that Peirce's adjacent terms and subjacent indices manage to impart their respective measures of information about relations. I will begin with the two examples illustrated in Figures 7 and 8, where I have drawn in the corresponding lines of identity between the subjacent marks of reference: <math>\dagger, \ddagger, \parallel, \S, \P.\!</math> |
| | | | |
| − | = <1, 0, 1, 1, 0, 0, 0>
| + | <br> |
| | | | |
| − | Since we are going to be regarding these tuples as "column vectors",
| + | {| align="center" cellpadding="10" |
| − | it is convenient to arrange them into a table of the following form:
| + | | [[Image:LOR 1870 Figure 7.0.jpg]] || (7) |
| | + | |} |
| | | | |
| − | | 1 b m w
| + | {| align="center" cellpadding="10" |
| − | ---o---------
| + | | [[Image:LOR 1870 Figure 8.0.jpg]] || (8) |
| − | B | 1 0 0 1
| + | |} |
| − | C | 1 0 1 0
| |
| − | D | 1 0 0 1
| |
| − | E | 1 0 0 1
| |
| − | I | 1 0 1 0
| |
| − | J | 1 0 1 0
| |
| − | O | 1 1 1 0
| |
| | | | |
| − | Here are the 2-adic relative terms again, followed by
| + | One way to approach the problem of “information fusion” in Peirce's syntax is to soften the distinction between adjacent terms and subjacent signs and to treat the types of constraints that they separately signify more on a par with each other. To that purpose, I will set forth a way of thinking about relational composition that emphasizes the set-theoretic constraints involved in the construction of a composite. |
| − | their representation as coefficient matrices, in this
| |
| − | case bordered by row and column labels to remind us
| |
| − | what the coefficient values are meant to signify.
| |
| | | | |
| − | 'l' = B:C +, C:B +, D:O +, E:I +, I:E +, O:D =
| + | For example, suppose that we are given the relations <math>L \subseteq X \times Y</math> and <math>M \subseteq Y \times Z.</math> Table 9 and Figure 10 present two ways of picturing the constraints that are involved in constructing the relational composition <math>L \circ M \subseteq X \times Z.</math> |
| | | | |
| − | 'l'| B C D E I J O
| + | <br> |
| − | ---o---------------
| |
| − | B | 0 1 0 0 0 0 0
| |
| − | C | 1 0 0 0 0 0 0
| |
| − | D | 0 0 0 0 0 0 1
| |
| − | E | 0 0 0 0 1 0 0
| |
| − | I | 0 0 0 1 0 0 0
| |
| − | J | 0 0 0 0 0 0 0
| |
| − | O | 0 0 1 0 0 0 0
| |
| | | | |
| − | 's' = C:O +, E:D +, I:O +, J:D +, J:O =
| + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| | + | |+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Relational Composition}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>Y\!</math> |
| | + | | |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>M\!</math> |
| | + | | |
| | + | | <math>Y\!</math> |
| | + | | <math>Z\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L \circ M\!</math> |
| | + | | <math>X\!</math> |
| | + | | |
| | + | | <math>Z\!</math> |
| | + | |} |
| | | | |
| − | 's'| B C D E I J O
| + | <br> |
| − | ---o---------------
| |
| − | B | 0 0 0 0 0 0 0
| |
| − | C | 0 0 0 0 0 0 1
| |
| − | D | 0 0 0 0 0 0 0
| |
| − | E | 0 0 1 0 0 0 0
| |
| − | I | 0 0 0 0 0 0 1
| |
| − | J | 0 0 1 0 0 0 1
| |
| − | O | 0 0 0 0 0 0 0
| |
| | | | |
| − | Here are the matrix representations of
| + | The way to read Table 9 is to imagine that you are playing a game that involves placing tokens on the squares of a board that is marked in just this way. The rules are that you have to place a single token on each marked square in the middle of the board in such a way that all of the indicated constraints are satisfied. That is to say, you have to place a token whose denomination is a value in the set <math>X\!</math> on each of the squares marked <math>{}^{\backprime\backprime} X {}^{\prime\prime},</math> and similarly for the squares marked <math>{}^{\backprime\backprime} Y {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} Z {}^{\prime\prime},</math> meanwhile leaving all of the blank squares empty. Furthermore, the tokens placed in each row and column have to obey the relational constraints that are indicated at the heads of the corresponding row and column. Thus, the two tokens from <math>X\!</math> have to denominate the very same value from <math>X,\!</math> and likewise for <math>Y\!</math> and <math>Z,\!</math> while the pairs of tokens on the rows marked <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} M {}^{\prime\prime}</math> are required to denote elements that are in the relations <math>L\!</math> and <math>M,\!</math> respectively. The upshot is that when just this much is done, that is, when the <math>L,\!</math> <math>M,\!</math> and <math>\mathit{1}\!</math> relations are satisfied, then the row marked <math>{}^{\backprime\backprime} L \circ M {}^{\prime\prime}</math> will automatically bear the tokens of a pair of elements in the composite relation <math>L \circ M.\!</math> |
| − | the products that we calculated before: | |
| | | | |
| − | 'l'1 = "lover of anybody" =
| + | Figure 10 shows a different way of viewing the same situation. |
| | | | |
| − | | 0 1 0 0 0 0 0 | | 1 | | 1 |
| + | <br> |
| − | | 1 0 0 0 0 0 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 0 0 1 0 0 | | 1 | = | 1 |
| |
| − | | 0 0 0 1 0 0 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 1 | | 1 |
| |
| | | | |
| − | 'l'b = "lover of a black" =
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 10.jpg]] || (10) |
| | + | |} |
| | | | |
| − | | 0 1 0 0 0 0 0 | | 0 | | 0 |
| + | ===Commentary Note 10.3=== |
| − | | 1 0 0 0 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 0 0 0 0 1 | | 0 | | 1 |
| |
| − | | 0 0 0 0 1 0 0 | | 0 | = | 0 |
| |
| − | | 0 0 0 1 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 1 | | 0 |
| |
| | | | |
| − | 'l'm = "lover of a man" =
| + | I will devote some time to drawing out the relationships that exist among the different pictures of relations and relative terms that were shown above, or as redrawn here: |
| | | | |
| − | | 0 1 0 0 0 0 0 | | 0 | | 1 |
| + | <br> |
| − | | 1 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 0 0 0 0 1 | | 0 | | 1 |
| |
| − | | 0 0 0 0 1 0 0 | | 0 | = | 1 |
| |
| − | | 0 0 0 1 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 1 | | 0 |
| |
| | | | |
| − | 'l'w = "lover of a woman" =
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 7.0.jpg]] || (11) |
| | + | |} |
| | | | |
| − | | 0 1 0 0 0 0 0 | | 1 | | 0 | | + | {| align="center" cellpadding="10" |
| − | | 1 0 0 0 0 0 0 | | 0 | | 1 |
| + | | [[Image:LOR 1870 Figure 8.0.jpg]] || (12) |
| − | | 0 0 0 0 0 0 1 | | 1 | | 0 |
| + | |} |
| − | | 0 0 0 0 1 0 0 | | 1 | = | 0 |
| |
| − | | 0 0 0 1 0 0 0 | | 0 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 | | |
| − | | 0 0 1 0 0 0 0 | | 0 | | 1 | | |
| | | | |
| − | 's'1 = "servant of anybody" =
| + | Figures 11 and 12 present examples of relative multiplication in one of the styles of syntax that Peirce used, to which I added lines of identity to connect the corresponding marks of reference. These pictures are adapted to showing the anatomy of relative terms, while the forms of analysis illustrated in Table 13 and Figure 14 are designed to highlight the structures of the objective relations themselves. |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| + | <br> |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 1 | = | 1 |
| |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 1 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| | | | |
| − | 's'b = "servant of a black" =
| + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| | + | |+ style="height:30px" | <math>\text{Table 13.} ~~ \text{Relational Composition}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>Y\!</math> |
| | + | | |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>S\!</math> |
| | + | | |
| | + | | <math>Y\!</math> |
| | + | | <math>Z\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L \circ S\!</math> |
| | + | | <math>X\!</math> |
| | + | | |
| | + | | <math>Z\!</math> |
| | + | |} |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| + | <br> |
| − | | 0 0 0 0 0 0 1 | | 0 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 0 | = | 0 |
| |
| − | | 0 0 0 0 0 0 1 | | 0 | | 1 |
| |
| − | | 0 0 1 0 0 0 1 | | 0 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| | | | |
| − | 's'm = "servant of a man" =
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 14.jpg]] || (14) |
| | + | |} |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| + | There are many ways that Peirce might have gotten from his 1870 Notation for the Logic of Relatives to his more evolved systems of Logical Graphs. It is interesting to speculate on how the metamorphosis might have been accomplished by way of transformations that act on these nascent forms of syntax and that take place not too far from the pale of its means, that is, as nearly as possible according to the rules and the permissions of the initial system itself. |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 0 | = | 0 |
| |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 1 0 0 0 1 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| | | | |
| − | 's'w = "servant of a woman" =
| + | In Existential Graphs, a relation is represented by a node whose degree is the adicity of that relation, and which is adjacent via lines of identity to the nodes that represent its correlative relations, including as a special case any of its terminal individual arguments. |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| + | In the 1870 Logic of Relatives, implicit lines of identity are invoked by the subjacent numbers and marks of reference only when a correlate of some relation is the relate of some relation. Thus, the principal relate, which is not a correlate of any explicit relation, is not singled out in this way. |
| − | | 0 0 0 0 0 0 1 | | 0 | | 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 1 | = | 1 |
| |
| − | | 0 0 0 0 0 0 1 | | 0 | | 0 |
| |
| − | | 0 0 1 0 0 0 1 | | 0 | | 1 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| |
| | | | |
| − | 'ls' = "lover of a servant of ---" =
| + | Remarkably enough, the comma modifier itself provides us with a mechanism to abstract the logic of relations from the logic of relatives, and thus to forge a possible link between the syntax of relative terms and the more graphical depiction of the objective relations themselves. |
| | | | |
| − | | 0 1 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 |
| + | Figure 15 demonstrates this possibility, posing a transitional case between the style of syntax in Figure 11 and the picture of composition in Figure 14. |
| − | | 1 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 | | 0 0 0 0 0 0 0 |
| |
| − | | 0 0 0 0 0 0 1 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 |
| |
| − | | 0 0 0 0 1 0 0 | | 0 0 1 0 0 0 0 | = | 0 0 0 0 0 0 1 |
| |
| − | | 0 0 0 1 0 0 0 | | 0 0 0 0 0 0 1 | | 0 0 1 0 0 0 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 1 | | 0 0 0 0 0 0 0 |
| |
| − | | 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 |
| |
| | | | |
| − | 'sl' = "servant of a lover of ---" =
| + | <br> |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 0 1 0 0 0 0 0 | | 0 0 0 0 0 0 0 | | + | {| align="center" cellpadding="10" |
| − | | 0 0 0 0 0 0 1 | | 1 0 0 0 0 0 0 | | 0 0 1 0 0 0 0 |
| + | | [[Image:LOR 1870 Figure 15.jpg]] || (15) |
| − | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 | | 0 0 0 0 0 0 0 |
| + | |} |
| − | | 0 0 1 0 0 0 0 | | 0 0 0 0 1 0 0 | = | 0 0 0 0 0 0 1 |
| |
| − | | 0 0 0 0 0 0 1 | | 0 0 0 1 0 0 0 | | 0 0 1 0 0 0 0 | | |
| − | | 0 0 1 0 0 0 1 | | 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 1 | | |
| − | | 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 |
| |
| | | | |
| | + | In this composite sketch the diagonal extension <math>\mathit{1}\!</math> of the universe <math>\mathbf{1}\!</math> is invoked up front to anchor an explicit line of identity for the leading relate of the composition, while the terminal argument <math>\mathrm{w}\!</math> has been generalized to the whole universe <math>\mathbf{1},\!</math> in effect, executing an act of abstraction. This type of universal bracketing isolates the composing of the relations <math>L\!</math> and <math>S\!</math> to form the composite <math>L \circ S.\!</math> The three relational domains <math>X, Y, Z\!</math> may be distinguished from one another, or else rolled up into a single universe of discourse, as one prefers. |
| | | | |
| | + | ===Commentary Note 10.4=== |
| | | | |
| − | LOR. Commentary Note 8.6
| + | From now on I will use the forms of analysis exemplified in the last set of Figures and Tables as a routine bridge between the logic of relative terms and the logic of their extended relations. For future reference, we may think of Table 13 as illustrating the ''spreadsheet'' model of relational composition, while Figure 14 may be thought of as making a start toward a ''hypergraph'' model of generalized compositions. I will explain the hypergraph model in some detail at a later point. The transitional form of analysis represented by Figure 15 may be called the ''universal bracketing'' of relatives as relations. |
| | | | |
| | + | ===Commentary Note 10.5=== |
| | | | |
| | + | We have sufficiently covered the application of the comma functor, or the diagonal extension, to absolute terms, so let us return to where we were in working our way through CP 3.73 and see whether we can validate Peirce's statements about the “commifications” of 2-adic relative terms that yield their 3-adic diagonal extensions. |
| | | | |
| − | The foregoing has hopefully filled in enough background that we
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | can begin to make sense of the more mysterious parts of CP 3.73.
| + | | |
| | + | <p>But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one.</p> |
| | | | |
| − | | Thus far, we have considered the multiplication of relative terms only. | + | <p>Then:</p> |
| − | | Since our conception of multiplication is the application of a relation, | + | |- |
| − | | we can only multiply absolute terms by considering them as relatives. | + | | align="center" | <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> |
| | + | |- |
| | | | | | |
| − | | Now the absolute term "man" is really exactly equivalent to
| + | <p>will denote a lover of a woman that is a servant of that woman.</p> |
| − | | the relative term "man that is ---", and so with any other. | + | |
| − | | I shall write a comma after any absolute term to show that
| + | <p>The comma here after <math>\mathit{l}\!</math> should not be considered as altering at all the meaning of <math>\mathit{l}\!</math>, but as only a subjacent sign, serving to alter the arrangement of the correlates.</p> |
| − | | it is so regarded as a relative term. | + | |
| | + | <p>(Peirce, CP 3.73).</p> |
| | + | |} |
| | + | |
| | + | Just to plant our feet on a more solid stage, let's apply this idea to the Othello example. For this performance only, just to make the example more interesting, let us assume that <math>\mathrm{Jeste ~ (J)}\!</math> is secretly in love with <math>\mathrm{Desdemona ~ (D)}.\!</math> |
| | + | |
| | + | Then we begin with the modified data set: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Then: | + | <math>\begin{array}{*{15}{c}} |
| | + | \mathrm{w} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | \\[6pt] |
| | + | \mathit{l} |
| | + | & = & \mathrm{B}\!:\!\mathrm{C} |
| | + | & +\!\!, & \mathrm{C}\!:\!\mathrm{B} |
| | + | & +\!\!, & \mathrm{D}\!:\!\mathrm{O} |
| | + | & +\!\!, & \mathrm{E}\!:\!\mathrm{I} |
| | + | & +\!\!, & \mathrm{I}\!:\!\mathrm{E} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{D} |
| | + | \\[6pt] |
| | + | \mathit{s} |
| | + | & = & \mathrm{C}\!:\!\mathrm{O} |
| | + | & +\!\!, & \mathrm{E}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{I}\!:\!\mathrm{O} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | And next we derive the following results: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | "man that is black" | + | <math>\begin{array}{l} |
| | + | \mathit{l}, ~= |
| | + | \\[6pt] |
| | + | \text{lover that is}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~} ~= |
| | + | \\[6pt] |
| | + | (\mathrm{B}\!:\!\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{D}) |
| | + | \\[12pt] |
| | + | \mathit{l},\!\mathit{s}\mathrm{w} ~= |
| | + | \\[6pt] |
| | + | (\mathrm{B}\!:\!\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{D}) |
| | + | \\ |
| | + | \times |
| | + | \\ |
| | + | (\mathrm{C}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{O}) |
| | + | \\ |
| | + | \times |
| | + | \\ |
| | + | (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Now what are we to make of that? |
| | + | |
| | + | If we operate in accordance with Peirce's example of <math>\mathfrak{g}\mathit{o}\mathrm{h}</math> as the “giver of a horse to an owner of that horse”, then we may assume that the associative law and the distributive law are in force, allowing us to derive this equation: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | will be written | + | <math>\begin{array}{lll} |
| | + | \mathit{l},\!\mathit{s}\mathrm{w} |
| | + | & = & |
| | + | \mathit{l},\!\mathit{s}(\mathrm{B} ~~+\!\!,~~ \mathrm{D} ~~+\!\!,~~ \mathrm{E}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathit{l},\!\mathit{s}\mathrm{B} ~~+\!\!,~~ \mathit{l},\!\mathit{s}\mathrm{D} ~~+\!\!,~~ \mathit{l},\!\mathit{s}\mathrm{E} |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | Evidently what Peirce means by the associative principle, as it applies to this type of product, is that a product of elementary relatives having the form <math>(\mathrm{R}\!:\!\mathrm{S}\!:\!\mathrm{T})(\mathrm{S}\!:\!\mathrm{T})(\mathrm{T})\!</math> is equal to <math>\mathrm{R}\!</math> but that no other form of product yields a non-null result. Scanning the implied terms of the triple product tells us that only the case <math>(\mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{D})(\mathrm{J}\!:\!\mathrm{D})(\mathrm{D}) = \mathrm{J}\!</math> is non-null. |
| | + | |
| | + | It follows that: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | m,b.
| + | <math>\begin{array}{lll} |
| − | |
| + | \mathit{l},\!\mathit{s}\mathrm{w} |
| − | | C.S. Peirce, CP 3.73 | + | & = & |
| | + | \text{lover and servant of a woman} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{lover that is a servant of a woman} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{lover of a woman that is a servant of that woman} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{J} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | In any system where elements are organized according to types,
| + | And so what Peirce says makes sense in this case. |
| − | there tend to be any number of ways in which elements of one
| |
| − | type are naturally associated with elements of another type.
| |
| − | If the association is anything like a logical equivalence,
| |
| − | but with the first type being "lower" and the second type
| |
| − | being "higher" in some sense, then one frequently speaks
| |
| − | of a "semantic ascent" from the lower to the higher type.
| |
| | | | |
| − | For instance, it is very common in mathematics to associate an element m
| + | ===Commentary Note 10.6=== |
| − | of a set M with the constant function f_m : X -> M such that f_m (x) = m
| |
| − | for all x in X, where X is an arbitrary set. Indeed, the correspondence
| |
| − | is so close that one often uses the same name "m" for the element m in M
| |
| − | and the function m = f_m : X -> M, relying on the context or an explicit
| |
| − | type indication to tell them apart.
| |
| | | | |
| − | For another instance, we have the "tacit extension" of a k-place relation
| + | As Peirce observes, it is not possible to work with relations in general without eventually abandoning all of one's algebraic principles, in due time the associative law and maybe even the distributive law, just as we already gave up the commutative law. It cannot be helped, as we cannot reflect on a law if not from a perspective outside it, at any rate, virtually so. |
| − | L c X_1 x ... x X_k to a (k+1)-place relation L' c X_1 x ... x X_k+1 that
| |
| − | we get by letting L' = L x X_k+1, that is, by maintaining the constraints
| |
| − | of L on the first k variables and letting the last variable wander freely.
| |
| | | | |
| − | What we have here, if I understand Peirce correctly, is another such
| + | This could be done from the standpoint of the combinator calculus, and there are places where Peirce verges on systems that are very similar, but here we are making a deliberate effort to stay within the syntactic neighborhood of Peirce's 1870 Logic of Relatives. Not too coincidentally, it is for the sake of making smoother transitions between narrower and wider regimes of algebraic law that we have been developing the paradigm of Figures and Tables indicated above. |
| − | type of natural extension, sometimes called the "diagonal extension".
| |
| − | This associates a k-adic relative or a k-adic relation, counting the
| |
| − | absolute term and the set whose elements it denotes as the cases for
| |
| − | k = 0, with a series of relatives and relations of higher adicities.
| |
| | | | |
| − | A few examples will suffice to anchor these ideas.
| + | For the next few episodes, then, I will examine the examples that Peirce gives at the next level of complication in the multiplication of relative terms, for example, the three that are repeated below. |
| | | | |
| − | Absolute terms:
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 8.0.jpg]] || (16) |
| | + | |} |
| | | | |
| − | m = "man" = C +, I +, J +, O
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 17.0.jpg]] || (17) |
| | + | |} |
| | | | |
| − | n = "noble" = C +, D +, O
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 18.jpg]] || (18) |
| | + | |} |
| | | | |
| − | w = "woman" = B +, D +, E
| + | ===Commentary Note 10.7=== |
| | | | |
| − | Diagonal extensions:
| + | Here is what I get when I try to analyze Peirce's “giver of a horse to a lover of a woman” example along the same lines as the dyadic compositions. |
| | | | |
| − | m, = "man that is ---" = C:C +, I:I +, J:J +, O:O
| + | We may begin with the mark-up shown in Figure 19. |
| | | | |
| − | n, = "noble that is ---" = C:C +, D:D +, O:O
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 8.0.jpg]] || (19) |
| | + | |} |
| | | | |
| − | w, = "woman that is ---" = B:B +, D:D +, E:E
| + | If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a triadic ''giving'' relation <math>G \subseteq T \times U \times V\!</math> with a dyadic ''loving'' relation <math>L \subseteq U \times W\!</math> so as to obtain a specialized sort of triadic relation <math>(G \circ L) \subseteq T \times V \times W.\!</math> The applicable constraints on tuples are shown in Table 20. |
| | | | |
| − | Sample products:
| + | <br> |
| | | | |
| − | m,n = "man that is noble"
| + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:75%" |
| | + | |+ style="height:30px" | <math>\text{Table 20.} ~~ \text{Composite of Triadic and Dyadic Relations}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black; border-bottom:1px solid black; width:20%" | |
| | + | | style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:20%" | <math>\mathit{1}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>G\!</math> |
| | + | | <math>T\!</math> |
| | + | | <math>U\!</math> |
| | + | | <math>V\!</math> |
| | + | | |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L\!</math> |
| | + | | |
| | + | | <math>U\!</math> |
| | + | | |
| | + | | <math>W\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>G \circ L</math> |
| | + | | <math>T\!</math> |
| | + | | |
| | + | | <math>V\!</math> |
| | + | | <math>W\!</math> |
| | + | |} |
| | | | |
| − | = (C:C +, I:I +, J:J +, O:O)(C +, D +, O)
| + | <br> |
| | | | |
| − | = C +, O
| + | The hypergraph picture of the abstract composition is given in Figure 21. |
| | | | |
| − | n,m = "noble that is man"
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 21.jpg]] || (21) |
| | + | |} |
| | | | |
| − | = (C:C +, D:D +, O:O)(C +, I +, J +, O)
| + | ===Commentary Note 10.8=== |
| | | | |
| − | = C +, O
| + | There's a critical transition point in sight of Peirce's 1870 Logic of Relatives and it's a point that turns on the teridentity relation. |
| | | | |
| − | n,w = "noble that is woman"
| + | In taking up the next example of relational composition, let's substitute the relation <math>\mathit{t} = \text{taker of}\, \underline{~~ ~~}\!</math> for Peirce's relation <math>\mathit{o} = \text{owner of}\, \underline{~~ ~~},\!</math> simply for the sake of avoiding conflicts in the symbols we use. In this way, Figure 17 is transformed into Figure 22. |
| | | | |
| − | = (C:C +, D:D +, O:O)(B +, D +, E)
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 22.jpg]] || (22) |
| | + | |} |
| | | | |
| − | = D
| + | The hypergraph picture of the abstract composition is given in Figure 23. |
| | | | |
| − | w,n = "woman that is noble"
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 23.jpg]] || (23) |
| | + | |} |
| | | | |
| − | = (B:B +, D:D +, E:E)(C +, D +, O)
| + | If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a triadic “giving” relation <math>G \subseteq X \times Y \times Z\!</math> with a dyadic “taking” relation <math>T \subseteq Y \times Z\!</math> in such a way as to determine a certain dyadic relation <math>(G \circ T) \subseteq X \times Z.\!</math> Table 24 schematizes the associated constraints on tuples. |
| | | | |
| − | = D
| + | <br> |
| | | | |
| | + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| | + | |+ style="height:30px" | <math>\text{Table 24.} ~~ \text{Another Brand of Composition}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>G\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>Y\!</math> |
| | + | | <math>Z\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>T\!</math> |
| | + | | |
| | + | | <math>Y\!</math> |
| | + | | <math>Z\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>G \circ T</math> |
| | + | | <math>X\!</math> |
| | + | | |
| | + | | <math>Z\!</math> |
| | + | |} |
| | | | |
| | + | <br> |
| | | | |
| − | LOR. Note 9
| + | So we see that the notorious teridentity relation, which I have left equivocally denoted by the same symbol as the identity relation <math>\mathit{1},\!</math> is already implicit in Peirce's discussion at this point. |
| | | | |
| | + | ===Commentary Note 10.9=== |
| | | | |
| | + | The use of the concepts of identity and teridentity is not to identify a thing-in-itself with itself, much less twice or thrice over — there is no need and therefore no utility in that. I can imagine Peirce asking, on Kantian principles if not entirely on Kantian premisses, <i>Where is the manifold to be unified?</i> The manifold that demands unification does not reside in the object but in the phenomena, that is, in the appearances that might have been appearances of different objects but that happen to be constrained by these identities to being just so many aspects, facets, parts, roles, or signs of one and the same object. |
| | | | |
| − | | The Signs for Multiplication (cont.)
| + | For example, notice how the various identity concepts actually functioned in the last example, where they had the opportunity to show their behavior in something like their natural habitat. |
| − | |
| + | |
| − | | It is obvious that multiplication into
| + | The use of the teridentity concept in the case of the “giver of a horse to a taker of it” is to say that the thing appearing with respect to its quality under an absolute term, <i>a horse</i>, the thing appearing with respect to its existence as the correlate of a dyadic relative, <i>a potential possession</i>, and the thing appearing with respect to its synthesis as the correlate of a triadic relative, <i>a gift</i>, are one and the same thing. |
| − | | a multiplicand indicated by a comma is
| |
| − | | commutative <1>, that is,
| |
| − | |
| |
| − | | 's','l' = 'l','s'.
| |
| − | |
| |
| − | | This multiplication is effectively the same as
| |
| − | | that of Boole in his logical calculus. Boole's
| |
| − | | unity is my 1, that is, it denotes whatever is.
| |
| − | |
| |
| − | | <1>. It will often be convenient to speak of the whole operation of
| |
| − | | affixing a comma and then multiplying as a commutative multiplication,
| |
| − | | the sign for which is the comma. But though this is allowable, we shall
| |
| − | | fall into confusion at once if we ever forget that in point of fact it is
| |
| − | | not a different multiplication, only it is multiplication by a relative
| |
| − | | whose meaning -- or rather whose syntax -- has been slightly altered;
| |
| − | | and that the comma is really the sign of this modification of the
| |
| − | | foregoing term.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.74
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| | | | |
| | + | ===Commentary Note 10.10=== |
| | | | |
| | + | The last of the three examples involving the composition of triadic relatives with dyadic relatives is shown again in Figure 25. |
| | | | |
| − | LOR. Commentary Note 9.1 | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 18.jpg]] || (25) |
| | + | |} |
| | | | |
| | + | The hypergraph picture of the abstract composition is given in Figure 26. |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 26.jpg]] || (26) |
| | + | |} |
| | | | |
| − | Let us backtrack a few years, and consider how Boole explained his
| + | This example illustrates the way that Peirce analyzes the logical conjunction, we might even say the ''parallel conjunction'', of a pair of dyadic relatives in terms of the comma extension and the same style of composition that we saw in the last example, that is, according to a pattern of anaphora that invokes the teridentity relation. |
| − | twin conceptions of "selective operations" and "selective symbols".
| |
| | | | |
| − | | Let us then suppose that the universe of our discourse
| + | If we lay out this analysis of conjunction on the spreadsheet model of relational composition, the gist of it is the diagonal extension of a dyadic ''loving'' relation <math>L \subseteq X \times Y\!</math> to the corresponding triadic ''being and loving'' relation <math>L \subseteq X \times X \times Y,\!</math> which is then composed in a specific way with a dyadic ''serving'' relation <math>S \subseteq X \times Y\!</math> so as to determine the dyadic relation <math>L,\!S \subseteq X \times Y.\!</math> Table 27 schematizes the associated constraints on tuples. |
| − | | is the actual universe, so that words are to be used in
| |
| − | | the full extent of their meaning, and let us consider the
| |
| − | | two mental operations implied by the words "white" and "men".
| |
| − | | The word "men" implies the operation of selecting in thought
| |
| − | | from its subject, the universe, all men; and the resulting
| |
| − | | conception, 'men', becomes the subject of the next operation.
| |
| − | | The operation implied by the word "white" is that of selecting
| |
| − | | from its subject, "men", all of that class which are white.
| |
| − | | The final resulting conception is that of "white men".
| |
| − | |
| |
| − | | Now it is perfectly apparent that if the operations above described
| |
| − | | had been performed in a converse order, the result would have been the
| |
| − | | same. Whether we begin by forming the conception of "'men'", and then by
| |
| − | | a second intellectual act limit that conception to "white men", or whether
| |
| − | | we begin by forming the conception of "white objects", and then limit it to
| |
| − | | such of that class as are "men", is perfectly indifferent so far as the result
| |
| − | | is concerned. It is obvious that the order of the mental processes would be
| |
| − | | equally indifferent if for the words "white" and "men" we substituted any
| |
| − | | other descriptive or appellative terms whatever, provided only that their
| |
| − | | meaning was fixed and absolute. And thus the indifference of the order
| |
| − | | of two successive acts of the faculty of Conception, the one of which
| |
| − | | furnishes the subject upon which the other is supposed to operate,
| |
| − | | is a general condition of the exercise of that faculty. It is
| |
| − | | a law of the mind, and it is the real origin of that law of
| |
| − | | the literal symbols of Logic which constitutes its formal
| |
| − | | expression (1) Chap. II, [namely, xy = yx].
| |
| − | |
| |
| − | | It is equally clear that the mental operation above described is of such
| |
| − | | a nature that its effect is not altered by repetition. Suppose that by
| |
| − | | a definite act of conception the attention has been fixed upon men, and
| |
| − | | that by another exercise of the same faculty we limit it to those of the
| |
| − | | race who are white. Then any further repetition of the latter mental act,
| |
| − | | by which the attention is limited to white objects, does not in any way
| |
| − | | modify the conception arrived at, viz., that of white men. This is also
| |
| − | | an example of a general law of the mind, and it has its formal expression
| |
| − | | in the law ((2) Chap. II) of the literal symbols [namely, x^2 = x].
| |
| − | |
| |
| − | | Boole, 'Laws of Thought', pp. 44-45.
| |
| − | |
| |
| − | | George Boole,
| |
| − | |'An Investigation of the Laws of Thought,
| |
| − | | On Which are Founded the Mathematical
| |
| − | | Theories of Logic and Probabilities',
| |
| − | | Reprinted, Dover, New York, NY, 1958.
| |
| − | | Originally published, Macmillan, 1854.
| |
| | | | |
| | + | <br> |
| | | | |
| | + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| | + | |+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Conjunction Via Composition}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L,\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>Y\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>S\!</math> |
| | + | | |
| | + | | <math>X\!</math> |
| | + | | <math>Y\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L,\!S</math> |
| | + | | <math>X\!</math> |
| | + | | |
| | + | | <math>Y\!</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 9.2
| + | <br> |
| | | | |
| | + | ===Commentary Note 10.11=== |
| | | | |
| | + | Let us return to the point where we left off unpacking the contents of CP 3.73. Peirce remarks that the comma operator can be iterated at will: |
| | | | |
| − | In setting up his discussion of selective operations and | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | their corresponding selective symbols, Boole writes this:
| + | | |
| | + | <p>In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates.</p> |
| | | | |
| − | | The operation which we really perform is one of 'selection according to
| + | <p>So:</p> |
| − | | a prescribed principle or idea'. To what faculties of the mind such an
| + | |- |
| − | | operation would be referred, according to the received classification of
| + | | align="center" | <math>\mathrm{m},\!,\!\mathrm{b},\!\mathrm{r}</math> |
| − | | its powers, it is not important to inquire, but I suppose that it would be
| + | |- |
| − | | considered as dependent upon the two faculties of Conception or Imagination,
| |
| − | | and Attention. To the one of these faculties might be referred the formation
| |
| − | | of the general conception; to the other the fixing of the mental regard upon
| |
| − | | those individuals within the prescribed universe of discourse which answer to
| |
| − | | the conception. If, however, as seems not improbable, the power of Attention
| |
| − | | is nothing more than the power of continuing the exercise of any other faculty
| |
| − | | of the mind, we might properly regard the whole of the mental process above
| |
| − | | described as referrible to the mental faculty of Imagination or Conception, | |
| − | | the first step of the process being the conception of the Universe itself, | |
| − | | and each succeeding step limiting in a definite manner the conception | |
| − | | thus formed. Adopting this view, I shall describe each such step,
| |
| − | | or any definite combination of such steps, as a 'definite act
| |
| − | | of conception'. | |
| | | | | | |
| − | | Boole, 'Laws of Thought', p. 43. | + | <p>interpreted like</p> |
| | + | |- |
| | + | | align="center" | <math>\mathfrak{g}\mathit{o}\mathrm{h}</math> |
| | + | |- |
| | | | | | |
| − | | George Boole,
| + | <p>means a man that is a rich individual and is a black that is that rich individual.</p> |
| − | |'An Investigation of the Laws of Thought,
| |
| − | | On Which are Founded the Mathematical
| |
| − | | Theories of Logic and Probabilities',
| |
| − | | Reprinted, Dover, New York, NY, 1958.
| |
| − | | Originally published, Macmillan, 1854.
| |
| | | | |
| | + | <p>But this has no other meaning than:</p> |
| | + | |- |
| | + | | align="center" | <math>\mathrm{m},\!\mathrm{b},\!\mathrm{r}</math> |
| | + | |- |
| | + | | |
| | + | <p>or a man that is a black that is rich.</p> |
| | | | |
| | + | <p>Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number.</p> |
| | | | |
| − | LOR. Commentary Note 9.3
| + | <p>(Peirce, CP 3.73).</p> |
| | + | |} |
| | | | |
| | + | Again, let us check whether this makes sense on the stage of our small but dramatic model. Let's say that Desdemona and Othello are rich, and, among the persons of the play, only they. With this premiss we obtain a sample of absolute terms that is sufficiently ample to work through our example: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{*{15}{c}} |
| | + | \mathbf{1} |
| | + | & = & \mathrm{B} |
| | + | & +\!\!, & \mathrm{C} |
| | + | & +\!\!, & \mathrm{D} |
| | + | & +\!\!, & \mathrm{E} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{b} |
| | + | & = & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m} |
| | + | & = & \mathrm{C} |
| | + | & +\!\!, & \mathrm{I} |
| | + | & +\!\!, & \mathrm{J} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{r} |
| | + | & = & \mathrm{D} |
| | + | & +\!\!, & \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | In algebra, an "idempotent element" x is one that obeys the
| + | One application of the comma operator yields the following 2-adic relatives: |
| − | "idempotent law", that is, it satisfies the equation xx = x.
| |
| − | Under most circumstances, it is usual to write this x^2 = x.
| |
| | | | |
| − | If the algebraic system in question falls under the additional laws
| + | {| align="center" cellspacing="6" width="90%" |
| − | that are necessary to carry out the requisite transformations, then
| + | | |
| − | x^2 = x is convertible into x - x^2 = 0, and this into x(1 - x) = 0.
| + | <math>\begin{array}{*{15}{c}} |
| | + | \mathbf{1,} |
| | + | & = & \mathrm{B}\!:\!\mathrm{B} |
| | + | & +\!\!, & \mathrm{C}\!:\!\mathrm{C} |
| | + | & +\!\!, & \mathrm{D}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{E}\!:\!\mathrm{E} |
| | + | & +\!\!, & \mathrm{I}\!:\!\mathrm{I} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{J} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{b,} |
| | + | & = & \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m,} |
| | + | & = & \mathrm{C}\!:\!\mathrm{C} |
| | + | & +\!\!, & \mathrm{I}\!:\!\mathrm{I} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{J} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{r,} |
| | + | & = & \mathrm{D}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | If the algebraic system in question happens to be a boolean algebra,
| + | Another application of the comma operator generates the following 3-adic relatives: |
| − | then the equation x(1 - x) = 0 says that x & ~x is identically false,
| |
| − | in effect, a statement of the classical principle of non-contradiction.
| |
| | | | |
| − | We have already seen how Boole found rationales for the commutative law and
| + | {| align="center" cellspacing="6" width="90%" |
| − | the idempotent law by contemplating the properties of "selective operations".
| + | | |
| | + | <math>\begin{array}{*{9}{c}} |
| | + | \mathbf{1,\!,} |
| | + | & = & \mathrm{B}\!:\!\mathrm{B}\!:\!\mathrm{B} |
| | + | & +\!\!, & \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{C} |
| | + | & +\!\!, & \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{E}\!:\!\mathrm{E}\!:\!\mathrm{E} |
| | + | \\ |
| | + | & & |
| | + | & +\!\!, & \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{I} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{J} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{b,\!,} |
| | + | & = & \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{m,\!,} |
| | + | & = & \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{C} |
| | + | & +\!\!, & \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{I} |
| | + | & +\!\!, & \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{J} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O} |
| | + | \\[6pt] |
| | + | \mathrm{r,\!,} |
| | + | & = & \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{D} |
| | + | & +\!\!, & \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | It is time to bring these threads together, which we can do by considering the
| + | Assuming the associativity of multiplication among 2-adic relatives, we may compute the product <math>~\mathrm{m},\mathrm{b},\mathrm{r}~</math> by a brute force method as follows: |
| − | so-called "idempotent representation" of sets. This will give us one of the
| |
| − | best ways to understand the significance that Boole attached to selective
| |
| − | operations. It will also link up with the statements that Peirce makes
| |
| − | about his adicity-augmenting comma operation.
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathrm{m},\mathrm{b},\mathrm{r} |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O})(\mathrm{O}\!:\!\mathrm{O})(\mathrm{D} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O})(\mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| | + | This says that a man that is black that is rich is Othello, which is true on the premisses of our present universe of discourse. |
| | | | |
| − | LOR. Commentary Note 9.4
| + | Following the standard associative combinations of <math>\mathfrak{g}\mathit{o}\mathrm{h},</math> the product <math>~\mathrm{m},\!,\mathrm{b},\mathrm{r}~</math> is multiplied out along the following lines, where the trinomials of the form <math>\mathrm{(X\!:\!Y\!:\!Z)(Y\!:\!Z)(Z)}\!</math> are the only ones that produce a non-null result, namely, <math>\mathrm{(X\!:\!Y\!:\!Z)(Y\!:\!Z)(Z) = X}.\!</math> |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | \mathrm{m},\!,\mathrm{b},\mathrm{r} |
| | + | & = & |
| | + | (\mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O})(\mathrm{O}\!:\!\mathrm{O})(\mathrm{D} ~+\!\!,~ \mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | (\mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O})(\mathrm{O}\!:\!\mathrm{O})(\mathrm{O}) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| | + | So we have that <math>\mathrm{m},\!,\mathrm{b},\mathrm{r} ~=~ \mathrm{m},\mathrm{b},\mathrm{r}.</math> |
| | | | |
| − | Boole rationalized the properties of what we now dub "boolean multiplication",
| + | In closing, observe that the teridentity relation has turned up again in this context, as the second comma-ing of the universal term itself: |
| − | roughly equivalent to logical conjunction, in terms of the laws that apply to
| |
| − | selective operations. Peirce, in his turn, taking a very significant step of
| |
| − | analysis that has seldom been recognized for what it would lead to, much less
| |
| − | followed, does not consider this multiplication to be a fundamental operation,
| |
| − | but derives it as a by-product of relative multiplication by a comma relative.
| |
| − | Thus, Peirce makes logical conjunction a special case of relative composition.
| |
| | | | |
| − | This opens up a very wide field of investigation,
| + | {| align="center" cellspacing="6" width="90%" |
| − | "the operational significance of logical terms", | + | | |
| − | one might say, but it will be best to advance
| + | <math>\begin{array}{l} |
| − | bit by bit, and to lean on simple examples.
| + | \mathbf{1},\!, ~=~ |
| | + | \mathrm{B}\!:\!\mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{O} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | Back to Venice, and the close-knit party
| + | ===Commentary Note 10.12=== |
| − | of absolutes and relatives that we were
| |
| − | entertaining when last we were there.
| |
| | | | |
| − | Here is the list of absolute terms that we were considering before,
| + | Potential ambiguities in Peirce's two versions of the “rich black man” example can be resolved by providing them with explicit graphical markups, as shown in Figures 28 and 29. |
| − | to which I have thrown in 1, the universe of "anybody or anything",
| |
| − | just for good measure:
| |
| | | | |
| − | 1 = "anybody" = B +, C +, D +, E +, I +, J +, O
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 28.jpg]] || (28) |
| | + | |- |
| | + | | [[Image:LOR 1870 Figure 29.jpg]] || (29) |
| | + | |} |
| | | | |
| − | m = "man" = C +, I +, J +, O
| + | On the other hand, as the forms of relational composition become more complex, the corresponding algebraic products of elementary relatives, for example, <math>\mathrm{(x\!:\!y\!:\!z)(y\!:\!z)(z)},\!</math> will not always determine unique results without the addition of more information about the intended linking of terms. |
| | | | |
| − | n = "noble" = C +, D +, O
| + | ==Selection 11== |
| | | | |
| − | w = "woman" = B +, D +, E
| + | ===The Signs for Multiplication (concl.)=== |
| | | | |
| − | Here is the list of "comma inflexions" or "diagonal extensions" of these terms:
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>The conception of multiplication we have adopted is that of the application of one relation to another. So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second.</p> |
| | | | |
| − | 1, = "anybody that is ---" = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
| + | <p>Even ordinary numerical multiplication involves the same idea, for <math>~2 \times 3~</math> is a pair of triplets, and <math>~3 \times 2~</math> is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives.</p> |
| | | | |
| − | m, = "man that is ---" = C:C +, I:I +, J:J +, O:O
| + | <p>If we have an equation of the form:</p> |
| | + | |- |
| | + | | align="center" | <math>xy ~=~ z</math> |
| | + | |- |
| | + | | |
| | + | <p>and there are just as many <math>x\!</math>'s per <math>y\!</math> as there are, ''per'' things, things of the universe, then we have also the arithmetical equation:</p> |
| | + | |- |
| | + | | align="center" | <math>[x][y] ~=~ [z].</math> |
| | + | |- |
| | + | | |
| | + | <p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]</math> |
| | + | |- |
| | + | | |
| | + | <p>holds arithmetically.</p> |
| | | | |
| − | n, = "noble that is ---" = C:C +, D:D +, O:O
| + | <p>So if men are just as apt to be black as things in general:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]</math> |
| | + | |- |
| | + | | |
| | + | <p>where the difference between <math>[\mathrm{m}]\!</math> and <math>[\mathrm{m,}]\!</math> must not be overlooked.</p> |
| | | | |
| − | w, = "woman that is ---" = B:B +, D:D +, E:E
| + | <p>It is to be observed that:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathit{1}] ~=~ \mathfrak{1}.</math> |
| | + | |- |
| | + | | |
| | + | <p>Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of ''expectation''.</p> |
| | | | |
| − | One observes that the diagonal extension of 1
| + | <p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.</p> |
| − | is the same thing as the identity relation !1!.
| |
| | | | |
| − | Inspired by this identification of "1," with "!1!", and because
| + | <p>(Peirce, CP 3.76).</p> |
| − | the affixed commas of the diagonal extensions tend to get lost
| + | |} |
| − | in the ordinary commas of punctuation, I will experiment with
| |
| − | using the alternative notations:
| |
| | | | |
| − | m, = !m!
| + | ===Commentary Note 11.1=== |
| − | n, = !n!
| |
| − | w, = !w!
| |
| | | | |
| − | Working within our smaller sample of absolute terms,
| + | We have reached a suitable place to pause in our reading of Peirce's text — actually, it's more like a place to run as fast as we can along a parallel track — where I can pay off a few of the expository IOUs I've been using to pave the way to this point. |
| − | we have already computed the sorts of products that | |
| − | apply the diagonal extension of an absolute term to
| |
| − | another absolute term, for instance, these products:
| |
| | | | |
| − | m,n = !m!n = "man that is noble" = C +, O
| + | The more pressing debts that come to mind are concerned with the matter of Peirce's “number of” function that maps a term <math>t\!</math> into a number <math>[t],\!</math> and with my justification for calling a certain style of illustration the ''hypergraph picture'' of relational composition. As it happens, there is a thematic relation between these topics, and so I can make my way forward by addressing them together. |
| − | n,m = !n!m = "noble that is man" = C +, O
| |
| − | n,w = !n!w = "noble that is woman" = D
| |
| − | w,n = !w!n = "woman that is noble" = D
| |
| | | | |
| − | This exercise gave us a bit of practical insight into
| + | At this point we have two good pictures of how to compute the relational compositions of arbitrary dyadic relations, namely, the bigraph representation and the matrix representation, each of which has its differential advantages in different types of situations. |
| − | why the commutative law holds for logical conjunction.
| |
| | | | |
| − | Further insight into the laws that govern this realm of logic,
| + | But we do not have a comparable picture of how to compute the richer variety of relational compositions that involve triadic or any higher adicity relations. As a matter of fact, we run into a non-trivial classification problem simply to enumerate the different types of compositions that arise in these cases. |
| − | and the underlying reasons why they apply, might be gained by
| |
| − | systematically working through the whole variety of different
| |
| − | products that are generated by the operational means in sight,
| |
| − | namely, the products indicated by {1, m, n, w}<,>{1, m, n, w}.
| |
| | | | |
| − | But before we try to explore this territory more systematically,
| + | Therefore, let us inaugurate a systematic study of relational composition, general enough to articulate the “generative potency” of Peirce's 1870 Logic of Relatives. |
| − | let us equip ourselves with the sorts of graphical and matrical | |
| − | representations that we discovered to provide us with such able
| |
| − | assists to the intuition in so many of our previous adventures.
| |
| | | | |
| | + | ===Commentary Note 11.2=== |
| | | | |
| | + | Let's bring together the various things that Peirce has said about the “number of function” up to this point in the paper. |
| | | | |
| − | LOR. Commentary Note 9.5
| + | ====NOF 1==== |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (''men''), the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus <math>[t].\!</math></p> |
| | | | |
| | + | <p>(Peirce, CP 3.65).</p> |
| | + | |} |
| | | | |
| − | Peirce's comma operation, in its application to an absolute term,
| + | ====NOF 2==== |
| − | is tantamount to the representation of that term's denotation as
| |
| − | an idempotent transformation, which is commonly represented as a
| |
| − | diagonal matrix. This is why I call it the "diagonal extension".
| |
| | | | |
| − | An idempotent element x is given by the abstract condition that xx = x,
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | but we commonly encounter such elements in more concrete circumstances, | + | | |
| − | acting as operators or transformations on other sets or spaces, and in
| + | <p>But not only do the significations of <math>~=~</math> and <math>~<~</math> here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write <math>~5 < 7~</math> is to say that <math>~5~</math> is part of <math>~7~</math>, just as to write <math>~\mathrm{f} < \mathrm{m}~</math> is to say that Frenchmen are part of men. Indeed, if <math>~\mathrm{f} < \mathrm{m}~</math>, then the number of Frenchmen is less than the number of men, and if <math>~\mathrm{v} = \mathrm{p}~</math>, then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.</p> |
| − | that action they will often be represented as matrices of coefficients. | |
| | | | |
| − | Let's see how all of this looks from the graphical and matrical perspectives.
| + | <p>(Peirce, CP 3.66).</p> |
| | + | |} |
| | | | |
| − | Absolute terms:
| + | ====NOF 3==== |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of ''taking together'' involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.</p> |
| | + | |
| | + | <p>Addition being taken in this sense, ''nothing'' is to be denoted by ''zero'', for then</p> |
| | + | |- |
| | + | | align="center" | <math>x ~+\!\!,~ 0 ~=~ x</math> |
| | + | |- |
| | + | | |
| | + | <p>whatever is denoted by <math>~x~</math>; and this is the definition of ''zero''. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of ''zero'' and that of nothing, and because we shall thus have</p> |
| | + | |- |
| | + | | align="center" | <math>[0] ~=~ 0.</math> |
| | + | |- |
| | + | | |
| | + | <p>(Peirce, CP 3.67).</p> |
| | + | |} |
| | + | |
| | + | ====NOF 4==== |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p> |
| | | | |
| − | 1 = "anybody" = B +, C +, D +, E +, I +, J +, O
| + | <p>Even ordinary numerical multiplication involves the same idea, for <math>~2 \times 3~</math> is a pair of triplets, and <math>~3 \times 2~</math> is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives.</p> |
| | | | |
| − | m = "man" = C +, I +, J +, O
| + | <p>If we have an equation of the form:</p> |
| | + | |- |
| | + | | align="center" | <math>xy ~=~ z</math> |
| | + | |- |
| | + | | |
| | + | <p>and there are just as many <math>x\!</math>'s per <math>y\!</math> as there are, ''per'' things, things of the universe, then we have also the arithmetical equation:</p> |
| | + | |- |
| | + | | align="center" | <math>[x][y] ~=~ [z].</math> |
| | + | |- |
| | + | | |
| | + | <p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]</math> |
| | + | |- |
| | + | | |
| | + | <p>holds arithmetically.</p> |
| | | | |
| − | n = "noble" = C +, D +, O
| + | <p>So if men are just as apt to be black as things in general:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]</math> |
| | + | |- |
| | + | | |
| | + | <p>where the difference between <math>[\mathrm{m}]\!</math> and <math>[\mathrm{m,}]\!</math> must not be overlooked.</p> |
| | | | |
| − | w = "woman" = B +, D +, E
| + | <p>It is to be observed that:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathit{1}] ~=~ \mathfrak{1}.</math> |
| | + | |- |
| | + | | |
| | + | <p>Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of ''expectation''.</p> |
| | | | |
| − | Previously, we represented absolute terms as column vectors.
| + | <p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.</p> |
| − | The above four terms are given by the columns of this table:
| |
| | | | |
| − | | 1 m n w
| + | <p>(Peirce, CP 3.76).</p> |
| − | ---o---------
| + | |} |
| − | B | 1 0 0 1
| |
| − | C | 1 1 1 0
| |
| − | D | 1 0 1 1
| |
| − | E | 1 0 0 1
| |
| − | I | 1 1 0 0
| |
| − | J | 1 1 0 0
| |
| − | O | 1 1 1 0
| |
| | | | |
| − | One way to represent sets in the bigraph picture
| + | ===Commentary Note 11.3=== |
| − | is simply to mark the nodes in some way, like so:
| |
| | | | |
| − | B C D E I J O
| + | Before I can discuss Peirce's “number of” function in greater detail I will need to deal with an expositional difficulty that I have been very carefully dancing around all this time, but one that will no longer abide its assigned place under the rug. |
| − | 1 + + + + + + +
| |
| | | | |
| − | B C D E I J O
| + | Functions have long been understood, from well before Peirce's time to ours, as special cases of dyadic relations, so the “number of” function itself is already to be numbered among the types of dyadic relatives that we've been explicitly mentioning and implicitly using all this time. But Peirce's way of talking about a dyadic relative term is to list the “relate” first and the “correlate” second, a convention that goes over into functional terms as making the functional value first and the functional argument second, whereas almost anyone brought up in our present time frame has difficulty thinking of a function any other way than as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second. |
| − | m o + o o + + +
| |
| | | | |
| − | B C D E I J O
| + | All of these syntactic wrinkles can be ironed out in a very smooth way, given a sufficiently general context of flexible enough interpretive conventions, but not without introducing an order of anachronism into Peirce's presentation that I am presently trying to avoid as much as possible. Thus, I will need to experiment with various styles of compromise formation. |
| − | n o + + o o o +
| |
| | | | |
| − | B C D E I J O
| + | The interpretation of Peirce's 1870 “Logic of Relatives” can be facilitated by introducing a few items of background material on relations in general, as regarded from a combinatorial point of view. |
| − | w + o + + o o o
| |
| | | | |
| − | Diagonal extensions of the absolute terms:
| + | ===Commentary Note 11.4=== |
| | | | |
| − | 1, = "anybody that is ---" = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
| + | The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations that are given by equivalence relations, functions, and so on. |
| | | | |
| − | m, = "man that is ---" = C:C +, I:I +, J:J +, O:O
| + | The first obstacle to get past is the order convention that Peirce's orientation to relative terms causes him to use for functions. To focus on a concrete example of immediate use in this discussion, let's take the “number of” function that Peirce denotes by means of square brackets and re-formulate it as a dyadic relative term <math>v\!</math> as follows: |
| | | | |
| − | n, = "noble that is ---" = C:C +, D:D +, O:O
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.\!</math> |
| | + | |} |
| | | | |
| − | w, = "woman that is ---" = B:B +, D:D +, E:E
| + | To set the dyadic relative term <math>v\!</math> within a suitable context of interpretation, let us suppose that <math>v\!</math> corresponds to a relation <math>V \subseteq \mathbb{R} \times S,\!</math> where <math>\mathbb{R}\!</math> is the set of real numbers and <math>S\!</math> is a suitable syntactic domain, here described as a set of ''terms''. The dyadic relation <math>V\!</math> is at first sight a function from <math>S\!</math> to <math>\mathbb{R}.\!</math> There is, however, a very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain <math>S\!</math> we happen to choose, so we may eventually be forced to treat the dyadic relation <math>V\!</math> as a partial function from <math>S\!</math> to <math>\mathbb{R}.\!</math> All things considered, then, let me try out the following impedimentaria of strategies and compromises. |
| | | | |
| − | Naturally enough, the diagonal extensions are represented by diagonal matrices:
| + | First, I adapt the functional arrow notation so that it allows us to detach the functional orientation from the order in which the names of domains are written on the page. Second, I change the notation for ''partial functions'', or ''pre-functions'', to one that is less likely to be confounded. This gives the scheme: |
| | | | |
| − | !1!| B C D E I J O | + | {| align="center" cellspacing="6" width="90%" |
| − | ---o---------------
| + | | <math>q : X \to Y\!</math> means that <math>q\!</math> is functional at <math>X.\!</math> |
| − | B | 1 0 0 0 0 0 0
| + | |- |
| − | C | 0 1 0 0 0 0 0
| + | | <math>q : X \leftarrow Y\!</math> means that <math>q\!</math> is functional at <math>Y.\!</math> |
| − | D | 0 0 1 0 0 0 0
| + | |- |
| − | E | 0 0 0 1 0 0 0
| + | | <math>q : X \rightharpoonup Y\!</math> means that <math>q\!</math> is pre-functional at <math>X.\!</math> |
| − | I | 0 0 0 0 1 0 0
| + | |- |
| − | J | 0 0 0 0 0 1 0
| + | | <math>q : X \leftharpoonup Y\!</math> means that <math>q\!</math> is pre-functional at <math>Y.\!</math> |
| − | O | 0 0 0 0 0 0 1
| + | |} |
| | | | |
| − | !m!| B C D E I J O | + | Until it becomes necessary to stipulate otherwise, let us assume that <math>v\!</math> is a function in <math>\mathbb{R}\!</math> of <math>S,\!</math> written <math>v : \mathbb{R} \leftarrow S,\!</math> amounting to the functional alias of the dyadic relation <math>V \subseteq \mathbb{R} \times S\!</math> and associated with the dyadic relative term <math>v\!</math> whose relate lies in the set <math>\mathbb{R}\!</math> of real numbers and whose correlate lies in the set <math>S\!</math> of syntactic terms. |
| − | ---o---------------
| |
| − | B | 0 0 0 0 0 0 0
| |
| − | C | 0 1 0 0 0 0 0
| |
| − | D | 0 0 0 0 0 0 0
| |
| − | E | 0 0 0 0 0 0 0
| |
| − | I | 0 0 0 0 1 0 0
| |
| − | J | 0 0 0 0 0 1 0
| |
| − | O | 0 0 0 0 0 0 1
| |
| | | | |
| − | !n!| B C D E I J O
| + | '''Note.''' See the article [[Relation Theory]] for the definitions of ''functions'' and ''pre-functions'' used in this section. |
| − | ---o---------------
| |
| − | B | 0 0 0 0 0 0 0
| |
| − | C | 0 1 0 0 0 0 0
| |
| − | D | 0 0 1 0 0 0 0
| |
| − | E | 0 0 0 0 0 0 0
| |
| − | I | 0 0 0 0 0 0 0
| |
| − | J | 0 0 0 0 0 0 0
| |
| − | O | 0 0 0 0 0 0 1
| |
| | | | |
| − | !w!| B C D E I J O
| + | ===Commentary Note 11.5=== |
| − | ---o---------------
| |
| − | B | 1 0 0 0 0 0 0
| |
| − | C | 0 0 0 0 0 0 0
| |
| − | D | 0 0 1 0 0 0 0
| |
| − | E | 0 0 0 1 0 0 0
| |
| − | I | 0 0 0 0 0 0 0
| |
| − | J | 0 0 0 0 0 0 0
| |
| − | O | 0 0 0 0 0 0 0
| |
| | | | |
| − | Cast into the bigraph picture of 2-adic relations,
| + | The right form of diagram can be a great aid in rendering complex matters comprehensible, so let's extract the overly compressed bits of the “[[Relation Theory]]” article that we need to illuminate Peirce's 1870 “Logic Of Relatives” and draw what icons we can within the current frame. |
| − | the diagonal extension of an absolute term takes on | |
| − | a very distinctive sort of "straight-laced" character:
| |
| | | | |
| − | B C D E I J O
| + | For the immediate present, we may start with dyadic relations and describe the customary species of relations and functions in terms of their local and numerical incidence properties. |
| − | u o o o o o o o
| |
| − | | | | | | | |
| |
| − | 1, | | | | | | |
| |
| − | | | | | | | |
| |
| − | u o o o o o o o
| |
| − | B C D E I J O
| |
| | | | |
| − | B C D E I J O
| + | Let <math>P \subseteq X \times Y\!</math> be an arbitrary dyadic relation. The following properties of <math>P\!</math> can be defined: |
| − | u o o o o o o o
| |
| − | | | | |
| |
| − | m, | | | |
| |
| − | | | | |
| |
| − | u o o o o o o o
| |
| − | B C D E I J O
| |
| | | | |
| − | B C D E I J O
| + | {| align="center" cellspacing="6" width="90%" |
| − | u o o o o o o o
| + | | |
| − | | | |
| + | <math>\begin{array}{lll} |
| − | n, | | |
| + | P ~\text{is total at}~ X |
| − | | | |
| + | & \iff & |
| − | u o o o o o o o
| + | P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X. |
| − | B C D E I J O
| + | \\[6pt] |
| | + | P ~\text{is total at}~ Y |
| | + | & \iff & |
| | + | P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y. |
| | + | \\[6pt] |
| | + | P ~\text{is tubular at}~ X |
| | + | & \iff & |
| | + | P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X. |
| | + | \\[6pt] |
| | + | P ~\text{is tubular at}~ Y |
| | + | & \iff & |
| | + | P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y. |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | B C D E I J O
| + | If <math>P \subseteq X \times Y\!</math> is tubular at <math>X,\!</math> then <math>P\!</math> is known as a ''partial function'' or a ''pre-function'' from <math>X\!</math> to <math>Y,\!</math> frequently signalized by renaming <math>P\!</math> with an alternate lower case name, say <math>{}^{\backprime\backprime} p {}^{\prime\prime},~\!</math> and writing <math>p : X \rightharpoonup Y.\!</math> |
| − | u o o o o o o o
| |
| − | | | |
| |
| − | w, | | |
| |
| − | | | |
| |
| − | u o o o o o o o
| |
| − | B C D E I J O
| |
| | | | |
| | + | Just by way of formalizing the definition: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | P ~\text{is a pre-function}~ P : X \rightharpoonup Y |
| | + | & \iff & |
| | + | P ~\text{is tubular at}~ X. |
| | + | \\[6pt] |
| | + | P ~\text{is a pre-function}~ P : X \leftharpoonup Y |
| | + | & \iff & |
| | + | P ~\text{is tubular at}~ Y. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 9.6
| + | To illustrate these properties, let us fashion a generic enough example of a dyadic relation, <math>E \subseteq X \times Y,~\!</math> where <math>X = Y = \{ 0, 1, \ldots, 8, 9 \},\!</math> and where the bigraph picture of <math>E\!</math> looks like this: |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 30.jpg]] || (30) |
| | + | |} |
| | | | |
| | + | If we scan along the <math>X\!</math> dimension from <math>0\!</math> to <math>9\!</math> we see that the incidence degrees of the <math>X\!</math> nodes with the <math>Y\!</math> domain are <math>0, 1, 2, 3, 1, 1, 1, 2, 0, 0,\!</math> in that order. |
| | | | |
| − | Just to be doggedly persistent about it all, here is what
| + | If we scan along the <math>Y\!</math> dimension from <math>0\!</math> to <math>9\!</math> we see that the incidence degrees of the <math>Y\!</math> nodes with the <math>X\!</math> domain are <math>0, 0, 3, 2, 1, 1, 2, 1, 1, 0,\!</math> in that order. |
| − | ought to be a sufficient sample of products involving the
| |
| − | multiplication of a comma relative onto an absolute term,
| |
| − | presented in both graphical and matrical representations.
| |
| | | | |
| − | Example 1. Anything That Is Anything
| + | Thus, <math>E\!</math> is not total at either <math>X\!</math> or <math>Y,\!</math> since there are nodes in both <math>X\!</math> and <math>Y\!</math> having incidence degrees less than <math>1.\!</math> |
| | | | |
| − | 1,1 = 1
| + | Also, <math>E\!</math> is not tubular at either <math>X\!</math> or <math>Y,\!</math> since there are nodes in both <math>X\!</math> and <math>Y\!</math> having incidence degrees greater than <math>1.\!</math> |
| | | | |
| − | "anything that is anything" = "anything"
| + | Clearly, then, the relation <math>E\!</math> cannot qualify as a pre-function, much less as a function on either of its relational domains. |
| | | | |
| − | B C D E I J O
| + | ===Commentary Note 11.6=== |
| − | + + + + + + + 1
| |
| − | | | | | | | |
| |
| − | | | | | | | | 1,
| |
| − | | | | | | | |
| |
| − | o o o o o o o =
| |
| | | | |
| − | + + + + + + + 1
| + | Let's continue working our way through the above definitions, constructing appropriate examples as we go. |
| − | B C D E I J O
| |
| | | | |
| − | | 1 0 0 0 0 0 0 | | 1 | | 1 |
| + | <math>E_1\!</math> exemplifies the quality of ''totality at <math>X.\!</math>'' |
| − | | 0 1 0 0 0 0 0 | | 1 | | 1 |
| |
| − | | 0 0 1 0 0 0 0 | | 1 | | 1 |
| |
| − | | 0 0 0 1 0 0 0 | | 1 | = | 1 |
| |
| − | | 0 0 0 0 1 0 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 1 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| | | | |
| − | Example 2. Anything That Is Man
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 31.jpg]] || (31) |
| | + | |} |
| | | | |
| − | 1,m = m
| + | <math>E_2\!</math> exemplifies the quality of ''totality at <math>Y.\!</math>'' |
| | | | |
| − | "anything that is man" = "man" | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 32.jpg]] || (32) |
| | + | |} |
| | | | |
| − | B C D E I J O
| + | <math>E_3\!</math> exemplifies the quality of ''tubularity at <math>X.\!</math>'' |
| − | o + o o + + + m
| |
| − | | | | | | | |
| |
| − | | | | | | | | 1,
| |
| − | | | | | | | |
| |
| − | o o o o o o o =
| |
| | | | |
| − | o + o o + + + m
| + | {| align="center" cellpadding="10" |
| − | B C D E I J O
| + | | [[Image:LOR 1870 Figure 33.jpg]] || (33) |
| | + | |} |
| | | | |
| − | | 1 0 0 0 0 0 0 | | 0 | | 0 |
| + | <math>E_4\!</math> exemplifies the quality of ''tubularity at <math>Y.\!</math>'' |
| − | | 0 1 0 0 0 0 0 | | 1 | | 1 |
| |
| − | | 0 0 1 0 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 0 1 0 0 0 | | 0 | = | 0 |
| |
| − | | 0 0 0 0 1 0 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 1 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| | | | |
| − | Example 3. Man That Is Anything
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 34.jpg]] || (34) |
| | + | |} |
| | | | |
| − | m,1 = m
| + | So <math>E_3\!</math> is a pre-function <math>e_3 : X \rightharpoonup Y,\!</math> and <math>E_4\!</math> is a pre-function <math>e_4 : X \leftharpoonup Y.\!</math> |
| | | | |
| − | "man that is anything" = "man"
| + | ===Commentary Note 11.7=== |
| | | | |
| − | B C D E I J O
| + | We come now to the very special cases of dyadic relations that are known as ''functions''. It will serve a dual purpose on behalf of the present exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the [[Relation Theory]] material. |
| − | + + + + + + + 1
| |
| − | | | | |
| |
| − | | | | | m,
| |
| − | | | | |
| |
| − | o o o o o o o =
| |
| | | | |
| − | o + o o + + + m
| + | To begin, let's recall the definition of a ''local flag'': |
| − | B C D E I J O
| |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 | | + | {| align="center" cellspacing="6" width="90%" |
| − | | 0 1 0 0 0 0 0 | | 1 | | 1 | | + | | <math>L_{x \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.\!</math> |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| + | |} |
| − | | 0 0 0 0 0 0 0 | | 1 | = | 0 |
| |
| − | | 0 0 0 0 1 0 0 | | 1 | | 1 |
| |
| − | | 0 0 0 0 0 1 0 | | 1 | | 1 | | |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| | | | |
| − | Example 4. Man That Is Noble
| + | In the case of a dyadic relation <math>L \subseteq X_1 \times X_2 = X \times Y,\!</math> it is possible to simplify the notation for local flags in a couple of ways. First, it is often easier in the dyadic case to refer to <math>L_{u \,\text{at}\, 1}\!</math> as <math>L_{u \,\text{at}\, X}\!</math> and <math>L_{v \,\text{at}\, 2}\!</math> as <math>L_{v \,\text{at}\, Y}.\!</math> Second, the notation may be streamlined even further by writing <math>L_{u \,\text{at}\, 1}\!</math> as <math>u \star L\!</math> and <math>L_{v \,\text{at}\, 2}\!</math> as <math>L \star v.\!</math> |
| | | | |
| − | m,n = "man that is noble"
| + | In light of these considerations, the local flags of a dyadic relation <math>L \subseteq X \times Y\!</math> may be formulated as follows: |
| | | | |
| − | B C D E I J O
| + | {| align="center" cellspacing="6" width="90%" |
| − | o + + o o o + n
| + | | |
| − | | | | |
| + | <math>\begin{array}{lll} |
| − | | | | | m,
| + | u \star L |
| − | | | | |
| + | & = & |
| − | o o o o o o o =
| + | L_{u \,\text{at}\, X} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \{ (u, y) \in L \} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{the ordered pairs in}~ L ~\text{that are incident with}~ u \in X. |
| | + | \\[9pt] |
| | + | L \star v |
| | + | & = & |
| | + | L_{v \,\text{at}\, Y} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \{ (x, v) \in L \} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{the ordered pairs in}~ L ~\text{that are incident with}~ v \in Y. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | o + o o o o + m,n
| + | The following definitions are also useful: |
| − | B C D E I J O
| |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 | | + | {| align="center" cellspacing="6" width="90%" |
| − | | 0 1 0 0 0 0 0 | | 1 | | 1 |
| + | | |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| + | <math>\begin{array}{lll} |
| − | | 0 0 0 0 0 0 0 | | 0 | = | 0 |
| + | u \cdot L |
| − | | 0 0 0 0 1 0 0 | | 0 | | 0 |
| + | & = & |
| − | | 0 0 0 0 0 1 0 | | 0 | | 0 |
| + | \mathrm{proj}_2 (u \star L) |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 | | + | \\[6pt] |
| | + | & = & |
| | + | \{ y \in Y : (u, y) \in L \} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{the elements of}~ Y ~\text{that are}~ L\text{-related to}~ u. |
| | + | \\[9pt] |
| | + | L \cdot v |
| | + | & = & |
| | + | \mathrm{proj}_1 (L \star v) |
| | + | \\[6pt] |
| | + | & = & |
| | + | \{ x \in X : (x, v) \in L \} |
| | + | \\[6pt] |
| | + | & = & |
| | + | \text{the elements of}~ X ~\text{that are}~ L\text{-related to}~ v. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | Example 5. Noble That Is Man
| + | A sufficient illustration is supplied by the earlier example <math>E.\!</math> |
| | | | |
| − | n,m = "noble that is man"
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 30.jpg]] || (35) |
| | + | |} |
| | | | |
| − | B C D E I J O
| + | The local flag <math>E_{3 \,\text{at}\, X}\!</math> is displayed here: |
| − | o + o o + + + m
| |
| − | | | |
| |
| − | | | | n,
| |
| − | | | |
| |
| − | o o o o o o o =
| |
| | | | |
| − | o + o o o o + n,m
| + | {| align="center" cellpadding="10" |
| − | B C D E I J O
| + | | [[Image:LOR 1870 Figure 36 ISW.jpg]] || (36) |
| | + | |} |
| | | | |
| − | | 0 0 0 0 0 0 0 | | 0 | | 0 |
| + | The local flag <math>E_{2 \,\text{at}\, Y}\!</math> is displayed here: |
| − | | 0 1 0 0 0 0 0 | | 1 | | 1 |
| |
| − | | 0 0 1 0 0 0 0 | | 0 | | 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 0 | = | 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 0 0 0 0 0 | | 1 | | 0 |
| |
| − | | 0 0 0 0 0 0 1 | | 1 | | 1 |
| |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 37 ISW.jpg]] || (37) |
| | + | |} |
| | | | |
| | + | ===Commentary Note 11.8=== |
| | | | |
| − | LOR. Commentary Note 9.7
| + | Next let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions. |
| | | | |
| | + | For example, <math>L\!</math> is said to be <math>{}^{\backprime\backprime} c\text{-regular at}~ j \, {}^{\prime\prime}\!</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}\!</math> is equal to <math>c\!</math> for all <math>x \in X_j,\!</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c\!</math> for all <math>{x \in X_j}.\!</math> |
| | | | |
| | + | In a similar fashion, it is possible to define the numerical incidence properties <math>{}^{\backprime\backprime}(< c)\text{-regular at}~ j \, {}^{\prime\prime},\!</math> <math>{}^{\backprime\backprime}(> c)\text{-regular at}~ j \, {}^{\prime\prime},\!</math> and so on. For ease of reference, a few of these definitions are recorded below. |
| | | | |
| − | From this point forward we may think of idempotents, selectives,
| + | {| align="center" cellspacing="6" width="90%" |
| − | and zero-one diagonal matrices as being roughly equivalent notions.
| + | | |
| − | The only reason that I say "roughly" is that we are comparing ideas
| + | <math>\begin{array}{lll} |
| − | at different levels of abstraction when we propose these connections. | + | L ~\text{is}~ c\text{-regular at}~ j |
| | + | & \iff & |
| | + | |L_{x \,\text{at}\, j}| = c ~\text{for all}~ x \in X_j. |
| | + | \\[6pt] |
| | + | L ~\text{is}~ (< c)\text{-regular at}~ j |
| | + | & \iff & |
| | + | |L_{x \,\text{at}\, j}| < c ~\text{for all}~ x \in X_j. |
| | + | \\[6pt] |
| | + | L ~\text{is}~ (> c)\text{-regular at}~ j |
| | + | & \iff & |
| | + | |L_{x \,\text{at}\, j}| > c ~\text{for all}~ x \in X_j. |
| | + | \\[6pt] |
| | + | L ~\text{is}~ (\le c)\text{-regular at}~ j |
| | + | & \iff & |
| | + | |L_{x \,\text{at}\, j}| \le c ~\text{for all}~ x \in X_j. |
| | + | \\[6pt] |
| | + | L ~\text{is}~ (\ge c)\text{-regular at}~ j |
| | + | & \iff & |
| | + | |L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | We have covered the way that Peirce uses his invention of the
| + | Clearly, if any relation is <math>(\le c)\text{-regular}\!</math> on one of its domains <math>X_j~\!</math> and also <math>(\ge c)\text{-regular}\!</math> on the same domain, then it must be <math>(= c)\text{-regular}\!</math> on that domain, in effect, <math>c\text{-regular}\!</math> at <math>j.\!</math> |
| − | comma modifier to assimilate boolean multiplication, logical
| |
| − | conjunction, or what we may think of as "serial selection"
| |
| − | under his more general account of relative multiplication.
| |
| | | | |
| − | But the comma functor has its application to relative terms
| + | For example, let <math>G = \{ r, s, t \}\!</math> and <math>H = \{ 1, \ldots, 9 \},\!</math> and consider the dyadic relation <math>F \subseteq G \times H\!</math> that is bigraphed here: |
| − | of any arity, not just the zeroth arity of absolute terms,
| |
| − | and so there will be a lot more to explore on this point. | |
| − | But now I must return to the anchorage of Peirce's text,
| |
| − | and hopefully get a chance to revisit this topic later. | |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 38.jpg]] || (38) |
| | + | |} |
| | | | |
| | + | We observe that <math>F\!</math> is 3-regular at <math>G\!</math> and 1-regular at <math>H.\!</math> |
| | | | |
| − | LOR. Note 10
| + | ===Commentary Note 11.9=== |
| | | | |
| | + | Among the variety of conceivable regularities affecting dyadic relations we pay special attention to the <math>c\!</math>-regularity conditions where <math>c\!</math> is equal to 1. |
| | | | |
| | + | Let <math>P \subseteq X \times Y\!</math> be an arbitrary dyadic relation. The following properties of <math>P\!</math> can be defined: |
| | | | |
| − | | The Signs for Multiplication (cont.) | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | The sum 'x' + 'x' generally denotes no logical term. | + | <math>\begin{array}{lll} |
| − | | But 'x',_oo + 'x',_oo may be considered as denoting
| + | P ~\text{is total at}~ X |
| − | | some two 'x's. | + | & \iff & |
| | + | P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X. |
| | + | \\[6pt] |
| | + | P ~\text{is total at}~ Y |
| | + | & \iff & |
| | + | P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y. |
| | + | \\[6pt] |
| | + | P ~\text{is tubular at}~ X |
| | + | & \iff & |
| | + | P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X. |
| | + | \\[6pt] |
| | + | P ~\text{is tubular at}~ Y |
| | + | & \iff & |
| | + | P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y. |
| | + | \end{array}\!</math> |
| | + | |} |
| | + | |
| | + | We have already looked at dyadic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and special-purpose notations for working with tubular relations: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | It is natural to write: | + | <math>\begin{array}{lll} |
| | + | P ~\text{is a pre-function}~ P : X \rightharpoonup Y |
| | + | & \iff & |
| | + | P ~\text{is tubular at}~ X. |
| | + | \\[6pt] |
| | + | P ~\text{is a pre-function}~ P : X \leftharpoonup Y |
| | + | & \iff & |
| | + | P ~\text{is tubular at}~ Y. |
| | + | \end{array}\!</math> |
| | + | |} |
| | + | |
| | + | We arrive by way of this winding stair at the special stamps of dyadic relations <math>P \subseteq X \times Y\!</math> that are variously described as ''1-regular'', ''total and tubular'', or ''total prefunctions'' on specified domains, either <math>X\!</math> or <math>Y\!</math> or both, and that are more often celebrated as ''functions'' on those domains. |
| | + | |
| | + | If <math>P\!</math> is a pre-function <math>P : X \rightharpoonup Y\!</math> that happens to be total at <math>X,\!</math> then <math>P\!</math> is known as a ''function'' from <math>X\!</math> to <math>Y,\!</math> typically indicated as <math>{P : X \to Y}.\!</math> |
| | + | |
| | + | To say that a relation <math>P \subseteq X \times Y\!</math> is ''totally tubular'' at <math>X\!</math> is to say that <math>P\!</math> is 1-regular at <math>X.\!</math> Thus, we may formalize the following definitions: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | 'x' + 'x' = !2!.'x'
| + | <math>\begin{array}{lll} |
| − | |
| + | P ~\text{is a function}~ P : X \to Y |
| − | | and
| + | & \iff & |
| − | |
| + | P ~\text{is}~ 1\text{-regular at}~ X. |
| − | | 'x',_oo + 'x',_oo = !2!.'x',_oo
| + | \\[6pt] |
| − | |
| + | P ~\text{is a function}~ P : X \leftarrow Y |
| − | | where the dot shows that this multiplication is invertible.
| + | & \iff & |
| − | |
| + | P ~\text{is}~ 1\text{-regular at}~ Y. |
| − | | We may also use the antique figures so that:
| + | \end{array}\!</math> |
| − | |
| + | |} |
| − | | !2!.'x',_oo = `2`'x'
| + | |
| − | |
| + | For example, let <math>X = Y = \{ 0, \ldots, 9 \}\!</math> and let <math>F \subseteq X \times Y\!</math> be the dyadic relation depicted in the bigraph below: |
| − | | just as
| |
| − | |
| |
| − | | !1!_oo = `1`.
| |
| − | |
| |
| − | | Then `2` alone will denote some two things.
| |
| − | |
| |
| − | | But this multiplication is not in general commutative,
| |
| − | | and only becomes so when it affects a relative which
| |
| − | | imparts a relation such that a thing only bears it
| |
| − | | to 'one' thing, and one thing 'alone' bears it to
| |
| − | | a thing.
| |
| − | |
| |
| − | | For instance, the lovers of two women are not
| |
| − | | the same as two lovers of women, that is:
| |
| − | |
| |
| − | | 'l'`2`.w
| |
| − | |
| |
| − | | and | |
| − | |
| |
| − | | `2`.'l'w
| |
| − | |
| |
| − | | are unequal;
| |
| − | |
| |
| − | | but the husbands of two women are the
| |
| − | | same as two husbands of women, that is:
| |
| − | |
| |
| − | | 'h'`2`.w = `2`.'h'w
| |
| − | |
| |
| − | | and in general:
| |
| − | |
| |
| − | | 'x',`2`.'y' = `2`.'x','y'.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.75
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 39.jpg]] || (39) |
| | + | |} |
| | | | |
| | + | We observe that <math>F\!</math> is a function at <math>Y\!</math> and we record this fact in either of the manners <math>F : X \leftarrow Y\!</math> or <math>F : Y \to X.\!</math> |
| | | | |
| − | LOR. Commentary Note 10.1
| + | ===Commentary Note 11.10=== |
| | | | |
| | + | In the case of a dyadic relation <math>F \subseteq X \times Y\!</math> that has the qualifications of a function <math>f : X \to Y,\!</math> there are a number of further differentia that arise: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | f ~\text{is surjective} & \iff & f ~\text{is total at}~ Y. |
| | + | \\[6pt] |
| | + | f ~\text{is injective} & \iff & f ~\text{is tubular at}~ Y. |
| | + | \\[6pt] |
| | + | f ~\text{is bijective} & \iff & f ~\text{is}~ 1\text{-regular at}~ Y. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | What Peirce is attempting to do in CP 3.75 is absolutely amazing,
| + | For example, the function <math>f : X \to Y\!</math> depicted below is neither total at <math>Y\!</math> nor tubular at <math>Y,\!</math> and so it cannot enjoy any of the properties of being surjective, injective, or bijective. |
| − | and I personally did not see anything on par with it again until | |
| − | I began to study the application of mathematical category theory
| |
| − | to computation and logic, back in the mid 1980's. To completely
| |
| − | evaluate the success of this attempt, we would have to return to
| |
| − | Peirce's earlier paper "Upon the Logic of Mathematics" (1867) to
| |
| − | pick up some of the ideas about arithmetic that he set out there.
| |
| | | | |
| − | Another branch of the investigation would require that we examine
| + | {| align="center" cellpadding="10" |
| − | more careully the entire syntactic mechanics of "subjacent signs"
| + | | [[Image:LOR 1870 Figure 40.jpg]] || (40) |
| − | that Peirce uses to establish linkages among relational domains.
| + | |} |
| − | It is important to note that these types of indices constitute
| |
| − | a diacritical, interpretive, syntactic category under which
| |
| − | Peirce also places the comma functor.
| |
| | | | |
| − | The way that I would currently approach both of these branches
| + | An easy way to extract a surjective function from any function is to reset its codomain to its range. For example, the range of the function <math>f\!</math> above is <math>Y^\prime = \{ 0, 2, 5, 6, 7, 8, 9 \}.\!</math> Thus, if we form a new function <math>g : X \to Y^\prime\!</math> that looks just like <math>f\!</math> on the domain <math>X\!</math> but is assigned the codomain <math>Y^\prime,\!</math> then <math>g\!</math> is surjective, and is described as mapping ''onto'' <math>Y^\prime.\!</math> |
| − | of the investigation would be to open up a wider context for
| |
| − | the study of relational compositions, attempting to get at | |
| − | the essence of what is going on we when relate relations, | |
| − | possibly complex, to other relations, possibly simple.
| |
| | | | |
| − | But that will take another cup of java ('c'j) ---
| + | {| align="center" cellpadding="10" |
| − | or maybe two, `2`'c'j = (!2!.'c',_oo)j ...
| + | | [[Image:LOR 1870 Figure 41.jpg]] || (41) |
| | + | |} |
| | | | |
| | + | The function <math>h : Y^\prime \to Y\!</math> is injective. |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 42.jpg]] || (42) |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 10.2
| + | The function <math>m : X \to Y\!</math> is bijective. |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 43.jpg]] || (43) |
| | + | |} |
| | | | |
| | + | ===Commentary Note 11.11=== |
| | | | |
| − | To say that a relative term "imparts a relation"
| + | The preceding exercises were intended to beef-up our “functional” literacy skills to the point where we can read our functional alphabets backwards and forwards and recognize the local functionalities that may be immanent in relative terms no matter where they locate themselves within the domains of relations. These skills will serve us in good stead as we work to build a catwalk from Peirce's platform of 1870 to contemporary scenes on the logic of relatives, and back again. |
| − | is to say that it conveys information about the
| |
| − | space of tuples in a cartesian product, that is,
| |
| − | it determines a particular subset of that space.
| |
| | | | |
| − | When we study the combinations of relative terms, from the most
| + | By way of extending a few very tentative planks, let us experiment with the following definitions: |
| − | elementary forms of composition to the most complex patterns of
| |
| − | correlation, we are considering the ways that these constraints,
| |
| − | determinations, and informations, as imparted by relative terms,
| |
| − | can be compounded in the formation of syntax.
| |
| | | | |
| − | Let us go back and look more carefully at just how it happens that
| + | {| align="center" cellspacing="6" width="90%" |
| − | Peirce's jacent terms and subjacent indices manage to impart their
| + | | |
| − | respective measures of information about relations.
| + | <p>A relative term <math>p\!</math> and the corresponding relation <math>P \subseteq X \times Y\!</math> are both called ''functional on relates'' if and only if <math>P\!</math> is a function at <math>X,\!</math> in symbols, <math>{P : X \to Y}.\!</math></p> |
| | + | |- |
| | + | | |
| | + | <p>A relative term <math>p\!</math> and the corresponding relation <math>P \subseteq X \times Y\!</math> are both called ''functional on correlates'' if and only if <math>P\!</math> is a function at <math>Y,\!</math> in symbols, <math>P : X \leftarrow Y.\!</math></p> |
| | + | |} |
| | | | |
| − | I will begin with the two examples illustrated in Figures 1 and 2,
| + | When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like <math>P : X \times Y,\!</math> <math>P : X \to Y,\!</math> <math>P : X \leftarrow Y,\!</math> as the case may be, when and if it serves to clarify matters. |
| − | where I have drawn in the corresponding lines of identity between
| |
| − | the subjacent marks of reference #, $, %. | |
| | | | |
| − | o-------------------------------------------------o
| + | From this current, perhaps transient, perspective, it appears that our next task is to examine how the known properties of relations are modified when an aspect of functionality is spied in the mix. Let us then return to our various ways of looking at relational composition, and see what changes and what stays the same when the relations in question happen to be functions of various different kinds at some of their domains. Here is one generic picture of relational composition, cast in a style that hews pretty close to the line of potentials inherent in Peirce's syntax of this period. |
| − | | |
| |
| − | | |
| |
| − | | 'l'__# #'s'__$ $w |
| |
| − | | o o o o |
| |
| − | | \ / \ / |
| |
| − | | \ / o |
| |
| − | | \ / $ |
| |
| − | | o |
| |
| − | | # |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 1. Lover of a Servant of a Woman
| |
| | | | |
| − | o-------------------------------------------------o
| + | <br> |
| − | | |
| |
| − | | |
| |
| − | | `g`__#__$ #'l'__% %w $h |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \/ o / |
| |
| − | | \ /\ % / |
| |
| − | | o ------o------ |
| |
| − | | # $ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 2. Giver of a Horse to a Lover of a Woman
| |
| | | | |
| − | One way to approach the problem of "information fusion"
| + | {| align="center" cellpadding="10" |
| − | in Peirce's syntax is to soften the distinction between
| + | | [[Image:LOR 1870 Figure 44.jpg]] || (44) |
| − | jacent terms and subjacent signs, and to treat the types
| + | |} |
| − | of constraints that they separately signify more on a par
| |
| − | with each other.
| |
| | | | |
| − | To that purpose, I will set forth a way of thinking about
| + | From this we extract the ''hypergraph picture'' of relational composition: |
| − | relational composition that emphasizes the set-theoretic | |
| − | constraints involved in the construction of a composite.
| |
| | | | |
| − | For example, suppose that we are given the relations L c X x Y, M c Y x Z.
| + | <br> |
| − | Table 3 and Figure 4 present a couple of ways of picturing the constraints
| |
| − | that are involved in constructing the relational composition L o M c X x Z.
| |
| | | | |
| − | Table 3. Relational Composition
| + | {| align="center" cellpadding="10" |
| − | o---------o---------o---------o---------o
| + | | [[Image:LOR 1870 Figure 45.jpg]] || (45) |
| − | | # !1! | !1! | !1! | | + | |} |
| − | o=========o=========o=========o=========o
| |
| − | | L # X | Y | | | |
| − | o---------o---------o---------o---------o
| |
| − | | M # | Y | Z | | |
| − | o---------o---------o---------o---------o
| |
| − | | L o M # X | | Z |
| |
| − | o---------o---------o---------o---------o
| |
| | | | |
| − | The way to read Table 3 is to imagine that you are
| + | All of the relevant information of these Figures can be compressed into the form of a spreadsheet, or constraint satisfaction table: |
| − | playing a game that involves placing tokens on the
| |
| − | squares of a board that is marked in just this way.
| |
| − | The rules are that you have to place a single token
| |
| − | on each marked square in the middle of the board in
| |
| − | such a way that all of the indicated constraints are
| |
| − | satisfied. That is to say, you have to place a token
| |
| − | whose denomination is a value in the set X on each of
| |
| − | the squares marked "X", and similarly for the squares
| |
| − | marked "Y" and "Z", meanwhile leaving all of the blank
| |
| − | squares empty. Furthermore, the tokens placed in each
| |
| − | row and column have to obey the relational constraints
| |
| − | that are indicated at the heads of the corresponding
| |
| − | row and column. Thus, the two tokens from X have to
| |
| − | denominate the very same value from X, and likewise
| |
| − | for Y and Z, while the pairs of tokens on the rows
| |
| − | marked "L" and "M" are required to denote elements
| |
| − | that are in the relations L and M, respectively.
| |
| − | The upshot is that when just this much is done,
| |
| − | that is, when the L, M, and !1! relations are
| |
| − | satisfied, then the row marked "L o M" will
| |
| − | automatically bear the tokens of a pair of
| |
| − | elements in the composite relation L o M.
| |
| | | | |
| − | o-------------------------------------------------o
| + | <br> |
| − | | |
| |
| − | | L L o M M |
| |
| − | | @ @ @ |
| |
| − | | / \ / \ / \ |
| |
| − | | o o o o o o |
| |
| − | | X Y X Z Y Z |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ / \ / |
| |
| − | | \ / \__ __/ \ / |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 4. Relational Composition
| |
| | | | |
| − | Figure 4 merely shows a different way of viewing the same situation.
| + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| | + | |+ style="height:30px" | <math>\text{Table 46.} ~~ \text{Relational Composition}~ P \circ Q\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>\mathit{1}\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>P\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>Y\!</math> |
| | + | | |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>Q\!</math> |
| | + | | |
| | + | | <math>Y\!</math> |
| | + | | <math>Z\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>P \circ Q</math> |
| | + | | <math>X\!</math> |
| | + | | |
| | + | | <math>Z\!</math> |
| | + | |} |
| | | | |
| | + | <br> |
| | | | |
| | + | So the following presents itself as a reasonable plan of study: Let's see how much easy mileage we can get in our exploration of functions by adopting the above templates as a paradigm. |
| | | | |
| − | LOR. Commentary Note 10.3
| + | ===Commentary Note 11.12=== |
| | | | |
| | + | Since functions are special cases of dyadic relations and since the space of dyadic relations is closed under relational composition — that is, the composition of two dyadic relations is again a dyadic relation — we know that the relational composition of two functions has to be a dyadic relation. If the relational composition of two functions is necessarily a function, too, then we would be justified in speaking of ''functional composition'' and also in saying that the space of functions is closed under this functional form of composition. |
| | | | |
| | + | Just for novelty's sake, let's try to prove this for relations that are functional on correlates. |
| | | | |
| − | I will devote some time to drawing out the relationships
| + | The task is this — We are given a pair of dyadic relations: |
| − | that exist among the different pictures of relations and
| |
| − | relative terms that were shown above, or as redrawn here:
| |
| | | | |
| − | o-------------------------------------------------o
| + | {| align="center" cellspacing="6" width="90%" |
| − | | |
| + | | <math>P \subseteq X \times Y \quad \text{and} \quad Q \subseteq Y \times Z\!</math> |
| − | | |
| + | |} |
| − | | 'l'__$ $'s'__% %w | | |
| − | | o o o o | | |
| − | | \ / \ / |
| |
| − | | \ / o |
| |
| − | | \ / % |
| |
| − | | o | | |
| − | | $ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 1. Lover of a Servant of a Woman
| |
| | | | |
| − | o-------------------------------------------------o
| + | <math>P\!</math> and <math>Q\!</math> are assumed to be functional on correlates, a premiss that we express as follows: |
| − | | |
| |
| − | | |
| |
| − | | `g`__$__% $'l'__* *w %h |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \/ o / |
| |
| − | | \ /\ * / |
| |
| − | | o ------o------ |
| |
| − | | $ % |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 2. Giver of a Horse to a Lover of a Woman
| |
| | | | |
| − | Table 3. Relational Composition
| + | {| align="center" cellspacing="6" width="90%" |
| − | o---------o---------o---------o---------o
| + | | <math>P : X \gets Y \quad \text{and} \quad Q : Y \gets Z\!</math> |
| − | | # !1! | !1! | !1! |
| + | |} |
| − | o=========o=========o=========o=========o
| |
| − | | L # X | Y | | | |
| − | o---------o---------o---------o---------o
| |
| − | | S # | Y | Z |
| |
| − | o---------o---------o---------o---------o
| |
| − | | L o S # X | | Z |
| |
| − | o---------o---------o---------o---------o
| |
| | | | |
| − | o-------------------------------------------------o
| + | We are charged with deciding whether the relational composition <math>P \circ Q \subseteq X \times Z\!</math> is also functional on correlates, in symbols, whether <math>{P \circ Q : X \gets Z}.\!</math> |
| − | | |
| |
| − | | L L o S S |
| |
| − | | @ @ @ |
| |
| − | | / \ / \ / \ |
| |
| − | | o o o o o o |
| |
| − | | X Y X Z Y Z |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ / \ / |
| |
| − | | \ / \__ __/ \ / |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 4. Relational Composition
| |
| | | | |
| − | Figures 1 and 2 exhibit examples of relative multiplication
| + | It always helps to begin by recalling the pertinent definitions. |
| − | in one of Peirce's styles of syntax, to which I subtended
| |
| − | lines of identity to mark the anaphora of the correlates.
| |
| − | These pictures are adapted to showing the anatomy of the
| |
| − | relative terms, while the forms of analysis illustrated
| |
| − | in Table 3 and Figure 4 are designed to highlight the
| |
| − | structures of the objective relations themselves.
| |
| | | | |
| − | There are many ways that Peirce might have gotten from his 1870 Notation
| + | For a dyadic relation <math>L \subseteq X \times Y,\!</math> we have: |
| − | for the Logic of Relatives to his more evolved systems of Logical Graphs.
| |
| − | For my part, I find it interesting to speculate on how the metamorphosis | |
| − | might have been accomplished by way of transformations that act on these
| |
| − | nascent forms of syntax and that take place not too far from the pale of
| |
| − | its means, that is, as nearly as possible according to the rules and the
| |
| − | permissions of the initial system itself.
| |
| | | | |
| − | In Existential Graphs, a relation is represented by a node
| + | {| align="center" cellspacing="6" width="90%" |
| − | whose degree is the adicity of that relation, and which is
| + | | |
| − | adjacent via lines of identity to the nodes that represent
| + | <math>\begin{array}{lll} |
| − | its correlative relations, including as a special case any
| + | L ~\text{is a function}~ L : X \gets Y |
| − | of its terminal individual arguments.
| + | & \iff & |
| | + | L ~\text{is}~ 1\text{-regular at}~ Y. |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | In the 1870 Logic of Relatives, implicit lines of identity are invoked by
| + | As for the definition of relational composition, it is enough to consider the coefficient of the composite relation on an arbitrary ordered pair, <math>i\!:\!j.</math> For that, we have the following formula, where the summation indicated is logical disjunction: |
| − | the subjacent numbers and marks of reference only when a correlate of some | |
| − | relation is the relate of some relation. Thus, the principal relate, which
| |
| − | is not a correlate of any explicit relation, is not singled out in this way.
| |
| | | | |
| − | Remarkably enough, the comma modifier itself provides us with a mechanism
| + | {| align="center" cellspacing="6" width="90%" |
| − | to abstract the logic of relations from the logic of relatives, and thus
| + | | <math>(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}\!</math> |
| − | to forge a possible link between the syntax of relative terms and the
| + | |} |
| − | more graphical depiction of the objective relations themselves.
| |
| | | | |
| − | Figure 5 demonstrates this possibility, posing a transitional case between
| + | So let's begin. |
| − | the style of syntax in Figure 1 and the picture of composition in Figure 4.
| |
| | | | |
| − | o-----------------------------------------------------------o
| + | {| align="center" cellspacing="6" width="90%" |
| − | | |
| + | | |
| − | | L o S |
| + | <p><math>P : X \gets Y,\!</math> or the fact that <math>P ~\text{is}~ 1\text{-regular at}~ Y,\!</math> means that there is exactly one ordered pair <math>i\!:\!k \in P</math> for each <math>k \in Y.\!</math></p> |
| − | | ____________@____________ |
| + | |- |
| − | | / \ |
| + | | |
| − | | / L S \ |
| + | <p><math>Q : Y \gets Z,\!</math> or the fact that <math>Q ~\text{is}~ 1\text{-regular at}~ Z,\!</math> means that there is exactly one ordered pair <math>k\!:\!j \in Q</math> for each <math>j \in Z.\!</math></p> |
| − | | / @ @ \ |
| + | |- |
| − | | / / \ / \ \ |
| + | | |
| − | | / / \ / \ \ |
| + | <p>As a result, there is exactly one ordered pair <math>i\!:\!j \in P \circ Q</math> for each <math>j \in Z,\!</math> which means that <math>P \circ Q ~\text{is}~ 1\text{-regular at}~ Z,\!</math> and so we have the function <math>{P \circ Q : X \gets Z}.\!</math></p> |
| − | | o o o o o o |
| + | |} |
| − | | X X Y Y Z Z |
| |
| − | | 1,__# #'l'__$ $'s'__% %1 |
| |
| − | | o o o o o o |
| |
| − | | \ / \ / \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | | | |
| − | o-----------------------------------------------------------o
| |
| − | Figure 5. Anything that is a Lover of a Servant of Anything
| |
| | | | |
| − | In this composite sketch, the diagonal extension of the universe 1
| + | And we are done. |
| − | is invoked up front to anchor an explicit line of identity for the
| |
| − | leading relate of the composition, while the terminal argument "w"
| |
| − | has been generalized to the whole universe 1, in effect, executing
| |
| − | an act of abstraction. This type of universal bracketing isolates
| |
| − | the composing of the relations L and S to form the composite L o S.
| |
| − | The three relational domains X, Y, Z may be distinguished from one
| |
| − | another, or else rolled up into a single universe of discourse, as
| |
| − | one prefers.
| |
| | | | |
| | + | ===Commentary Note 11.13=== |
| | | | |
| | + | As we make our way toward the foothills of Peirce's 1870 Logic of Relatives, there are several pieces of equipment that we must not leave the plains without, namely, the utilities variously known as ''arrows'', ''morphisms'', ''homomorphisms'', ''structure-preserving maps'', among other names, depending on the altitude of abstraction we happen to be traversing at the moment in question. As a moderate to middling but not too beaten track, let's examine a few ways of defining morphisms that will serve us in the present discussion. |
| | | | |
| − | LOR. Commentary Note 10.4
| + | Suppose we are given three functions <math>J, K, L~\!</math> that satisfy the following conditions: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lcccl} |
| | + | J & : & X & \gets & Y |
| | + | \\[6pt] |
| | + | K & : & X & \gets & X \times X |
| | + | \\[6pt] |
| | + | L & : & Y & \gets & Y \times Y |
| | + | \end{array}</math> |
| | + | |- |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | J(L(u, v)) & = & K(Ju, Jv) |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| | + | Our sagittarian leitmotif can be rubricized in the following slogan: |
| | | | |
| − | From now on I will use the forms of analysis exemplified in the last set of
| + | {| align="center" cellspacing="12" width="90%" |
| − | Figures and Tables as a routine bridge between the logic of relative terms
| + | | <math>\textit{The~image~of~the~ligature~is~the~compound~of~the~images.}</math> |
| − | and the logic of their extended relations. For future reference, we may
| + | |- |
| − | think of Table 3 as illustrating the "solitaire" or "spreadsheet" model
| + | | (Where <math>J\!</math> is the ''image'', <math>K\!</math> is the ''compound'', and <math>L\!</math> is the ''ligature''.) |
| − | of relational composition, while Figure 4 may be thought of as making | + | |} |
| − | a start toward the "hyper(di)graph" model of generalized compositions.
| |
| − | I will explain the hypergraph model in some detail at a later point.
| |
| − | The transitional form of analysis represented by Figure 5 may be
| |
| − | called the "universal bracketing" of relatives as relations.
| |
| | | | |
| | + | Figure 47 presents us with a picture of the situation in question. |
| | | | |
| | + | <br> |
| | | | |
| − | LOR. Commentary Note 10.5 | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 47.jpg]] || (47) |
| | + | |} |
| | | | |
| | + | Table 48 gives the constraint matrix version of the same thing. |
| | | | |
| | + | <br> |
| | | | |
| − | We have sufficiently covered the application of the comma functor,
| + | {| align="center" cellpadding="10" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| − | or the diagonal extension, to absolute terms, so let us return to
| + | |+ style="height:30px" | <math>\text{Table 48.} ~~ \text{Arrow Equation:} ~~ J(L(u, v)) = K(Ju, Jv)\!</math> |
| − | where we were in working our way through CP 3.73, and see whether
| + | |- |
| − | we can validate Peirce's statements about the "commifications" of
| + | | style="border-right:1px solid black; border-bottom:1px solid black; width:25%" | |
| − | 2-adic relative terms that yield their 3-adic diagonal extensions.
| + | | style="border-bottom:1px solid black; width:25%" | <math>J\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>J\!</math> |
| | + | | style="border-bottom:1px solid black; width:25%" | <math>J\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>K\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>X\!</math> |
| | + | | <math>X\!</math> |
| | + | |- |
| | + | | style="border-right:1px solid black" | <math>L\!</math> |
| | + | | <math>Y\!</math> |
| | + | | <math>Y\!</math> |
| | + | | <math>Y\!</math> |
| | + | |} |
| | | | |
| − | | But not only may any absolute term be thus regarded as
| + | <br> |
| − | | a relative term, but any relative term may in the same
| |
| − | | way be regarded as a relative with one correlate more.
| |
| − | | It is convenient to take this additional correlate as
| |
| − | | the first one.
| |
| − | |
| |
| − | | Then:
| |
| − | |
| |
| − | | 'l','s'w
| |
| − | |
| |
| − | | will denote a lover of a woman
| |
| − | | that is a servant of that woman.
| |
| − | |
| |
| − | | The comma here after 'l' should not be considered
| |
| − | | as altering at all the meaning of 'l', but as only
| |
| − | | a subjacent sign, serving to alter the arrangement
| |
| − | | of the correlates.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.73
| |
| | | | |
| − | Just to plant our feet on a more solid stage,
| + | One way to read this Table is in terms of the informational redundancies that it schematizes. In particular, it can be read to say that when one satisfies the constraint in the <math>L\!</math> row, along with all the constraints in the <math>J\!</math> columns, then the constraint in the <math>K\!</math> row is automatically true. That is one way of understanding the equation: <math>J(L(u, v)) ~=~ K(Ju, Jv).</math> |
| − | let's apply this idea to the Othello example.
| |
| | | | |
| − | For this performance only, just to make the example more interesting,
| + | ===Commentary Note 11.14=== |
| − | let us assume that Jeste (J) is secretly in love with Desdemona (D).
| |
| | | | |
| − | Then we begin with the modified data set:
| + | Now, as promised, let's look at a more homely example of a morphism, say, any one of the mappings <math>J : \mathbb{R} \to \mathbb{R}\!</math> (roughly speaking) that are commonly known as ''logarithm functions'', where you get to pick your favorite base. In this case, <math>K(r, s) = r + s~\!</math> and <math>L(u, v) = u \cdot v,\!</math> and the defining formula <math>J(L(u, v)) = K(Ju, Jv)\!</math> comes out looking like <math>J(u \cdot v) = J(u) + J(v),\!</math> writing a dot <math>(\cdot)~\!</math> and a plus sign <math>(+)\!</math> for the ordinary binary operations of arithmetical multiplication and arithmetical summation, respectively. |
| | | | |
| − | w = "woman" = B +, D +, E
| + | <br> |
| | | | |
| − | 'l' = "lover of ---" = B:C +, C:B +, D:O +, E:I +, I:E +, J:D +, O:D
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 49.jpg]] || (49) |
| | + | |} |
| | | | |
| − | 's' = "servant of ---" = C:O +, E:D +, I:O +, J:D +, J:O | + | Thus, where the ''image'' <math>J\!</math> is the logarithm map, the ''compound'' <math>K\!</math> is the numerical sum, and the ''ligature'' <math>L\!</math> is the numerical product, one has the following rule of thumb: |
| | | | |
| − | And next we derive the following results:
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <p><math>\textit{The~image~of~the~product~is~the~sum~of~the~images.}</math></p> |
| | + | |- |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | J(u \cdot v) & = & J(u) + J(v) |
| | + | \\[12pt] |
| | + | J(L(u, v)) & = & K(Ju, Jv) |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 'l', = "lover that is --- of ---"
| + | ===Commentary Note 11.15=== |
| | | | |
| − | = B:B:C +, C:C:B +, D:D:O +, E:E:I +, I:I:E +, J:J:D +, O:O:D
| + | I'm going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving maps, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce's “number of” function on logical terms. |
| | | | |
| − | 'l','s'w = (B:B:C +, C:C:B +, D:D:O +, E:E:I +, I:I:E +, J:J:D +, O:O:D) | + | The ''structure'' that is preserved by a structure-preserving map is just the structure that we all know and love as a triadic relation. Very typically, it will be the type of triadic relation that defines the type of binary operation that obeys the rules of a mathematical structure that is known as a ''group'', that is, a structure that satisfies the axioms for closure, associativity, identities, and inverses. |
| | | | |
| − | x (C:O +, E:D +, I:O +, J:D +, J:O)
| + | For example, in the previous case of the logarithm map <math>J,\!</math> we have the data: |
| | | | |
| − | x (B +, D +, E)
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lcccll} |
| | + | J & : & \mathbb{R} & \gets & \mathbb{R} |
| | + | & \text{(properly restricted)} |
| | + | \\[6pt] |
| | + | K & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} |
| | + | & \text{where}~ K(r, s) = r + s |
| | + | \\[6pt] |
| | + | L & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} |
| | + | & \text{where}~ L(u, v) = u \cdot v |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | Now what are we to make of that?
| + | Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in braces as a name for the triadic relation that constitutes or defines the corresponding group, then we have the following set-up: |
| | | | |
| − | If we operate in accordance with Peirce's example of `g`'o'h
| + | {| align="center" cellspacing="6" width="90%" |
| − | as the "giver of a horse to an owner of that horse", then we
| + | | |
| − | may assume that the associative law and the distributive law
| + | <math>\begin{matrix} |
| − | are by default in force, allowing us to derive this equation:
| + | J |
| | + | & : & |
| | + | [+] \gets [\,\cdot\,] |
| | + | \\[6pt] |
| | + | [+] |
| | + | & \subseteq & |
| | + | \mathbb{R} \times \mathbb{R} \times \mathbb{R} |
| | + | \\[6pt] |
| | + | [\,\cdot\,] |
| | + | & \subseteq & |
| | + | \mathbb{R} \times \mathbb{R} \times \mathbb{R} |
| | + | \end{matrix}</math> |
| | + | |} |
| | | | |
| − | 'l','s'w = 'l','s'(B +, D +, E)
| + | In many cases, one finds that both group operations are indicated by the same sign, typically <math>\cdot\!</math> , <math>*\!</math> , <math>+\!</math> , or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like these two variants: |
| | | | |
| − | = 'l','s'B +, 'l','s'D +, 'l','s'E
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | <p><math>\textit{The~image~of~the~sum~is~the~sum~of~the~images.}</math></p> |
| | + | |- |
| | + | | <p><math>\textit{The~image~of~the~product~is~the~sum~of~the~images.}</math></p> |
| | + | |} |
| | | | |
| − | Evidently what Peirce means by the associative principle,
| + | Figure 50 presents a generic picture for groups <math>G\!</math> and <math>H.\!</math> |
| − | as it applies to this type of product, is that a product
| |
| − | of elementary relatives having the form (R:S:T)(S:T)(T)
| |
| − | is equal to R but that no other form of product yields
| |
| − | a non-null result. Scanning the implied terms of the
| |
| − | triple product tells us that only the following case
| |
| − | is non-null: J = (J:J:D)(J:D)(D). It follows that:
| |
| | | | |
| − | 'l','s'w = "lover and servant of a woman"
| + | <br> |
| | | | |
| − | = "lover that is a servant of a woman"
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 50.jpg]] || (50) |
| | + | |} |
| | | | |
| − | = "lover of a woman that is a servant of that woman"
| + | In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, <math>J(L(u, v)) = K(Ju, Jv),\!</math> takes on the shape <math>J(u + v) = Ju + Jv,\!</math> which looks very analogous to the distributive multiplication of a sum <math>(u + v)\!</math> by a factor <math>J.\!</math> Hence another popular name for a morphism: a ''linear'' map. |
| | | | |
| − | = J
| + | ===Commentary Note 11.16=== |
| | | | |
| − | And so what Peirce says makes sense in this case.
| + | We have enough material on morphisms now to go back and cast a more studied eye on what Peirce is doing with that “number of” function, whose application to a logical term <math>t\!</math> is indicated by writing the term in square brackets, as <math>[t].\!</math> It is convenient to have a prefix notation for the function that maps a term <math>t\!</math> to a number <math>[t]\!</math> but Peirce has previously reserved <math>\mathit{n}\!</math> for the logical <math>\mathrm{not},\!</math> so let's use <math>v(t)\!</math> as a variant for <math>[t].\!</math> |
| | | | |
| | + | My plan will be nothing less plodding than to work through the statements that Peirce made in defining and explaining the “number of” function up to our present place in the paper, namely, the budget of points collected in [[Peirce%27s_1870_Logic_Of_Relatives#Commentary_Note_11.2|Section 11.2]]. |
| | | | |
| | + | '''NOF 1''' |
| | | | |
| − | LOR. Commentary Note 10.6
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (''men''), the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus <math>[t].\!</math></p> |
| | | | |
| | + | <p>(Peirce, CP 3.65).</p> |
| | + | |} |
| | | | |
| | + | The role of the “number of” function may be formalized by assigning it a name and a type as <math>v : S \to \mathbb{R},\!</math> where <math>S\!</math> is a suitable set of signs, a ''>syntactic domain'', containing all the logical terms whose numbers we need to evaluate in a given discussion, and where <math>\mathbb{R}\!</math> is the set of real numbers. |
| | | | |
| − | As Peirce observes, it is not possible to work with
| + | Transcribing Peirce's example: |
| − | relations in general without eventually abandoning
| |
| − | all of one's algebraic principles, in due time the
| |
| − | associative and maybe even the distributive, just
| |
| − | as we have already left behind the commutative.
| |
| − | It cannot be helped, as we cannot reflect on
| |
| − | a law if not from a perspective outside it,
| |
| − | that is to say, at any rate, virtually so.
| |
| | | | |
| − | One way to do this would be from the standpoint of the combinator calculus,
| + | {| width="100%" |
| − | and there are places where Peirce verges on systems that are very similar,
| + | | width="10%" | Let |
| − | but I am making a deliberate effort to remain here as close as possible
| + | | <math>\mathrm{m} = \text{man}\!</math> |
| − | within the syntactoplastic chronism of his 1870 Logic of Relatives.
| + | | width="10%" | |
| − | So let us make use of the smoother transitions that are afforded
| + | |- |
| − | by the paradigmatic Figures and Tables that I drew up earlier.
| + | | |
| | + | |- |
| | + | | and |
| | + | | <math>\mathit{t} = \text{tooth of}\,\underline{~~ ~~}.</math> |
| | + | | |
| | + | |- |
| | + | | |
| | + | |- |
| | + | | Then |
| | + | | <math>v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.\!</math> |
| | + | | |
| | + | |} |
| | | | |
| − | For the next few episodes, then, I will examine the examples
| + | Thus, in a universe of perfect human dentition, the number of the relative term <math>{}^{\backprime\backprime} \text{tooth of}\,\underline{~~ ~~} {}^{\prime\prime}\!</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math> |
| − | that Peirce gives at the next level of complication in the
| |
| − | multiplication of relative terms, for instance, the three
| |
| − | that I have redrawn below.
| |
| | | | |
| − | o-------------------------------------------------o
| + | The dyadic relative term <math>t\!</math> determines a dyadic relation <math>T \subseteq X \times Y,</math> where <math>X\!</math> contains all the teeth and <math>Y\!</math> contains all the people that happen to be under discussion. |
| − | | |
| |
| − | | |
| |
| − | | `g`__$__% $'l'__* *w %h |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \/ @ / |
| |
| − | | \ /\______ ______/ |
| |
| − | | @ @ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 6. Giver of a Horse to a Lover of a Woman
| |
| | | | |
| − | o-------------------------------------------------o
| + | A rough indication of the bigraph for <math>T\!</math> might be drawn as follows, showing just the first few items in the toothy part of <math>X\!</math> and the peoply part of <math>Y.\!</math> |
| − | | |
| |
| − | | |
| |
| − | | `g`__$__% $'o'__* *%h |
| |
| − | | o o o o oo |
| |
| − | | \ \ / \ // |
| |
| − | | \ \/ @/ |
| |
| − | | \ /\____ ____/ |
| |
| − | | @ @ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 7. Giver of a Horse to an Owner of It
| |
| | | | |
| − | o-------------------------------------------------o
| + | {| align="center" cellpadding="10" |
| − | | |
| + | | [[Image:LOR 1870 Figure 51.jpg]] || (51) |
| − | | | | + | |} |
| − | | 'l',__$__% $'s'__* *%w | | |
| − | | o o o o oo | | |
| − | | \ \ / \ // |
| |
| − | | \ \/ @/ |
| |
| − | | \ /\____ ____/ |
| |
| − | | @ @ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 8. Lover that is a Servant of a Woman
| |
| | | | |
| | + | Notice that the “number of” function <math>v : S \to \mathbb{R}</math> needs the data that is represented by this entire bigraph for <math>T\!</math> in order to compute the value <math>[t].\!</math> |
| | | | |
| | + | Finally, one observes that this component of <math>T\!</math> is a function in the direction <math>T : X \to Y,</math> since we are counting only teeth that occupy exactly one mouth of a tooth-bearing creature. |
| | | | |
| − | LOR. Commentary Note 10.7
| + | ===Commentary Note 11.17=== |
| | | | |
| | + | I think the reader is beginning to get an inkling of the crucial importance of the “number of” function in Peirce's way of looking at logic. Among other things it is one of the planks in the bridge from logic to the theories of probability, statistics, and information, in which setting logic forms but a limiting case at one scenic turnout on the expanding vista. It is, as a matter of necessity and a matter of fact, practically speaking at any rate, one way that Peirce forges a link between the ''eternal'', logical, or rational realm and the ''secular'', empirical, or real domain. |
| | | | |
| | + | With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text. |
| | | | |
| − | Here is what I get when I try to analyze Peirce's
| + | '''NOF 2''' |
| − | "giver of a horse to a lover of a woman" example
| |
| − | along the same lines as the 2-adic compositions.
| |
| | | | |
| − | We may begin with the mark-up shown in Figure 6.
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>But not only do the significations of <math>=\!</math> and <math><\!</math> here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write <math>5 < 7\!</math> is to say that <math>5\!</math> is part of <math>7\!</math>, just as to write <math>\mathrm{f} < \mathrm{m}~\!</math> is to say that Frenchmen are part of men. Indeed, if <math>\mathrm{f} < \mathrm{m}~\!</math>, then the number of Frenchmen is less than the number of men, and if <math>\mathrm{v} = \mathrm{p}\!</math>, then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.</p> |
| | | | |
| − | o-------------------------------------------------o
| + | <p>(Peirce, CP 3.66).</p> |
| − | | |
| + | |} |
| − | | |
| |
| − | | `g`__$__% $'l'__* *w %h |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \/ @ / | | |
| − | | \ /\______ ______/ |
| |
| − | | @ @ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 6. Giver of a Horse to a Lover of a Woman
| |
| | | | |
| − | If we analyze this in accord with the "spreadsheet" model
| + | Peirce is here remarking on the principle that the measure <math>\mathit{v}\!</math> on terms ''preserves'' or ''respects'' the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms. In these initiatory passages of the text, Peirce is using a single symbol <math><\!</math> to denote the usual linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes. Later, of course, he will introduce distinctive symbols for logical orders. The links among terms, sets, and numbers can be pursued in all directions, and Peirce has already indicated in an earlier paper how he would construct the integers from sets, that is, from the aggregate denotations of terms. I will try to get back to that another time. |
| − | of relational composition, the core of it is a particular | |
| − | way of composing a 3-adic "giving" relation G c T x U x V
| |
| − | with a 2-adic "loving" relation L c U x W so as to obtain
| |
| − | a specialized sort of 3-adic relation (G o L) c T x W x V.
| |
| − | The applicable constraints on tuples are shown in Table 9. | |
| | | | |
| − | Table 9. Composite of Triadic and Dyadic Relations
| + | We have a statement of the following form: |
| − | o---------o---------o---------o---------o---------o
| |
| − | | # !1! | !1! | !1! | !1! |
| |
| − | o=========o=========o=========o=========o=========o
| |
| − | | G # T | U | | V |
| |
| − | o---------o---------o---------o---------o---------o
| |
| − | | L # | U | W | |
| |
| − | o---------o---------o---------o---------o---------o
| |
| − | | G o L # T | | W | V |
| |
| − | o---------o---------o---------o---------o---------o
| |
| | | | |
| − | The hypergraph picture of the abstract composition is given in Figure 10.
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | If <math>\mathrm{f} < \mathrm{m},\!</math> then the number of Frenchmen is less than the number of men. |
| | + | |} |
| | | | |
| − | o---------------------------------------------------------------------o
| + | This goes into symbolic form as follows: |
| − | | |
| |
| − | | G o L |
| |
| − | | ___________@___________ |
| |
| − | | / \ \ |
| |
| − | | / G L \ \ |
| |
| − | | / @ @ \ \ |
| |
| − | | / /|\ / \ \ \ |
| |
| − | | / / | \ / \ \ \ |
| |
| − | | / / | \ / \ \ \ |
| |
| − | | / / | \ / \ \ \ |
| |
| − | | o o o o o o o o |
| |
| − | | T T U V U W W V |
| |
| − | | 1,_# #`g`_$____% $'l'______* *1 %1 |
| |
| − | | o o o o o o o o |
| |
| − | | \ / \ \ / \ / / |
| |
| − | | @ \ \/ @ / |
| |
| − | | !1! \ /\ !1! / |
| |
| − | | \ / \_______ _______/ |
| |
| − | | @ @ |
| |
| − | | !1! !1! |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 10. Anything that is a Giver of Anything to a Lover of Anything
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{matrix} |
| | + | \mathrm{f} < \mathrm{m} & \Rightarrow & [\mathrm{f}] < [\mathrm{m}]. |
| | + | \end{matrix}</math> |
| | + | |} |
| | | | |
| | + | In this setting the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the left is a logical ordering on syntactic terms while the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the right is an arithmetic ordering on real numbers. |
| | | | |
| − | LOR. Commentary Note 10.8
| + | The question that arises in this case is whether a map between two ordered sets is ''order-preserving''. In order to formulate the question in more general terms, we may begin with the following set-up: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | Let <math>X_1\!</math> be a set with the ordering <math><_1\!.</math> |
| | + | |- |
| | + | | Let <math>X_2\!</math> be a set with the ordering <math><_2\!.</math> |
| | + | |} |
| | | | |
| | + | An order relation is typically defined by a set of axioms that determines its properties. Since we have frequent occasion to view the same set in the light of several different order relations, we often resort to explicit specifications like <math>(X, <_1),\!</math> <math>(X, <_2),\!</math> and so on, to indicate a set with a given ordering. |
| | | | |
| − | In taking up the next example of relational composition,
| + | A map <math>F : (X_1, <_1) \to (X_2, <_2)</math> is ''order-preserving'' if and only if a statement of a particular form holds for all <math>x\!</math> and <math>y\!</math> in <math>(X_1, <_1),\!</math> namely, the following: |
| − | let's exchange the relation 't' = "trainer of ---" for
| |
| − | Peirce's relation 'o' = "owner of ---", simply for the
| |
| − | sake of avoiding conflicts in the symbols that we use.
| |
| − | In this way, Figure 7 is transformed into Figure 11.
| |
| | | | |
| − | o-------------------------------------------------o
| + | {| align="center" cellspacing="6" width="90%" |
| − | | |
| + | | |
| − | | |
| + | <math>\begin{matrix} |
| − | | `g`__$__% $'t'__* *%h | | + | x <_1 y & \Rightarrow & F(x) <_2 F(y). |
| − | | o o o o oo |
| + | \end{matrix}</math> |
| − | | \ \ / \ // |
| + | |} |
| − | | \ \/ @/ |
| |
| − | | \ /\____ ____/ |
| |
| − | | @ @ |
| |
| − | | | | |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 11. Giver of a Horse to a Trainer of It
| |
| | | | |
| − | Now here's an interesting point, in fact, a critical transition point,
| + | The “number of” map <math>v : (S, <_1) \to (\mathbb{R}, <_2)</math> has just this character, as exemplified in the case at hand: |
| − | that we see resting in potential but a stone's throw removed from the
| |
| − | chronism, the secular neigborhood, the temporal vicinity of Peirce's
| |
| − | 1870 LOR, and it's a vertex that turns on the teridentity relation.
| |
| | | | |
| − | The hypergraph picture of the abstract composition is given in Figure 12.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{matrix} |
| | + | \mathrm{f} & < & \mathrm{m} & \Rightarrow & [\mathrm{f}] & < & [\mathrm{m}] |
| | + | \\[6pt] |
| | + | \mathrm{f} & < & \mathrm{m} & \Rightarrow & v(\mathrm{f}) & < & v(\mathrm{m}) |
| | + | \end{matrix}</math> |
| | + | |} |
| | | | |
| − | o---------------------------------------------------------------------o
| + | Here, the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the left is read as ''proper inclusion'', in other words, ''subset of but not equal to'', while the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the right is read as the ordinary ''less than'' relation. |
| − | | |
| |
| − | | G o T |
| |
| − | | _________________@_________________ |
| |
| − | | / \ |
| |
| − | | / G T \ |
| |
| − | | / @ @ \ |
| |
| − | | / /|\ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | o o o o o o o |
| |
| − | | X X Y Z Y Z Z |
| |
| − | | 1,_# #`g`_$____% $'t'______% %1 |
| |
| − | | o o o o o o o |
| |
| − | | \ / \ \ / | / |
| |
| − | | \ / \ \/ | / |
| |
| − | | \ / \ /\ | / |
| |
| − | | \ / \ / \__________|__________/ |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 12. Anything that is a Giver of Anything to a Trainer of It
| |
| | | | |
| − | If we analyze this in accord with the "spreadsheet" model
| + | ===Commentary Note 11.18=== |
| − | of relational composition, the core of it is a particular
| |
| − | way of composing a 3-adic "giving" relation G c X x Y x Z
| |
| − | with a 2-adic "training" relation T c Y x Z in such a way
| |
| − | as to determine a certain 2-adic relation (G o T) c X x Z.
| |
| − | Table 13 schematizes the associated constraints on tuples.
| |
| | | | |
| − | Table 13. Another Brand of Composition
| + | An ''order-preserving map'' is a special case of a ''structure preserving map'', and the idea of ''preserving structure'', as used in mathematics, always means preserving ''some'' but not necessarily ''all'' the structure of the source domain in question. People sometimes express this by speaking of ''structure preservation in measure'', the implication being that any property that is amenable to being qualified in manner is potentially amenable to being quantified in degree, perhaps in such a way as to answer questions like “How structure-preserving is it?” |
| − | o---------o---------o---------o---------o
| |
| − | | # !1! | !1! | !1! |
| |
| − | o=========o=========o=========o=========o
| |
| − | | G # X | Y | Z |
| |
| − | o---------o---------o---------o---------o
| |
| − | | T # | Y | Z |
| |
| − | o---------o---------o---------o---------o
| |
| − | | G o T # X | | Z |
| |
| − | o---------o---------o---------o---------o
| |
| | | | |
| − | So we see that the notorious teridentity relation,
| + | Let's see how this remark applies to the order-preserving property of the “number of” mapping <math>v : S \to \mathbb{R}.</math> For any pair of absolute terms <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S,\!</math> we have the following implications, where <math>^{\backprime\backprime}-\!\!\!<\!^{\prime\prime}</math> denotes the logical subsumption relation on terms and <math>^{\backprime\backprime}\!\!\le\!^{\prime\prime}</math> denotes the ''less than or equal to'' relation on the real number domain <math>\mathbb{R}.</math> |
| − | which I have left equivocally denoted by the same
| |
| − | symbol as the identity relation !1!, is already
| |
| − | implicit in Peirce's discussion at this point.
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | x ~-\!\!\!< y & \Rightarrow & vx \le vy |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| | + | Equivalently: |
| | | | |
| − | LOR. Commentary Note 10.9
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lll} |
| | + | x ~-\!\!\!< y & \Rightarrow & [x] \le [y] |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| | + | Nowhere near the number of logical distinctions that exist on the left hand side of the implication arrow can be preserved as one passes to the linear ordering of real numbers on the right hand side of the implication arrow, but that is not required in order to call the map <math>v : S \to \mathbb{R}</math> ''order-preserving'', or what is known as an ''order morphism''. |
| | | | |
| | + | ===Commentary Note 11.19=== |
| | | | |
| − | The use of the concepts of identity and teridentity is not to identify
| + | Up to this point in the 1870 Logic of Relatives, Peirce has introduced the “number of” function on logical terms and discussed the extent to which its use as a measure, <math>v : S \to \mathbb{R}\!</math> such that <math>v : s \mapsto [s],\!</math> satisfies the relevant measure-theoretic principles, for starters, these two: |
| − | a thing in itself with itself, much less twice or thrice over, since | |
| − | there is no need and thus no utility in that. I can imagine Peirce
| |
| − | asking, on Kantian principles if not entirely on Kantian premisses,
| |
| − | "Where is the manifold to be unified?" The manifold that demands
| |
| − | unification does not reside in the object but in the phenomena,
| |
| − | that is, in the appearances that might have been appearances
| |
| − | of different objects but that happen to be constrained by
| |
| − | these identities to being just so many aspects, facets, | |
| − | parts, roles, or signs of one and the same object.
| |
| | | | |
| − | For example, notice how the various identity concepts actually
| + | {| align="center" cellspacing="6" width="90%" |
| − | functioned in the last example, where they had the opportunity
| + | | valign="top" | 1. |
| − | to show their behavior in something like their natural habitat. | + | | The “number of” map exhibits a certain type of ''uniformity property'', whereby the value of the measure on a uniformly qualified population is in fact actualized by each member of the population. |
| | + | |- |
| | + | | valign="top" | 2. |
| | + | | The “number of” map satisfies an ''order morphism principle'', whereby the illative partial ordering of logical terms is reflected up to a partial extent by the arithmetical linear ordering of their measures. |
| | + | |} |
| | | | |
| − | The use of the teridentity concept in the case
| + | Peirce next takes up the action of the “number of” map on the two types of, loosely speaking, ''additive'' operations that we normally consider in logic. |
| − | of the "giver of a horse to a trainer of it" is | |
| − | to stipulate that the thing appearing with respect
| |
| − | to its quality under the aspect of an absolute term,
| |
| − | a horse, and the thing appearing with respect to its
| |
| − | recalcitrance in the role of the correlate of a 2-adic
| |
| − | relative, a brute to be trained, and the thing appearing
| |
| − | with respect to its synthesis in the role of a correlate
| |
| − | of a 3-adic relative, a gift, are one and the same thing.
| |
| | | | |
| | + | '''NOF 3.1''' |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions.</p> |
| | | | |
| − | LOR. Commentary Note 10.10
| + | <p>(Peirce, CP 3.67).</p> |
| | + | |} |
| | | | |
| | + | The sign <math>^{\backprime\backprime} +\!\!, {}^{\prime\prime}</math> denotes what Peirce calls “the regular non-invertible addition”, corresponding to the inclusive disjunction of logical terms or the union of their extensions as sets. |
| | | | |
| | + | The sign <math>^{\backprime\backprime} + ^{\prime\prime}</math> denotes what Peirce calls “the invertible addition”, corresponding to the exclusive disjunction of logical terms or the symmetric difference of their extensions as sets. |
| | | | |
| − | Figure 8 depicts the last of the three examples involving
| + | '''NOF 3.2''' |
| − | the composition of 3-adic relatives with 2-adic relatives: | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>But the notation has other recommendations. The conception of ''taking together'' involved in these processes is strongly analogous to that of summation, the sum of <math>2\!</math> and <math>5,\!</math> for example, being the number of a collection which consists of a collection of two and a collection of five.</p> |
| | | | |
| − | o-------------------------------------------------o
| + | <p>(Peirce, CP 3.67).</p> |
| − | | |
| + | |} |
| − | | |
| |
| − | | 'l',__$__% $'s'__* *%w |
| |
| − | | o o o o oo |
| |
| − | | \ \ / \ // |
| |
| − | | \ \/ @/ | | |
| − | | \ /\____ ____/ |
| |
| − | | @ @ |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 8. Lover that is a Servant of a Woman
| |
| | | | |
| − | The hypergraph picture of the abstract composition is given in Figure 14.
| + | A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word ''collection'', and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion. |
| | | | |
| − | o---------------------------------------------------------------------o
| + | The “number of” map <math>v : S \to \mathbb{R}</math> evidently induces some sort of morphism with respect to logical sums. If this were straightforwardly true, we could write: |
| − | | |
| |
| − | | L , S |
| |
| − | | __________________^__________________ |
| |
| − | | / \ |
| |
| − | | / L_, S \ |
| |
| − | | / @ @ \ |
| |
| − | | / /|\ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | / / | \ / \ \ |
| |
| − | | o o o o o o o |
| |
| − | | X X X Y X Y Y |
| |
| − | | 1,_# #'l',_$_____% $'t'________% %1 |
| |
| − | | o o o o o o o |
| |
| − | | \ / \ \ / | / |
| |
| − | | \ / \ \ | / |
| |
| − | | \ / \ / \ | / |
| |
| − | | \ / \ / \___________|___________/ |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 14. Anything that's a Lover of Anything and that's a Servant of It
| |
| | | | |
| − | This example illustrates the way that Peirce analyzes the logical conjunction,
| + | {| align="center" cellspacing="6" width="90%" |
| − | we might even say the "parallel conjunction", of a couple of 2-adic relatives
| + | | |
| − | in terms of the comma extension and the same style of composition that we saw
| + | <math>\begin{matrix} |
| − | in the last example, that is, according to a pattern of anaphora that invokes
| + | ? & v(x ~+\!\!,~ y) & = & v(x) ~+~ v(y) & ? |
| − | the teridentity relation.
| + | \end{matrix}</math> |
| | + | |} |
| | | | |
| − | If we lay out this analysis of conjunction on the spreadsheet model
| + | Equivalently: |
| − | of relational composition, the gist of it is the diagonal extension
| |
| − | of a 2-adic "loving" relation L c X x Y to the corresponding 3-adic
| |
| − | "loving and being" relation L_, c X x X x Y, which is then composed
| |
| − | in a specific way with a 2-adic "serving" relation S c X x Y, so as
| |
| − | to determine the 2-adic relation L,S c X x Y. Table 15 schematizes
| |
| − | the associated constraints on tuples.
| |
| | | | |
| − | Table 15. Conjunction Via Composition
| + | {| align="center" cellspacing="6" width="90%" |
| − | o---------o---------o---------o---------o
| + | | |
| − | | # !1! | !1! | !1! |
| + | <math>\begin{matrix} |
| − | o=========o=========o=========o=========o
| + | ? & [x ~+\!\!,~ y] & = & [x] ~+~ [y] & ? |
| − | | L, # X | X | Y |
| + | \end{matrix}</math> |
| − | o---------o---------o---------o---------o
| + | |} |
| − | | S # | X | Y |
| |
| − | o---------o---------o---------o---------o
| |
| − | | L , S # X | | Y | | |
| − | o---------o---------o---------o---------o
| |
| | | | |
| | + | Of course, things are not quite that simple when it comes to inclusive disjunctions and set-theoretic unions, so it is usual to introduce the concept of a ''sub-additive measure'' to describe the principle that does hold here, namely, the following: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{matrix} |
| | + | v(x ~+\!\!,~ y) & \le & v(x) ~+~ v(y) |
| | + | \end{matrix}</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 10.11
| + | Equivalently: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{matrix} |
| | + | [x ~+\!\!,~ y] & \le & [x] ~+~ [y] |
| | + | \end{matrix}</math> |
| | + | |} |
| | | | |
| | + | This is why Peirce trims his discussion of this point with the following hedge: |
| | | | |
| − | I return to where we were in unpacking the contents of CP 3.73.
| + | '''NOF 3.3''' |
| − | Peirce remarks that the comma operator can be iterated at will:
| |
| | | | |
| − | | In point of fact, since a comma may be added in this way to any | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | | relative term, it may be added to one of these very relatives
| |
| − | | formed by a comma, and thus by the addition of two commas
| |
| − | | an absolute term becomes a relative of two correlates.
| |
| | | | | | |
| − | | So: | + | <p>Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.</p> |
| | + | |
| | + | <p>(Peirce, CP 3.67).</p> |
| | + | |} |
| | + | |
| | + | Finally, a morphism with respect to addition, even a contingently qualified one, must do the right stuff on behalf of the additive identity: |
| | + | |
| | + | '''NOF 3.4''' |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | | | | | |
| − | | m,,b,r
| + | <p>Addition being taken in this sense, ''nothing'' is to be denoted by ''zero'', for then</p> |
| | + | |- |
| | + | | align="center" | <math>x ~+\!\!,~ 0 ~=~ x</math> |
| | + | |- |
| | | | | | |
| − | | interpreted like | + | <p>whatever is denoted by <math>x\!</math>; and this is the definition of ''zero''. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of ''zero'' and that of nothing, and because we shall thus have</p> |
| | + | |- |
| | + | | align="center" | <math>[0] ~=~ 0.</math> |
| | + | |- |
| | | | | | |
| − | | `g`'o'h
| + | <p>(Peirce, CP 3.67).</p> |
| − | |
| + | |} |
| − | | means a man that is a rich individual and
| |
| − | | is a black that is that rich individual.
| |
| − | |
| |
| − | | But this has no other meaning than:
| |
| − | |
| |
| − | | m,b,r
| |
| − | |
| |
| − | | or a man that is a black that is rich.
| |
| − | |
| |
| − | | Thus we see that, after one comma is added, the
| |
| − | | addition of another does not change the meaning
| |
| − | | at all, so that whatever has one comma after it
| |
| − | | must be regarded as having an infinite number.
| |
| − | | | |
| − | | C.S. Peirce, CP 3.73
| |
| | | | |
| − | Again, let us check whether this makes sense
| + | With respect to the nullity <math>0\!</math> in <math>S\!</math> and the number <math>0\!</math> in <math>\mathbb{R},</math> we have: |
| − | on the stage of our small but dramatic model.
| |
| | | | |
| − | Let's say that Desdemona and Othello are rich,
| + | {| align="center" cellspacing="6" width="90%" |
| − | and, among the persons of the play, only they.
| + | | <math>v0 ~=~ [0] ~=~ 0.</math> |
| | + | |} |
| | | | |
| − | With this premiss we obtain a sample of absolute terms
| + | In sum, therefore, it can be said: ''It also serves that only preserves a due respect for the function of a vacuum in nature.'' |
| − | that is sufficiently ample to work through our example:
| |
| | | | |
| − | 1 = B +, C +, D +, E +, I +, J +, O
| + | ===Commentary Note 11.20=== |
| | | | |
| − | b = O
| + | We arrive at the last of Peirce's statements about the “number of” map that we singled out above: |
| | | | |
| − | m = C +, I +, J +, O
| + | '''NOF 4.1''' |
| | | | |
| − | r = D +, O
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p> |
| | | | |
| − | One application of the comma operator
| + | <p>Even ordinary numerical multiplication involves the same idea, for <math>~2 \times 3~</math> is a pair of triplets, and <math>~3 \times 2~</math> is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives.</p> |
| − | yields the following 2-adic relatives:
| |
| | | | |
| − | 1, = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
| + | <p>If we have an equation of the form:</p> |
| | + | |- |
| | + | | align="center" | <math>xy ~=~ z</math> |
| | + | |- |
| | + | | |
| | + | <p>and there are just as many <math>x\!</math>'s per <math>y\!</math> as there are ''per'' things, things of the universe, then we have also the arithmetical equation:</p> |
| | + | |- |
| | + | | align="center" | <math>[x][y] ~=~ [z].</math> |
| | + | |- |
| | + | | |
| | + | <p>(Peirce, CP 3.76).</p> |
| | + | |} |
| | | | |
| − | b, = O:O
| + | Peirce is here observing what we might call a ''contingent morphism''. Provided that a certain condition, to be named in short order, happens to be satisfied, we would find it holding that the “number of” map <math>v : S \to \mathbb{R}</math> such that <math>v(s) = [s]\!</math> serves to preserve the multiplication of relative terms, that is to say, the composition of relations, in the form: <math>[xy] = [x][y].\!</math> So let us try to uncross Peirce's manifestly chiasmatic encryption of the condition that is called on in support of this preservation. |
| | | | |
| − | m, = C:C +, I:I +, J:J +, O:O
| + | The proviso for the equation <math>[xy] = [x][y]\!</math> to hold is this: |
| | | | |
| − | r, = D:D +, O:O
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>There are just as many <math>x\!</math>'s per <math>y\!</math> as there are ''per'' things, things of the universe.</p> |
| | | | |
| − | Another application of the comma operator
| + | <p>(Peirce, CP 3.76).</p> |
| − | generates the following 3-adic relatives:
| + | |} |
| | | | |
| − | 1,, = B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O
| + | Returning to the example that Peirce gives: |
| | | | |
| − | b,, = O:O:O
| + | '''NOF 4.2''' |
| | | | |
| − | m,, = C:C:C +, I:I:I +, J:J:J +, O:O:O
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]</math> |
| | + | |- |
| | + | | |
| | + | <p>holds arithmetically.</p> |
| | | | |
| − | r,, = D:D:D +, O:O:O
| + | <p>(Peirce, CP 3.76).</p> |
| | + | |} |
| | | | |
| − | Assuming the associativity of multiplication among 2-adic relatives,
| + | Now that is something that we can sink our teeth into and trace the bigraph representation of the situation. It will help to recall our first examination of the “tooth of” relation and to adjust the picture we sketched of it on that occasion. |
| − | we may compute the product m,b,r by a brute force method as follows: | |
| | | | |
| − | m,b,r = (C:C +, I:I +, J:J +, O:O)(O:O)(D +, O)
| + | Transcribing Peirce's example: |
| | | | |
| − | = (C:C +, I:I +, J:J +, O:O)(O)
| + | {| width="100%" |
| | + | | width="10%" | Let |
| | + | | <math>\mathrm{m} = \text{man}\!</math> |
| | + | | width="10%" | |
| | + | |- |
| | + | | |
| | + | |- |
| | + | | and |
| | + | | <math>\mathit{t} = \text{tooth of}\,\underline{~~ ~~}.\!</math> |
| | + | | |
| | + | |- |
| | + | | |
| | + | |- |
| | + | | Then |
| | + | | <math>v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.\!</math> |
| | + | | |
| | + | |} |
| | | | |
| − | = O
| + | That is to say, the number of the relative term <math>\text{tooth of}\,\underline{~~ ~~}\!</math> is equal to the number of teeth of humans divided by the number of humans. In a universe of perfect human dentition this gives a quotient of <math>32.\!</math> |
| | | | |
| − | This avers that a man that is black that is rich is Othello,
| + | The dyadic relative term <math>t\!</math> determines a dyadic relation <math>T \subseteq X \times Y,</math> where <math>X\!</math> contains all the teeth and <math>Y\!</math> contains all the people that happen to be under discussion. |
| − | which is true on the premisses of our universe of discourse.
| |
| | | | |
| − | The stock associations of `g`'o'h lead us to multiply out the
| + | To make the case as simple as possible and still cover the point, suppose there are just four people in our universe of discourse and just two of them are French. The bigraphical composition below shows the pertinent facts of the case. |
| − | product m,,b,r along the following lines, where the trinomials
| |
| − | of the form (X:Y:Z)(Y:Z)(Z) are the only ones that produce any | |
| − | non-null result, specifically, of the form (X:Y:Z)(Y:Z)(Z) = X.
| |
| | | | |
| − | m,,b,r = (C:C:C +, I:I:I +, J:J:J +, O:O:O)(O:O)(D +, O)
| + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 52.jpg]] || (52) |
| | + | |} |
| | | | |
| − | = (O:O:O)(O:O)(O)
| + | In this picture the order of relational composition flows down the page. For convenience in composing relations, the absolute term <math>\mathrm{f} = \text{Frenchman}\!</math> is inflected by the comma functor to form the dyadic relative term <math>\mathrm{f,} = \text{Frenchman that is}\,\underline{~~ ~~},\!</math> which in turn determines the idempotent representation of Frenchmen as a subset of mankind, <math>F \subseteq Y \times Y.\!</math> |
| | | | |
| − | = O
| + | By way of a legend for the figure, we have the following data: |
| | | | |
| − | So we have that m,,b,r = m,b,r.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{lllr} |
| | + | \mathrm{m} |
| | + | & = & |
| | + | \mathrm{J} ~+\!\!,~ \mathrm{K} ~+\!\!,~ \mathrm{L} ~+\!\!,~ \mathrm{M} \qquad = & |
| | + | \mathbf{1} |
| | + | \\[6pt] |
| | + | \mathrm{f} |
| | + | & = & \mathrm{K} ~+\!\!,~ \mathrm{M} |
| | + | \\[6pt] |
| | + | \mathrm{f,} |
| | + | & = & \mathrm{K}\!:\!\mathrm{K} ~+\!\!,~ \mathrm{M}\!:\!\mathrm{M} |
| | + | \\[6pt] |
| | + | \mathit{t} |
| | + | & = & (T_{001} ~+\!\!,~ \dots ~+\!\!,~ T_{032}):J & ~+\!\!, |
| | + | \\[6pt] |
| | + | & & (T_{033} ~+\!\!,~ \dots ~+\!\!,~ T_{064}):K & ~+\!\!, |
| | + | \\[6pt] |
| | + | & & (T_{065} ~+\!\!,~ \dots ~+\!\!,~ T_{096}):L & ~+\!\!, |
| | + | \\[6pt] |
| | + | & & (T_{097} ~+\!\!,~ \dots ~+\!\!,~ T_{128}):M |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | In closing, observe that the teridentity relation has turned up again
| + | Now let's see if we can use this picture to make sense of the following statement: |
| − | in this context, as the second comma-ing of the universal term itself:
| |
| | | | |
| − | 1,, = B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O.
| + | '''NOF 4.3''' |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]</math> |
| | + | |- |
| | + | | |
| | + | <p>holds arithmetically.</p> |
| | | | |
| | + | <p>(Peirce, CP 3.76).</p> |
| | + | |} |
| | | | |
| − | LOR. Note 11
| + | In statistical terms, Peirce is saying this: If the population of Frenchmen is a ''fair sample'' of the general population with regard to the factor of dentition, then the morphic equation, |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>[\mathit{t}\mathrm{f}] = [\mathit{t}][\mathrm{f}],\!</math> |
| | + | |} |
| | | | |
| | + | whose transpose gives the equation, |
| | | | |
| − | | The Signs for Multiplication (concl.) | + | {| align="center" cellspacing="6" width="90%" |
| − | |
| + | | <math>[\mathit{t}] = \frac{[\mathit{t}\mathrm{f}]}{[\mathrm{f}]},\!</math> |
| − | | The conception of multiplication we have adopted is that of
| + | |} |
| − | | the application of one relation to another. So, a quaternion
| |
| − | | being the relation of one vector to another, the multiplication
| |
| − | | of quaternions is the application of one such relation to a second.
| |
| − | |
| |
| − | | Even ordinary numerical multiplication involves the same idea, for
| |
| − | | 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs, where
| |
| − | | "triplet of" and "pair of" are evidently relatives.
| |
| − | |
| |
| − | | If we have an equation of the form:
| |
| − | |
| |
| − | | xy = z
| |
| − | |
| |
| − | | and there are just as many x's per y as there are,
| |
| − | | 'per' things, things of the universe, then we have
| |
| − | | also the arithmetical equation:
| |
| − | |
| |
| − | | [x][y] = [z]. | |
| − | |
| |
| − | | For instance, if our universe is perfect men, and there
| |
| − | | are as many teeth to a Frenchman (perfect understood)
| |
| − | | as there are to any one of the universe, then:
| |
| − | |
| |
| − | | ['t'][f] = ['t'f]
| |
| − | |
| |
| − | | holds arithmetically.
| |
| − | |
| |
| − | | So if men are just as apt to be black as things in general:
| |
| − | |
| |
| − | | [m,][b] = [m,b]
| |
| − | |
| |
| − | | where the difference between [m] and [m,] must not be overlooked.
| |
| − | |
| |
| − | | It is to be observed that:
| |
| − | |
| |
| − | | [!1!] = `1`.
| |
| − | | | |
| − | | Boole was the first to show this connection between logic and
| |
| − | | probabilities. He was restricted, however, to absolute terms.
| |
| − | | I do not remember having seen any extension of probability to
| |
| − | | relatives, except the ordinary theory of 'expectation'.
| |
| − | |
| |
| − | | Our logical multiplication, then, satisfies the essential conditions
| |
| − | | of multiplication, has a unity, has a conception similar to that of
| |
| − | | admitted multiplications, and contains numerical multiplication as
| |
| − | | a case under it.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| | | | |
| | + | is every bit as true as the defining equation in this circumstance, namely, |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>[\mathit{t}] = \frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.\!</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 11.1
| + | ===Commentary Note 11.21=== |
| | | | |
| | + | One more example and one more general observation, and then we will be all caught up with our homework on Peirce's “number of” function. |
| | | | |
| | + | '''NOF 4.4''' |
| | | | |
| − | We have reached in our reading of Peirce's text a suitable place to pause --
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | actually, it is more like to run as fast as we can along a parallel track --
| + | | |
| − | where I can due quietus make of a few IOU's that I've used to pave my way.
| + | <p>So if men are just as apt to be black as things in general,</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],\!</math> |
| | + | |- |
| | + | | |
| | + | <p>where the difference between <math>[\mathrm{m}]\!</math> and <math>[\mathrm{m,}]\!</math> must not be overlooked.</p> |
| | + | |
| | + | <p>(Peirce, CP 3.76).</p> |
| | + | |} |
| | + | |
| | + | The protasis, “men are just as apt to be black as things in general”, is elliptic in structure, and presents us with a potential ambiguity. If we had no further clue to its meaning, it might be read as either of the following: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | valign="top" | 1. |
| | + | | Men are just as apt to be black as things in general are apt to be black. |
| | + | |- |
| | + | | valign="top" | 2. |
| | + | | Men are just as apt to be black as men are apt to be things in general. |
| | + | |} |
| | + | |
| | + | The second interpretation, if grammatical, is pointless to state, since it equates a proper contingency with an absolute certainty. So I think it is safe to assume this paraphrase of what Peirce intends: |
| | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <p>Men are just as likely to be black as things in general are likely to be black.</p> |
| | + | |} |
| | + | |
| | + | Stated in terms of the conditional probability: |
| | | | |
| − | The more pressing debts that come to mind are concerned with the matter
| + | {| align="center" cellspacing="6" width="90%" |
| − | of Peirce's "number of" function, that maps a term t into a number [t],
| + | | <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}).\!</math> |
| − | and with my justification for calling a certain style of illustration
| + | |} |
| − | by the name of the "hypergraph" picture of relational composition.
| |
| − | As it happens, there is a thematic relation between these topics,
| |
| − | and so I can make my way forward by addressing them together.
| |
| | | | |
| − | At this point we have two good pictures of how to compute the
| + | From the definition of conditional probability: |
| − | relational compositions of arbitrary 2-adic relations, namely,
| |
| − | the bigraph and the matrix representations, each of which has
| |
| − | its differential advantages in different types of situations.
| |
| | | | |
| − | But we do not have a comparable picture of how to compute the
| + | {| align="center" cellspacing="6" width="90%" |
| − | richer variety of relational compositions that involve 3-adic
| + | | <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ {\mathrm{P}(\mathrm{b}\mathrm{m}) \over \mathrm{P}(\mathrm{m})}.\!</math> |
| − | or any higher adicity relations. As a matter of fact, we run
| + | |} |
| − | into a non-trivial classification problem simply to enumerate
| |
| − | the different types of compositions that arise in these cases.
| |
| | | | |
| − | Therefore, let us inaugurate a systematic study of relational composition,
| + | Equivalently: |
| − | general enough to explicate the "generative potency" of Peirce's 1870 LOR.
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}).\!</math> |
| | + | |} |
| | | | |
| | + | Taking everything together, we obtain the following result: |
| | | | |
| − | LOR. Commentary Note 11.2
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b})\mathrm{P}(\mathrm{m}).\!</math> |
| | + | |} |
| | | | |
| | + | This, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man. It seems to be the most likely guess that this is the meaning of Peirce's statement about frequencies: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}].\!</math> |
| | + | |} |
| | | | |
| − | Let's bring together the various things that Peirce has said
| + | The terms of this equation can be normalized to produce the corresponding statement about probabilities: |
| − | about the "number of function" up to this point in the paper.
| |
| | | | |
| − | NOF 1.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>\mathrm{P}(\mathrm{m}\mathrm{b}) ~=~ \mathrm{P}(\mathrm{m})\mathrm{P}(\mathrm{b}).\!</math> |
| | + | |} |
| | | | |
| − | | I propose to assign to all logical terms, numbers;
| + | Let's see if this checks out. |
| − | | to an absolute term, the number of individuals it denotes;
| + | |
| − | | to a relative term, the average number of things so related
| + | Let <math>N\!</math> be the number of things in general. In terms of Peirce's “number of” function, then, we have the equation <math>[\mathbf{1}] = N.</math> On the assumption that <math>\mathrm{m}\!</math> and <math>\mathrm{b}\!</math> are associated with independent events, we obtain the following sequence of equations: |
| − | | to one individual. | + | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | | | | | |
| − | | Thus in a universe of perfect men ('men'),
| + | <math>\begin{array}{lll} |
| − | | the number of "tooth of" would be 32. | + | [\mathrm{m,}\mathrm{b}] |
| − | |
| + | & = & |
| − | | The number of a relative with two correlates would be the
| + | \mathrm{P}(\mathrm{m}\mathrm{b}) N |
| − | | average number of things so related to a pair of individuals;
| + | \\[6pt] |
| − | | and so on for relatives of higher numbers of correlates.
| + | & = & |
| − | |
| + | \mathrm{P}(\mathrm{m})\mathrm{P}(\mathrm{b}) N |
| − | | I propose to denote the number of a logical term by
| + | \\[6pt] |
| − | | enclosing the term in square brackets, thus ['t'].
| + | & = & |
| − | |
| + | \mathrm{P}(\mathrm{m})[\mathrm{b}] |
| − | | C.S. Peirce, CP 3.65
| + | \\[6pt] |
| | + | & = & |
| | + | [\mathrm{m,}][\mathrm{b}]. |
| | + | \end{array}</math> |
| | + | |} |
| | + | |
| | + | As a result, we have to interpret <math>[\mathrm{m,}]\!</math> = “the average number of men per things in general” as <math>\mathrm{P}(\mathrm{m})\!</math> = “the probability of a thing in general being a man”. This seems to make sense. |
| | | | |
| − | NOF 2.
| + | ===Commentary Note 11.22=== |
| | | | |
| − | | But not only do the significations of '=' and '<' here adopted fulfill all
| + | Let's look at that last example from a different angle. |
| − | | absolute requirements, but they have the supererogatory virtue of being very
| |
| − | | nearly the same as the common significations. Equality is, in fact, nothing
| |
| − | | but the identity of two numbers; numbers that are equal are those which are
| |
| − | | predicable of the same collections, just as terms that are identical are those
| |
| − | | which are predicable of the same classes. So, to write 5 < 7 is to say that 5
| |
| − | | is part of 7, just as to write f < m is to say that Frenchmen are part of men.
| |
| − | | Indeed, if f < m, then the number of Frenchmen is less than the number of men,
| |
| − | | and if v = p, then the number of Vice-Presidents is equal to the number of
| |
| − | | Presidents of the Senate; so that the numbers may always be substituted
| |
| − | | for the terms themselves, in case no signs of operation occur in the
| |
| − | | equations or inequalities.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.66
| |
| | | | |
| − | NOF 3. | + | '''NOF 4.4''' |
| | | | |
| − | | It is plain that both the regular non-invertible addition | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | | and the invertible addition satisfy the absolute conditions.
| |
| − | | But the notation has other recommendations. The conception
| |
| − | | of 'taking together' involved in these processes is strongly
| |
| − | | analogous to that of summation, the sum of 2 and 5, for example,
| |
| − | | being the number of a collection which consists of a collection of
| |
| − | | two and a collection of five. Any logical equation or inequality
| |
| − | | in which no operation but addition is involved may be converted
| |
| − | | into a numerical equation or inequality by substituting the
| |
| − | | numbers of the several terms for the terms themselves --
| |
| − | | provided all the terms summed are mutually exclusive.
| |
| | | | | | |
| − | | Addition being taken in this sense,
| + | <p>So if men are just as apt to be black as things in general,</p> |
| − | | 'nothing' is to be denoted by 'zero', | + | |- |
| − | | for then: | + | | align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],\!</math> |
| | + | |- |
| | | | | | |
| − | | x +, 0 = x
| + | <p>where the difference between <math>[\mathrm{m}]\!</math> and <math>[\mathrm{m,}]\!</math> must not be overlooked.</p> |
| − | | | + | |
| − | | whatever is denoted by x; and this is the definition
| + | <p>(Peirce, CP 3.76).</p> |
| − | | of 'zero'. This interpretation is given by Boole, and
| + | |} |
| − | | is very neat, on account of the resemblance between the
| + | |
| − | | ordinary conception of 'zero' and that of nothing, and
| + | In different lights the formula <math>[\mathrm{m,}\mathrm{b}] = [\mathrm{m,}][\mathrm{b}]\!</math> presents itself as an ''aimed arrow'', ''fair sample'', or ''stochastic independence'' condition. |
| − | | because we shall thus have
| + | |
| − | |
| + | The example apparently assumes a universe of ''things in general'', encompassing among other things the denotations of the absolute terms <math>\mathrm{m} = \text{man}\!</math> and <math>\mathrm{b} = \text{black}.\!</math> That suggests to me that we might well illustrate this case in relief, by returning to our earlier staging of ''Othello'' and seeing how well that universe of dramatic discourse observes the premiss that “men are just as apt to be black as things in general”. |
| − | | [0] = 0.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.67
| |
| | | | |
| − | NOF 4.
| + | Here are the relevant data: |
| | | | |
| − | | The conception of multiplication we have adopted is | + | {| align="center" cellspacing="6" width="90%" |
| − | | that of the application of one relation to another. ...
| |
| | | | | | |
| − | | Even ordinary numerical multiplication involves the same idea,
| + | <math>\begin{array}{*{15}{l}} |
| − | | for 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs,
| + | \mathrm{b} & = & \mathrm{O} |
| − | | where "triplet of" and "pair of" are evidently relatives.
| + | \\[6pt] |
| − | |
| + | \mathrm{m} & = & |
| − | | If we have an equation of the form:
| + | \mathrm{C} & +\!\!, & |
| − | |
| + | \mathrm{I} & +\!\!, & |
| − | | xy = z
| + | \mathrm{J} & +\!\!, & |
| − | |
| + | \mathrm{O} |
| − | | and there are just as many x's per y as there are,
| + | \\[6pt] |
| − | | 'per' things, things of the universe, then we have
| + | \mathbf{1} & = & |
| − | | also the arithmetical equation:
| + | \mathrm{B} & +\!\!, & |
| − | |
| + | \mathrm{C} & +\!\!, & |
| − | | [x][y] = [z].
| + | \mathrm{D} & +\!\!, & |
| − | |
| + | \mathrm{E} & +\!\!, & |
| − | | For instance, if our universe is perfect men, and there
| + | \mathrm{I} & +\!\!, & |
| − | | are as many teeth to a Frenchman (perfect understood)
| + | \mathrm{J} & +\!\!, & |
| − | | as there are to any one of the universe, then:
| + | \mathrm{O} |
| − | |
| + | \\[12pt] |
| − | | ['t'][f] = ['t'f]
| + | \mathrm{b,} & = & \mathrm{O\!:\!O} |
| − | |
| + | \\[6pt] |
| − | | holds arithmetically.
| + | \mathrm{m,} & = & |
| − | |
| + | \mathrm{C\!:\!C} & +\!\!, & |
| − | | So if men are just as apt to be black as things in general:
| + | \mathrm{I\!:\!I} & +\!\!, & |
| − | |
| + | \mathrm{J\!:\!J} & +\!\!, & |
| − | | [m,][b] = [m,b]
| + | \mathrm{O\!:\!O} |
| − | |
| + | \\[6pt] |
| − | | where the difference between [m] and [m,] must not be overlooked.
| + | \mathbf{1,} & = & |
| − | |
| + | \mathrm{B\!:\!B} & +\!\!, & |
| − | | It is to be observed that:
| + | \mathrm{C\!:\!C} & +\!\!, & |
| − | |
| + | \mathrm{D\!:\!D} & +\!\!, & |
| − | | [!1!] = `1`.
| + | \mathrm{E\!:\!E} & +\!\!, & |
| − | | | + | \mathrm{I\!:\!I} & +\!\!, & |
| − | | Boole was the first to show this connection between logic and
| + | \mathrm{J\!:\!J} & +\!\!, & |
| − | | probabilities. He was restricted, however, to absolute terms.
| + | \mathrm{O\!:\!O} |
| − | | I do not remember having seen any extension of probability to
| + | \end{array}</math> |
| − | | relatives, except the ordinary theory of 'expectation'.
| + | |} |
| − | |
| + | |
| − | | Our logical multiplication, then, satisfies the essential conditions
| + | The ''fair sampling'' condition is tantamount to this: “Men are just as apt to be black as things in general are apt to be black”. In other words, men are a fair sample of things in general with respect to the factor of being black. |
| − | | of multiplication, has a unity, has a conception similar to that of
| |
| − | | admitted multiplications, and contains numerical multiplication as
| |
| − | | a case under it.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| | | | |
| − | Before I can discuss Peirce's "number of" function in greater detail
| + | Should this hold, the consequence would be: |
| − | I will need to deal with an expositional difficulty that I have been
| |
| − | very carefully dancing around all this time, but that will no longer
| |
| − | abide its assigned place under the rug.
| |
| | | | |
| − | Functions have long been understood, from well before Peirce's time to ours,
| + | {| align="center" cellspacing="6" width="90%" |
| − | as special cases of 2-adic relations, so the "number of" function itself is
| + | | <math>[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}].</math> |
| − | already to be numbered among the types of 2-adic relatives that we've been
| + | |} |
| − | explictly mentioning and implicitly using all this time. But Peirce's way
| |
| − | of talking about a 2-adic relative term is to list the "relate" first and
| |
| − | the "correlate" second, a convention that goes over into functional terms
| |
| − | as making the functional value first and the functional antecedent second,
| |
| − | whereas almost anyone brought up in our present time frame has difficulty
| |
| − | thinking of a function any other way than as a set of ordered pairs where
| |
| − | the order in each pair lists the functional argument, or domain element,
| |
| − | first and the functional value, or codomain element, second.
| |
| | | | |
| − | It is possible to work all this out in a very nice way within a very general context
| + | When <math>[\mathrm{b}]\!</math> is not zero, we obtain the result: |
| − | of flexible conventions, but not without introducing an order of anachronisms into
| |
| − | Peirce's presentation that I am presently trying to avoid as much as possible.
| |
| − | Thus, I will need to experiment with various sorts of compromise formations.
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>[\mathrm{m,}] ~=~ \frac{[\mathrm{m,}\mathrm{b}]}{[\mathrm{b}]}.</math> |
| | + | |} |
| | | | |
| | + | As before, it is convenient to represent the absolute term <math>\mathrm{b} = \text{black}\!</math> by means of the corresponding idempotent term <math>\mathrm{b,} = \text{black that is}\,\underline{~~ ~~}.</math> |
| | | | |
| − | LOR. Commentary Note 11.3
| + | Consider the bigraph for the composition: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>\mathrm{m,}\mathrm{b} ~=~ \text{man that is black}.</math> |
| | + | |} |
| | | | |
| | + | This is represented below in the equivalent form: |
| | | | |
| − | Having spent a fair amount of time in earnest reflection on the issue,
| + | {| align="center" cellspacing="6" width="90%" |
| − | I cannot see a way to continue my interpretation of Peirce's 1870 LOR,
| + | | <math>\mathrm{m,}\mathrm{b,} ~=~ \text{man that is black that is}\,\underline{~~ ~~}.</math> |
| − | to master the distance between his conventions of presentation and my
| + | |} |
| − | present personal perspectives on relations, without introducing a few
| |
| − | interpretive anachronisms and other artifacts in the process, and the
| |
| − | only excuse that I can make for myself is that at least these will be
| |
| − | novel sorts of anachronisms and artifacts in comparison with the ones
| |
| − | that the reeder may alreedy have seen. A poor excuse, but all I have.
| |
| − | The least that I can do, then, and I'm something of an expert on that,
| |
| − | is to exposit my personal interpretive apparatus on a separate thread,
| |
| − | where it will not distract too much from the intellectual canon, that
| |
| − | is to opine, the "thinking panpipe" that we find in Peirce's 1870 LOR.
| |
| | | | |
| − | Ripped from the pages of my dissertation, then, I will lay out
| + | {| align="center" cellpadding="10" |
| − | some samples of background material on "Relations In General",
| + | | [[Image:LOR 1870 Figure 53.jpg]] || (53) |
| − | as spied from a combinatorial point of view, that I hope will
| + | |} |
| − | serve in reeding Peirce's text, if we draw on it judiciously.
| |
| | | | |
| | + | Thus we observe one of the more factitious facts affecting this very special universe of discourse, namely: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>\mathrm{m,}\mathrm{b} ~=~ \mathrm{b}.</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 11.4
| + | This is equivalent to the implication <math>\mathrm{b} \Rightarrow \mathrm{m}</math> that Peirce would have written in the form <math>\mathrm{b} ~-\!\!\!<~ \mathrm{m}.</math> |
| | | | |
| | + | That is enough to puncture any notion that <math>\mathrm{b}\!</math> and <math>\mathrm{m}\!</math> are statistically independent, but let us continue to develop the plot a bit more. Putting all the general formulas and particular facts together, we arrive at the following summation of the situation in the ''Othello'' case: |
| | | | |
| | + | If the fair sampling condition were true, it would have the following consequence: |
| | | | |
| − | The task before us now is to get very clear about the relationships
| + | {| align="center" cellspacing="6" width="90%" |
| − | among relative terms, relations, and the special cases of relations
| + | | <math>[\mathrm{m,}] ~=~ \frac{[\mathrm{m,}\mathrm{b}]}{[\mathrm{b}]} ~=~ \frac{[\mathrm{b}]}{[\mathrm{b}]} ~=~ \mathfrak{1}.</math> |
| − | that are constituted by equivalence relations, functions, and so on.
| + | |} |
| | | | |
| − | I am optimistic that the some of the tethering material that I spun
| + | On the contrary, we have the following fact: |
| − | along the "Relations In General" (RIG) thread will help us to track
| |
| − | the equivalential and functional properties of special relations in
| |
| − | a way that will not weigh too heavy on the rather capricious lineal
| |
| − | embedding of syntax in 1-dimensional strings on 2-dimensional pages.
| |
| − | But I cannot see far enough ahead to forsee all the consequences of
| |
| − | trying this tack, and so I cannot help but to be a bit experimental.
| |
| | | | |
| − | The first obstacle to get past is the order convention
| + | {| align="center" cellspacing="6" width="90%" |
| − | that Peirce's orientation to relative terms causes him
| + | | <math>[\mathrm{m,}] ~=~ \frac{[\mathrm{m,}\mathbf{1}]}{[\mathbf{1}]} ~=~ \frac{[\mathrm{m}]}{[\mathbf{1}]} ~=~ \frac{4}{7}.\!</math> |
| − | to use for functions. By way of making our discussion
| + | |} |
| − | concrete, and directing our attentions to an immediate
| |
| − | object example, let us say that we desire to represent
| |
| − | the "number of" function, that Peirce denotes by means
| |
| − | of square brackets, by means of a 2-adic relative term,
| |
| − | say 'v', where 'v'(t) = [t] = the number of the term t.
| |
| | | | |
| − | To set the 2-adic relative term 'v' within a suitable context of interpretation,
| + | In sum, it is not the case in the ''Othello'' example that “men are just as apt to be black as things in general”. |
| − | let us suppose that 'v' corresponds to a relation V c R x S, where R is the set
| |
| − | of real numbers and S is a suitable syntactic domain, here described as "terms".
| |
| − | Then the 2-adic relation V is evidently a function from S to R. We might think
| |
| − | to use the plain letter "v" to denote this function, as v : S -> R, but I worry
| |
| − | this may be a chaos waiting to happen. Also, I think that we should anticipate
| |
| − | the very great likelihood that we cannot always assign numbers to every term in
| |
| − | whatever syntactic domain S that we choose, so it is probably better to account
| |
| − | the 2-adic relation V as a partial function from S to R. All things considered,
| |
| − | then, let me try out the following impedimentaria of strategies and compromises.
| |
| | | | |
| − | First, I will adapt the functional arrow notation so that it allows us
| + | Expressed in terms of probabilities: <math>\mathrm{P}(\mathrm{m}) = \frac{4}{7}</math> and <math>\mathrm{P}(\mathrm{b}) = \frac{1}{7}.</math> |
| − | to detach the functional orientation from the order in which the names
| |
| − | of domains are written on the page. Second, I will need to change the | |
| − | notation for "pre-functions", or "partial functions", from one likely
| |
| − | confound to a slightly less likely confound. This gives the scheme:
| |
| | | | |
| − | q : X -> Y means that q is functional at X.
| + | If these were independent terms we would have: <math>\mathrm{P}(\mathrm{m}\mathrm{b}) = \frac{4}{49}.</math> |
| | | | |
| − | q : X <- Y means that q is functional at Y.
| + | In point of fact, however, we have: <math>\mathrm{P}(\mathrm{m}\mathrm{b}) = \mathrm{P}(\mathrm{b}) = \frac{1}{7}.</math> |
| | | | |
| − | q : X ~> Y means that q is pre-functional at X.
| + | Another way to see it is to observe that: <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) = \frac{1}{4}</math> while <math>\mathrm{P}(\mathrm{b}) = \frac{1}{7}.</math> |
| | | | |
| − | q : X <~ Y means that q is pre-functional at Y.
| + | ===Commentary Note 11.23=== |
| | | | |
| − | For now, I will pretend that v is a function in R of S, v : R <- S,
| + | Peirce's description of logical conjunction and conditional probability via the logic of relatives and the mathematics of relations is critical to understanding the relationship between logic and measurement, in effect, the qualitative and quantitative aspects of inquiry. To ground this connection firmly in mind, I will try to sum up as succinctly as possible, in more current notation, the lesson we ought to take away from Peirce's last “number of” example, since I know the account I have given so far may appear to have wandered widely. |
| − | amounting to the functional alias of the 2-adic relation V c R x S,
| |
| − | and associated with the 2-adic relative term 'v' whose relate lies
| |
| − | in the set R of real numbers and whose correlate lies in the set S | |
| − | of syntactic terms. | |
| | | | |
| | + | '''NOF 4.4''' |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>So if men are just as apt to be black as things in general,</p> |
| | + | |- |
| | + | | align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],\!</math> |
| | + | |- |
| | + | | |
| | + | <p>where the difference between <math>[\mathrm{m}]\!</math> and <math>[\mathrm{m,}]\!</math> must not be overlooked.</p> |
| | | | |
| − | LOR. Commentary Note 11.5
| + | <p>(Peirce, CP 3.76).</p> |
| | + | |} |
| | | | |
| | + | In different lights the formula <math>[\mathrm{m,}\mathrm{b}] = [\mathrm{m,}][\mathrm{b}]\!</math> presents itself as an ''aimed arrow'', ''fair sampling'', or ''statistical independence'' condition. The concept of independence was illustrated above by means of a case where independence fails. The details of that counterexample are summarized below. |
| | | | |
| | + | {| align="center" cellpadding="10" |
| | + | | [[Image:LOR 1870 Figure 53.jpg]] || (54) |
| | + | |} |
| | | | |
| − | It always helps me to draw lots of pictures of stuff,
| + | The condition that “men are just as apt to be black as things in general” is expressed in terms of conditional probabilities as <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) = \mathrm{P}(\mathrm{b}),\!</math> which means that the probability of the event <math>\mathrm{b}\!</math> given the event <math>\mathrm{m}\!</math> is equal to the unconditional probability of the event <math>\mathrm{b}.\!</math> |
| − | so let's extract the somewhat overly compressed bits
| |
| − | of the "Relations In General" thread that we'll need | |
| − | right away for the applications to Peirce's 1870 LOR,
| |
| − | and draw what icons we can within the frame of Ascii.
| |
| | | | |
| − | For the immediate present, we may start with 2-adic relations
| + | In the ''Othello'' example, it is enough to observe that <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) = \tfrac{1}{4}\!</math> while <math>\mathrm{P}(\mathrm{b}) = \tfrac{1}{7}\!</math> in order to recognize the bias or dependency of the sampling map. |
| − | and describe the customary species of relations and functions
| |
| − | in terms of their local and numerical incidence properties.
| |
| | | | |
| − | Let P c X x Y be an arbitrary 2-adic relation.
| + | The reduction of a conditional probability to an absolute probability, as <math>\mathrm{P}(A|Z) = \mathrm{P}(A),\!</math> is one of the ways we come to recognize the condition of independence, <math>\mathrm{P}(AZ) = \mathrm{P}(A)P(Z),\!</math> via the definition of conditional probability, <math>\mathrm{P}(A|Z) = \displaystyle{\mathrm{P}(AZ) \over \mathrm{P}(Z)}.\!</math> |
| − | The following properties of P can be defined:
| |
| | | | |
| − | P is "total" at X iff P is (>=1)-regular at X. | + | To recall the derivation, the definition of conditional probability plus the independence condition yields <math>\mathrm{P}(A|Z) = \displaystyle{\mathrm{P}(AZ) \over P(Z)} = \displaystyle{\mathrm{P}(A)\mathrm{P}(Z) \over \mathrm{P}(Z)},\!</math> in short, <math>\mathrm{P}(A|Z) = \mathrm{P}(A).\!</math> |
| | | | |
| − | P is "total" at Y iff P is (>=1)-regular at Y.
| + | As Hamlet discovered, there's a lot to be learned from turning a crank. |
| | | | |
| − | P is "tubular" at X iff P is (=<1)-regular at X.
| + | ===Commentary Note 11.24=== |
| | | | |
| − | P is "tubular" at Y iff P is (=<1)-regular at Y.
| + | We come to the end of the “number of” examples that we found on our agenda at this point in the text: |
| | | | |
| − | To illustrate these properties, let us fashion
| + | '''NOF 4.5''' |
| − | a "generic enough" example of a 2-adic relation,
| |
| − | E c X x Y, where X = Y = {0, 1, ..., 8, 9}, and
| |
| − | where the bigraph picture of E looks like this:
| |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | o o o o o o o o o o X
| + | | |
| − | \ |\ /|\ \ \ | |\
| + | <p>It is to be observed that</p> |
| − | \ | / | \ \ \ | | \ E
| + | |- |
| − | \|/ \| \ \ \| | \
| + | | align="center" | <math>[\mathit{1}] ~=~ 1.</math> |
| − | o o o o o o o o o o Y
| + | |- |
| − | 0 1 2 3 4 5 6 7 8 9
| + | | |
| | + | <p>Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of ''expectation''.</p> |
| | | | |
| − | If we scan along the X dimension we see that the "Y incidence degrees"
| + | <p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.</p> |
| − | of the X nodes 0 through 9 are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0, in order. | |
| | | | |
| − | If we scan along the Y dimension we see that the "X incidence degrees"
| + | <p>(Peirce, CP 3.76 and CE 2, 376).</p> |
| − | of the Y nodes 0 through 9 are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, in order.
| + | |} |
| | | | |
| − | Thus, E is not total at either X or Y,
| + | There are problems with the printing of the text at this point. Let us first recall the conventions we are using in this transcription, in particular, <math>\mathit{1}\!</math> for the italic 1 that signifies the dyadic identity relation and <math>\mathfrak{1}</math> for the “antique figure one” that Peirce defines as <math>\mathit{1}_\infty = \text{something}.</math> |
| − | since there are nodes in both X and Y
| |
| − | having incidence degrees that equal 0.
| |
| | | | |
| − | Also, E is not tubular at either X or Y,
| + | CP 3 gives <math>[\mathit{1}] = \mathfrak{1},</math> which I cannot make sense of. CE 2 gives the 1's in different styles of italics, but reading the equation as <math>[\mathit{1}] = 1,\!</math> makes the best sense if the “1” on the right hand side is read as the numeral “1” that denotes the natural number 1, and not as the absolute term “1” that denotes the universe of discourse. Read this way, <math>[\mathit{1}]\!</math> is the average number of things related by the identity relation <math>\mathit{1}\!</math> to one individual, and so it makes sense that <math>[\mathit{1}] = 1 \in \mathbb{N},</math> where <math>\mathbb{N}</math> is the set of non-negative integers <math>\{ 0, 1, 2, \ldots \}.</math> |
| − | since there exist nodes in both X and Y
| |
| − | having incidence degrees greater than 1.
| |
| | | | |
| − | Clearly, then, E cannot qualify as a pre-function
| + | With respect to the relative term <math>^{\backprime\backprime} \mathit{1} ^{\prime\prime}</math> in the syntactic domain <math>S\!</math> and the number <math>1\!</math> in the non-negative integers <math>\mathbb{N} \subset \mathbb{R},</math> we have: |
| − | or a function on either of its relational domains.
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | <math>v(\mathit{1}) ~=~ [\mathit{1}] ~=~ 1.</math> |
| | + | |} |
| | | | |
| | + | And so the “number of” mapping <math>v : S \to \mathbb{R}</math> has another one of the properties that would be required of an arrow <math>S \to \mathbb{R}.</math> |
| | | | |
| − | LOR. Commentary Note 11.6
| + | The manner in which these arrows and qualified arrows help us to construct a suspension bridge that unifies logic, semiotics, statistics, stochastics, and information theory will be one of the main themes I aim to elaborate throughout the rest of this inquiry. |
| | | | |
| | + | ==Selection 12== |
| | | | |
| | + | ===The Sign of Involution=== |
| | | | |
| − | Let's continue to work our way through the rest of the first
| + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | set of definitions, making up appropriate examples as we go.
| |
| − | | |
| − | | Let P c X x Y be an arbitrary 2-adic relation. | |
| − | | The following properties of P can be defined:
| |
| − | |
| |
| − | | P is "total" at X iff P is (>=1)-regular at X.
| |
| − | |
| |
| − | | P is "total" at Y iff P is (>=1)-regular at Y.
| |
| | | | | | |
| − | | P is "tubular" at X iff P is (=<1)-regular at X. | + | <p>I shall take involution in such a sense that <math>x^y\!</math> will denote everything which is an <math>x\!</math> for every individual of <math>y.\!</math> Thus <math>\mathit{l}^\mathrm{w}\!</math> will be a lover of every woman. Then <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math> will denote whatever stands to every woman in the relation of servant of every lover of hers; and <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math> will denote whatever is a servant of everything that is lover of a woman. So that</p> |
| | + | |- |
| | + | | align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.\!</math> |
| | + | |- |
| | | | | | |
| − | | P is "tubular" at Y iff P is (=<1)-regular at Y.
| + | <p>(Peirce, CP 3.77).</p> |
| | + | |} |
| | | | |
| − | E_1 exemplifies the quality of "totality at X".
| + | ===Commentary Note 12.1=== |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions: |
| − | o o o o o o o o o o X
| |
| − | \ \ |\ /|\ \ \ | |\ \ |
| |
| − | \ \ | / | \ \ \ | | \ \ | E_1
| |
| − | \ \|/ \| \ \ \| | \ \|
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| | | | |
| − | E_2 exemplifies the quality of "totality at Y".
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="40" | <math>X\!</math> is a set singled out in a particular discussion as the ''universe of discourse''. |
| | + | |- |
| | + | | height="40" | <math>W \subseteq X\!</math> is the 1-adic relation, or set, whose elements fall under the absolute term <math>\mathrm{w} = \text{woman}.\!</math> The elements of <math>W\!</math> are sometimes referred to as the ''denotation'' or the set-theoretic ''extension'' of the term <math>\mathrm{w}.\!</math> |
| | + | |- |
| | + | | height="40" | <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{~~ ~~}.\!</math> |
| | + | |- |
| | + | | height="40" | <math>S \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{s} = \text{servant of}\,\underline{~~ ~~}.\!</math> |
| | + | |} |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | {| align="center" cellspacing="6" width="90%" |
| − | o o o o o o o o o o X
| + | | height="40" | <math>\mathsf{W} = (\mathsf{W}_x) = \mathrm{Mat}(W) = \mathrm{Mat}(\mathrm{w})</math> is the 1-dimensional matrix representation of the set <math>W\!</math> and the term <math>\mathrm{w}.\!</math> |
| − | |\ \ |\ /|\ \ \ | |\ \ | + | |- |
| − | | \ \ | / | \ \ \ | | \ \ E_2
| + | | height="40" | <math>\mathsf{L} = (\mathsf{L}_{xy}) = \mathrm{Mat}(L) = \mathrm{Mat}(\mathit{l})~\!</math> is the 2-dimensional matrix representation of the relation <math>L\!</math> and the relative term <math>\mathit{l}.\!</math> |
| − | | \ \|/ \| \ \ \| | \ \
| + | |- |
| − | o o o o o o o o o o Y
| + | | height="40" | <math>\mathsf{S} = (\mathsf{S}_{xy}) = \mathrm{Mat}(S) = \mathrm{Mat}(\mathit{s})\!</math> is the 2-dimensional matrix representation of the relation <math>S\!</math> and the relative term <math>\mathit{s}.~\!</math> |
| − | 0 1 2 3 4 5 6 7 8 9
| + | |} |
| | | | |
| − | E_3 exemplifies the quality of "tubularity at X".
| + | Recalling a few definitions, the ''local flags'' of the relation <math>L\!</math> are given as follows: |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | {| align="center" cellspacing="6" width="90%" |
| − | o o o o o o o o o o X
| + | | |
| − | \ | / \ \ | |
| + | <math>\begin{array}{lll} |
| − | \ | / \ \ | | E_3
| + | u \star L |
| − | \|/ \ \| |
| + | & = & L_{u \,\text{at}\, 1} |
| − | o o o o o o o o o o Y
| + | \\[6pt] |
| − | 0 1 2 3 4 5 6 7 8 9
| + | & = & \{ (u, x) \in L \} |
| | + | \\[6pt] |
| | + | & = & \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}. |
| | + | \\[9pt] |
| | + | L \star v |
| | + | & = & L_{v \,\text{at}\, 2} |
| | + | \\[6pt] |
| | + | & = & \{ (x, v) \in L \} |
| | + | \\[6pt] |
| | + | & = & \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | E_4 exemplifies the quality of "tubularity at Y".
| + | The ''applications'' of the relation <math>L\!</math> are defined as follows: |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | {| align="center" cellspacing="6" width="90%" |
| − | o o o o o o o o o o X
| |
| − | /|\ \ \ |\
| |
| − | / | \ \ \ | \ E_4
| |
| − | / | \ \ \ | \
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| − | | |
| − | | If P c X x Y is tubular at X, then P is known as a "partial function"
| |
| − | | or a "pre-function" from X to Y, frequently signalized by renaming P
| |
| − | | with an alternative lower case name, say "p", and writing p : X ~> Y.
| |
| | | | | | |
| − | | Just by way of formalizing the definition:
| + | <math>\begin{array}{lll} |
| − | |
| + | u \cdot L |
| − | | P is a "pre-function" P : X ~> Y iff P is tubular at X.
| + | & = & \mathrm{proj}_2 (u \star L) |
| − | |
| + | \\[6pt] |
| − | | P is a "pre-function" P : X <~ Y iff P is tubular at Y.
| + | & = & \{ x \in X : (u, x) \in L \} |
| | + | \\[6pt] |
| | + | & = & \text{loved by}~ u. |
| | + | \\[9pt] |
| | + | L \cdot v |
| | + | & = & \mathrm{proj}_1 (L \star v) |
| | + | \\[6pt] |
| | + | & = & \{ x \in X : (x, v) \in L \} |
| | + | \\[6pt] |
| | + | & = & \text{lover of}~ v. |
| | + | \end{array}\!</math> |
| | + | |} |
| | | | |
| − | So, E_3 is a pre-function e_3 : X ~> Y,
| + | ===Commentary Note 12.2=== |
| − | and E_4 is a pre-function e_4 : X <~ Y.
| |
| | | | |
| | + | Let us make a few preliminary observations about the operation of ''logical involution'', as Peirce introduces it here: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| | + | | |
| | + | <p>I shall take involution in such a sense that <math>x^y\!</math> will denote everything which is an <math>x\!</math> for every individual of <math>y.\!</math> Thus <math>\mathit{l}^\mathrm{w}\!</math> will be a lover of every woman.</p> |
| | | | |
| − | LOR. Commentary Note 11.7
| + | <p>(Peirce, CP 3.77).</p> |
| | + | |} |
| | | | |
| | + | In ordinary arithmetic the ''involution'' <math>x^y,\!</math> or the ''exponentiation'' of <math>x\!</math> to the power of <math>y,\!</math> is the repeated application of the multiplier <math>x\!</math> for as many times as there are ones making up the exponent <math>y.\!</math> |
| | | | |
| | + | In analogous fashion, the logical involution <math>\mathit{l}^\mathrm{w}\!</math> is the repeated application of the term <math>\mathit{l}\!</math> for as many times as there are individuals under the term <math>\mathrm{w}.\!</math> According to Peirce's interpretive rules, the repeated applications of the base term <math>\mathit{l}\!</math> are distributed across the individuals of the exponent term <math>\mathrm{w}.\!</math> In particular, the base term <math>\mathit{l}\!</math> is not applied successively in the manner that would give something like “a lover of a lover of … a lover of a woman”. |
| | | | |
| − | We come now to the very special cases of 2-adic relations that are
| + | For example, suppose that a universe of discourse numbers among its contents just three women, <math>\mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.</math> This could be expressed in Peirce's notation by writing: |
| − | known as functions. It will serve a dual purpose on behalf of the
| |
| − | exposition if we take the class of functions as a source of object
| |
| − | examples to clarify the more abstruse concepts in the RIG material.
| |
| | | | |
| − | To begin, let's recall the definition of a local flag:
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>\mathrm{w} ~=~ \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}</math> |
| | + | |} |
| | | | |
| − | L_x@j = {<x_1, ..., x_j, ..., x_k> in L : x_j = x}.
| + | Under these circumstances the following equation would hold: |
| | | | |
| − | In the case of a 2-adic relation L c X_1 x X_2 = X x Y,
| + | {| align="center" cellspacing="6" width="90%" |
| − | we can reap the benefits of a radical simplification in
| + | | height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \mathit{l}^{(\mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime})} ~=~ (\mathit{l}\mathrm{W}^{\prime}), (\mathit{l}\mathrm{W}^{\prime\prime}), (\mathit{l}\mathrm{W}^{\prime\prime\prime}).</math> |
| − | the definitions of the local flags. Also in this case,
| + | |} |
| − | we tend to denote L_u@1 by "L_u@X" and L_v@2 by "L_v@Y".
| |
| | | | |
| − | In the light of these considerations, the local flags of
| + | This says that a lover of every woman in the given universe of discourse is a lover of <math>\mathrm{W}^{\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math> In other words, a lover of every woman in this context is a lover of <math>\mathrm{W}^{\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math> |
| − | a 2-adic relation L c X x Y may be formulated as follows: | |
| | | | |
| − | L_u@X = {<x, y> in L : x = u}
| + | The denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> is a subset of <math>X\!</math> that can be obtained as follows: For each flag of the form <math>L \star x\!</math> with <math>x \in W,\!</math> collect the elements <math>\mathrm{proj}_1 (L \star x)~\!</math> that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together: |
| | | | |
| − | = the set of all ordered pairs in L incident with u in X.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \mathrm{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x</math> |
| | + | |} |
| | | | |
| − | L_v@Y = {<x, y> in L : y = v}
| + | It is very instructive to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> at this point, not the least because it effectively dispels the mystery of the name ''involution''. First, let us make the following observation. To say that <math>j\!</math> is a lover of every woman is to say that <math>j\!</math> loves <math>k\!</math> if <math>k\!</math> is a woman. This can be rendered in symbols as follows: |
| | | | |
| − | = the set of all ordered pairs in L incident with v in Y.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>j ~\text{loves}~ k ~\Leftarrow~ k ~\text{is a woman}</math> |
| | + | |} |
| | | | |
| − | A sufficient illustration is supplied by the earlier example E.
| + | Reading the formula <math>\mathit{l}^\mathrm{w}\!</math> as “<math>j\!</math> loves <math>k\!</math> if <math>k\!</math> is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name. |
| − | | |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| − | o o o o o o o o o o X
| |
| − | \ |\ /|\ \ \ | |\
| |
| − | \ | / | \ \ \ | | \ E
| |
| − | \|/ \| \ \ \| | \
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| − | | |
| − | The local flag E_3@X is displayed here:
| |
| − | | |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| − | o o o o o o o o o o X
| |
| − | /|\
| |
| − | / | \ E_3@X
| |
| − | / | \
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| − | | |
| − | The local flag E_2@Y is displayed here:
| |
| − | | |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| − | o o o o o o o o o o X
| |
| − | \ | /
| |
| − | \ | / E_2@Y
| |
| − | \|/
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| | | | |
| | + | The operations defined by the formulas <math>x^y = z\!</math> and <math>(x\!\Leftarrow\!y) = z</math> for <math>x, y, z \in \mathbb{B} = \{ 0, 1 \}</math> are tabulated below: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math> |
| | + | \begin{array}{ccc} |
| | + | x^y & = & z \\ |
| | + | \hline |
| | + | 0^0 & = & 1 \\ |
| | + | 0^1 & = & 0 \\ |
| | + | 1^0 & = & 1 \\ |
| | + | 1^1 & = & 1 |
| | + | \end{array} |
| | + | \qquad\qquad\qquad |
| | + | \begin{array}{ccc} |
| | + | x\!\Leftarrow\!y & = & z \\ |
| | + | \hline |
| | + | 0\!\Leftarrow\!0 & = & 1 \\ |
| | + | 0\!\Leftarrow\!1 & = & 0 \\ |
| | + | 1\!\Leftarrow\!0 & = & 1 \\ |
| | + | 1\!\Leftarrow\!1 & = & 1 |
| | + | \end{array} |
| | + | </math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 11.8
| + | It is clear that these operations are isomorphic, amounting to the same operation of type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> All that remains is to see how this operation on coefficient values in <math>\mathbb{B}\!</math> induces the corresponding operations on sets and terms. |
| | | | |
| | + | The term <math>\mathit{l}^\mathrm{w}\!</math> determines a selection of individuals from the universe of discourse <math>X\!</math> that may be computed by means of the corresponding operation on coefficient matrices. If the terms <math>\mathit{l}\!</math> and <math>\mathrm{w}\!</math> are represented by the matrices <math>\mathsf{L} = \mathrm{Mat}(\mathit{l})</math> and <math>\mathsf{W} = \mathrm{Mat}(\mathrm{w}),</math> respectively, then the operation on terms that produces the term <math>\mathit{l}^\mathrm{w}\!</math> must be represented by a corresponding operation on matrices, say, <math>\mathsf{L}^\mathsf{W} = \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})},</math> that produces the matrix <math>\mathrm{Mat}(\mathit{l}^\mathrm{w}).</math> In other words, the involution operation on matrices must be defined in such a way that the following equations hold: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>\mathsf{L}^\mathsf{W} ~=~ \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})} ~=~ \mathrm{Mat}(\mathit{l}^\mathrm{w})\!</math> |
| | + | |} |
| | | | |
| − | Now let's re-examine the "numerical incidence properties" of relations,
| + | The fact that <math>\mathit{l}^\mathrm{w}\!</math> denotes the elements of a subset of <math>X\!</math> means that the matrix <math>\mathsf{L}^\mathsf{W}\!</math> is a 1-dimensional array of coefficients in <math>\mathbb{B}\!</math> that is indexed by the elements of <math>X.\!</math> The value of the matrix <math>\mathsf{L}^\mathsf{W}\!</math> at the index <math>{u \in X}\!</math> is written <math>(\mathsf{L}^\mathsf{W})_u\!</math> and computed as follows: |
| − | concentrating on the definitions of the assorted regularity conditions.
| |
| | | | |
| − | | For instance, L is said to be "c-regular at j" if and only if | + | {| align="center" cellspacing="6" width="90%" |
| − | | the cardinality of the local flag L_x@j is c for all x in X_j,
| + | | height="60" | <math>(\mathsf{L}^\mathsf{W})_u ~=~ \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}\!</math> |
| − | | coded in symbols, if and only if |L_x@j| = c for all x in X_j.
| + | |} |
| − | |
| |
| − | | In a similar fashion, one can define the NIP's "<c-regular at j",
| |
| − | | ">c-regular at j", and so on. For ease of reference, I record a
| |
| − | | few of these definitions here:
| |
| − | |
| |
| − | | L is c-regular at j iff |L_x@j| = c for all x in X_j. | |
| − | | | |
| − | | L is (<c)-regular at j iff |L_x@j| < c for all x in X_j.
| |
| − | |
| |
| − | | L is (>c)-regular at j iff |L_x@j| > c for all x in X_j.
| |
| − | |
| |
| − | | L is (=<c)-regular at j iff |L_x@j| =< c for all x in X_j.
| |
| − | |
| |
| − | | L is (>=c)-regular at j iff |L_x@j| >= c for all x in X_j.
| |
| | | | |
| − | Clearly, if any relation is (=<c)-regular on one
| + | ===Commentary Note 12.3=== |
| − | of its domains X_j and also (>=c)-regular on the
| |
| − | same domain, then it must be (=c)-regular on the
| |
| − | affected domain X_j, in effect, c-regular at j.
| |
| | | | |
| − | For example, let G = {r, s, t} and H = {1, ..., 9},
| + | We now have two ways of computing a logical involution that raises a dyadic relative term to the power of a monadic absolute term, for example, <math>\mathit{l}^\mathrm{w}\!</math> for “lover of every woman”. |
| − | and consider the 2-adic relation F c G x H that is
| |
| − | bigraphed here:
| |
| | | | |
| − | r s t
| + | The first method operates in the medium of set theory, expressing the denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> as the intersection of a set of relational applications: |
| − | o o o G
| |
| − | /|\ /|\ /|\
| |
| − | / | \ / | \ / | \ F
| |
| − | / | \ / | \ / | \
| |
| − | o o o o o o o o o H
| |
| − | 1 2 3 4 5 6 7 8 9
| |
| | | | |
| − | We observe that F is 3-regular at G and 1-regular at H.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x\!</math> |
| | + | |} |
| | | | |
| | + | The second method operates in the matrix representation, expressing the value of the matrix <math>\mathsf{L}^\mathsf{W}\!</math> with respect to an argument <math>u\!</math> as a product of coefficient powers: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>(\mathsf{L}^\mathsf{W})_u ~=~ \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}\!</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 11.9
| + | Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved. |
| | | | |
| | + | ====Example 6==== |
| | | | |
| | + | Consider a universe of discourse <math>X\!</math> that is subject to the following data: |
| | | | |
| − | Among the vast variety of conceivable regularities affecting 2-adic relations,
| + | {| align="center" cellspacing="6" width="90%" |
| − | we pay special attention to the c-regularity conditions where c is equal to 1.
| |
| − | | |
| − | | Let P c X x Y be an arbitrary 2-adic relation. | |
| − | | The following properties of P can be defined:
| |
| | | | | | |
| − | | P is "total" at X iff P is (>=1)-regular at X.
| + | <math>\begin{array}{*{15}{c}} |
| − | |
| + | X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \} |
| − | | P is "total" at Y iff P is (>=1)-regular at Y.
| + | \\[6pt] |
| − | |
| + | W & = & \{ & d, & f & \} |
| − | | P is "tubular" at X iff P is (=<1)-regular at X.
| + | \\[6pt] |
| − | | | + | L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} |
| − | | P is "tubular" at Y iff P is (=<1)-regular at Y.
| + | \end{array}</math> |
| | + | |} |
| | | | |
| − | We have already looked at 2-adic relations that
| + | Figure 55 shows the placement of <math>W\!</math> within <math>X\!</math> and the placement of <math>L\!</math> within <math>X \times X.\!</math> |
| − | separately exemplify each of these regularities.
| |
| | | | |
| − | Also, we introduced a few bits of additional terminology and
| + | {| align="center" cellpadding="10" width="100%" |
| − | special-purpose notations for working with tubular relations:
| + | | width="3%" | |
| | + | | width="47%" | [[Image:LOR 1870 Figure 55.jpg]] |
| | + | | width="50%" | (55) |
| | + | |} |
| | | | |
| − | | P is a "pre-function" P : X ~> Y iff P is tubular at X.
| + | To highlight the role of <math>W\!</math> more clearly, the Figure represents the absolute term <math>{}^{\backprime\backprime} \mathrm{w} {}^{\prime\prime}\!</math> by means of the relative term <math>{}^{\backprime\backprime} \mathrm{w}, \! {}^{\prime\prime}\!</math> that conveys the same information. |
| − | |
| |
| − | | P is a "pre-function" P : X <~ Y iff P is tubular at Y.
| |
| | | | |
| − | Thus, we arrive by way of this winding stair at the very special stamps
| + | Computing the denotation of <math>\mathit{l}^\mathrm{w}\!</math> by way of the set-theoretic formula, we can show our work as follows: |
| − | of 2-adic relations P c X x Y that are "total prefunctions" at X (or Y),
| |
| − | "total and tubular" at X (or Y), or "1-regular" at X (or Y), more often
| |
| − | celebrated as "functions" at X (or Y).
| |
| | | | |
| − | | If P is a pre-function P : X ~> Y that happens to be total at X, then P | + | {| align="center" cellspacing="6" width="90%" |
| − | | is known as a "function" from X to Y, typically indicated as P : X -> Y. | + | | height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x ~=~ L \cdot d ~\cap~ L \cdot f ~=~ \{ c, e \} \cap \{ e, g \} ~=~ \{ e \}</math> |
| − | |
| + | |} |
| − | | To say that a relation P c X x Y is totally tubular at X is to say that
| |
| − | | it is 1-regular at X. Thus, we may formalize the following definitions:
| |
| − | |
| |
| − | | P is a "function" p : X -> Y iff P is 1-regular at X.
| |
| − | | | |
| − | | P is a "function" p : X <- Y iff P is 1-regular at Y.
| |
| | | | |
| − | For example, let X = Y = {0, ..., 9} and let F c X x Y be
| + | With the above Figure in mind, we can visualize the computation of <math>(\mathsf{L}^\mathsf{W})_u = \textstyle\prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}\!</math> as follows: |
| − | the 2-adic relation that is depicted in the bigraph below:
| |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | {| align="center" cellspacing="6" width="90%" |
| − | o o o o o o o o o o X
| + | | valign="top" | 1. |
| − | \ / /|\ \ | |\ \
| + | | Pick a specific <math>u\!</math> in the bottom row of the Figure. |
| − | \ / | \ \ | | \ \ F
| + | |- |
| − | / \ / | \ \ | | \ \
| + | | valign="top" | 2. |
| − | o o o o o o o o o o Y
| + | | Pan across the elements <math>v\!</math> in the middle row of the Figure. |
| − | 0 1 2 3 4 5 6 7 8 9 | + | |- |
| | + | | valign="top" | 3. |
| | + | | If <math>u\!</math> links to <math>v\!</math> then <math>\mathsf{L}_{uv} = 1,\!</math> otherwise <math>{\mathsf{L}_{uv} = 0}.\!</math> |
| | + | |- |
| | + | | valign="top" | 4. |
| | + | | If <math>v\!</math> in the middle row links to <math>v\!</math> in the top row then <math>\mathsf{W}_v = 1,\!</math> otherwise <math>\mathsf{W}_v = 0.\!</math> |
| | + | |- |
| | + | | valign="top" | 5. |
| | + | | Compute the value <math>\mathsf{L}_{uv}^{\mathsf{W}_v} = (\mathsf{L}_{uv} \Leftarrow \mathsf{W}_v)\!</math> for each <math>v\!</math> in the middle row. |
| | + | |- |
| | + | | valign="top" | 6. |
| | + | | If any of the values <math>\mathsf{L}_{uv}^{\mathsf{W}_v}\!</math> is <math>0\!</math> then the product <math>\textstyle\prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}\!</math> is <math>0,\!</math> otherwise it is <math>1.\!</math> |
| | + | |} |
| | | | |
| − | We observe that F is a function at Y,
| + | As a general observation, we know that the value of <math>(\mathsf{L}^\mathsf{W})_u\!</math> goes to <math>0~\!</math> just as soon as we find a <math>v \in X\!</math> such that <math>\mathsf{L}_{uv} = 0\!</math> and <math>\mathsf{W}_v = 1,\!</math> in other words, such that <math>(u, v) \notin L\!</math> but <math>v \in W.\!</math> If there is no such <math>v\!</math> then <math>(\mathsf{L}^\mathsf{W})_u = 1.\!</math> |
| − | and we record this fact in either of
| |
| − | the manners F : X <- Y or F : Y -> X.
| |
| | | | |
| | + | Running through the program for each <math>u \in X,\!</math> the only case that produces a non-zero result is <math>(\mathsf{L}^\mathsf{W})_e = 1.\!</math> That portion of the work can be sketched as follows: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>(\mathsf{L}^\mathsf{W})_e ~=~ \prod_{v \in X} \mathsf{L}_{ev}^{\mathsf{W}_v} ~=~ 0^0 \cdot 0^0 \cdot 0^0 \cdot 1^1 \cdot 1^0 \cdot 1^1 \cdot 0^0 \cdot 0^0 \cdot 0^0 ~=~ 1\!</math> |
| | + | |} |
| | | | |
| − | LOR. Commentary Note 11.10
| + | ===Commentary Note 12.4=== |
| | | | |
| | + | Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to a law of exponents in ordinary arithmetic, namely, <math>(a^b)^c = a^{bc}.\!</math> |
| | | | |
| − | | + | {| align="center" cellspacing="6" width="90%" <!--QUOTE--> |
| − | In the case of a 2-adic relation F c X x Y that has
| |
| − | the qualifications of a function f : X -> Y, there
| |
| − | are a number of further differentia that arise:
| |
| − | | |
| − | | f is "surjective" iff f is total at Y.
| |
| | | | | | |
| − | | f is "injective" iff f is tubular at Y. | + | <p>Then <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math> will denote whatever stands to every woman in the relation of servant of every lover of hers; and <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math> will denote whatever is a servant of everything that is lover of a woman. So that</p> |
| | + | |- |
| | + | | align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.\!</math> |
| | + | |- |
| | | | | | |
| − | | f is "bijective" iff f is 1-regular at Y. | + | <p>(Peirce, CP 3.77).</p> |
| | + | |} |
| | | | |
| − | For example, or more precisely, contra example,
| + | Articulating the compound relative term <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math> in set-theoretic terms is fairly immediate: |
| − | the function f : X -> Y that is depicted below | |
| − | is neither total at Y nor tubular at Y, and so
| |
| − | it cannot enjoy any of the properties of being
| |
| − | sur-, or in-, or bi-jective.
| |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | {| align="center" cellspacing="6" width="90%" |
| − | o o o o o o o o o o X
| + | | height="60" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})} ~=~ \bigcap_{x \in LW} \mathrm{proj}_1 (S \star x) ~=~ \bigcap_{x \in LW} S \cdot x\!</math> |
| − | | \ | / \ \ | | \ / | + | |} |
| − | | \ | / \ \ | | \ f
| |
| − | | \|/ \ \| | / \
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| | | | |
| − | A cheap way of getting a surjective function out of any function
| + | On the other hand, translating the compound relative term <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math> into a set-theoretic equivalent is less immediate, the hang-up being that we have yet to define the case of logical involution that raises a dyadic relative term to the power of a dyadic relative term. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example. |
| − | is to reset its codomain to its range. For example, the range | |
| − | of the function f above is Y'= {0, 2, 5, 6, 7, 8, 9}. Thus, | |
| − | if we form a new function g : X -> Y' that looks just like
| |
| − | f on the domain X but is assigned the codomain Y', then
| |
| − | g is surjective, and is described as mapping "onto" Y'.
| |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | ====Example 7==== |
| − | o o o o o o o o o o X
| |
| − | | \ | / \ \ | | \ /
| |
| − | | \ | / \ \ | | \ g
| |
| − | | \|/ \ \| | / \
| |
| − | o o o o o o o Y'
| |
| − | 0 2 5 6 7 8 9
| |
| | | | |
| − | The function h : Y' -> Y is injective.
| + | {| align="center" cellspacing="6" width="90%" |
| | + | | |
| | + | <math>\begin{array}{*{15}{c}} |
| | + | X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i\ & \} |
| | + | \\[6pt] |
| | + | L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} |
| | + | \\[6pt] |
| | + | S & = & \{ & b\!:\!a, & b\!:\!c, & d\!:\!c, & d\!:\!d, & d\!:\!e, & f\!:\!e, & f\!:\!f, & f\!:\!g, & h\!:\!g, & h\!:\!i\ & \} |
| | + | \end{array}</math> |
| | + | |} |
| | | | |
| − | 0 2 5 6 7 8 9
| + | {| align="center" cellpadding="10" width="100%" |
| − | o o o o o o o Y'
| + | | width="3%" | |
| − | | | \ / | \ / | + | | width="47%" | [[Image:LOR 1870 Figure 56.jpg]] |
| − | | | \ | \ h | + | | width="50%" | (56) |
| − | | | / \ | / \ | + | |} |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| | | | |
| − | The function m : X -> Y is bijective.
| + | There is a “servant of every lover of” link between <math>u\!</math> and <math>v\!</math> if and only if <math>u \cdot S ~\supseteq~ L \cdot v.\!</math> But the vacuous inclusions, that is, the cases where <math>L \cdot v = \varnothing,\!</math> have the effect of adding non-intuitive links to the mix. |
| | | | |
| − | 0 1 2 3 4 5 6 7 8 9
| + | The computational requirements are evidently met by the following formula: |
| − | o o o o o o o o o o X
| |
| − | | | | \ / \ / | \ /
| |
| − | | | | \ \ | \ m
| |
| − | | | | / \ / \ | / \
| |
| − | o o o o o o o o o o Y
| |
| − | 0 1 2 3 4 5 6 7 8 9
| |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>(\mathsf{S}^\mathsf{L})_{xy} ~=~ \prod_{p \in X} \mathsf{S}_{xp}^{\mathsf{L}_{py}}\!</math> |
| | + | |} |
| | | | |
| | + | In other words, <math>(\mathsf{S}^\mathsf{L})_{xy} = 0\!</math> if and only if there exists a <math>{p \in X}\!</math> such that <math>\mathsf{S}_{xp} = 0\!</math> and <math>\mathsf{L}_{py} = 1.\!</math> |
| | | | |
| − | LOR. Commentary Note 11.11
| + | ===Commentary Note 12.5=== |
| | | | |
| | + | The equation <math>(\mathit{s}^\mathit{l})^\mathrm{w} = \mathit{s}^{\mathit{l}\mathrm{w}}\!</math> can be verified by establishing the corresponding equation in matrices: |
| | | | |
| | + | {| align="center" cellspacing="6" width="90%" |
| | + | | height="60" | <math>(\mathsf{S}^\mathsf{L})^\mathsf{W} ~=~ \mathsf{S}^{\mathsf{L}\mathsf{W}}</math> |
| | + | |} |
| | | | |
| − | The preceding exercises were intended to beef-up our
| + | If <math>\mathsf{A}</math> and <math>\mathsf{B}</math> are two 1-dimensional matrices over the same index set <math>X\!</math> then <math>\mathsf{A} = \mathsf{B}</math> if and only if <math>\mathsf{A}_x = \mathsf{B}_x</math> for every <math>x \in X.</math> Thus, a routine way to check the validity of <math>(\mathsf{S}^\mathsf{L})^\mathsf{W} = \mathsf{S}^{\mathsf{L}\mathsf{W}}</math> is to check whether the following equation holds for arbitrary <math>x \in X.</math> |
| − | functional literacy skills to the point where we can
| |
| − | read our functional alphabets backwards and forwards
| |
| − | and to ferret out the local functionalites that may | |
| − | be immanent in relative terms no matter where they
| |
| − | locate themselves within the domains of relations.
| |
| − | I am hopeful that these skills will serve us in
| |
| − | good stead as we work to build a catwalk from
| |
| − | Peirce's platform to contemporary scenes on
| |
| − | the logic of relatives, and back again. | |
| | | | |
| − | By way of extending a few very tentative plancks,
| + | {| align="center" cellspacing="6" width="90%" |
| − | let us experiment with the following definitions:
| + | | height="60" | <math>((\mathsf{S}^\mathsf{L})^\mathsf{W})_x ~=~ (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x</math> |
| | + | |} |
| | | | |
| − | 1. A relative term 'p' and the corresponding relation P c X x Y are both
| + | Taking both ends toward the middle, we proceed as follows: |
| − | called "functional on relates" if and only if P is a function at X,
| |
| − | in symbols, P : X -> Y.
| |
| | | | |
| − | 2. A relative term 'p' and the corresponding relation P c X x Y are both
| + | {| align="center" cellspacing="6" width="90%" |
| − | called "functional on correlates" if and only if P is function at Y,
| + | | height="200" | |
| − | in symbols, P : X <- Y.
| + | <math> |
| | + | \begin{array}{*{7}{l}} |
| | + | ((\mathsf{S}^\mathsf{L})^\mathsf{W})_x |
| | + | & = & \displaystyle |
| | + | \prod_{p \in X} (\mathsf{S}^\mathsf{L})_{xp}^{\mathsf{W}_p} |
| | + | & = & \displaystyle |
| | + | \prod_{p \in X} (\prod_{q \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp}})^{\mathsf{W}_p} |
| | + | & = & \displaystyle |
| | + | \prod_{p \in X} \prod_{q \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp}\mathsf{W}_p} |
| | + | \\[36px] |
| | + | (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x |
| | + | & = & \displaystyle |
| | + | \prod_{q \in X} \mathsf{S}_{xq}^{(\mathsf{L}\mathsf{W})_q} |
| | + | & = & \displaystyle |
| | + | \prod_{q \in X} \mathsf{S}_{xq}^{\sum_{p \in X} \mathsf{L}_{qp} \mathsf{W}_p} |
| | + | & = & \displaystyle |
| | + | \prod_{q \in X} \prod_{p \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp} \mathsf{W}_p} |
| | + | \end{array} |
| | + | </math> |
| | + | |} |
| | | | |
| − | When a relation happens to be a function, it may be excusable
| + | The products commute, so the equation holds. In essence, the matrix identity turns on the fact that the law of exponents <math>(a^b)^c = a^{bc}\!</math> in ordinary arithmetic holds when the values <math>a, b, c\!</math> are restricted to the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math> Interpreted as a logical statement, the law of exponents <math>(a^b)^c = a^{bc}\!</math> amounts to a theorem of propositional calculus that is otherwise expressed in the following ways: |
| − | to use the same name for it in both applications, writing out | |
| − | explicit type markers like P : X x Y, P : X -> Y, P : X <- Y,
| |
| − | as the case may be, when and if it serves to clarify matters.
| |
| | | | |
| − | From this current, perhaps transient, perspective, it appears that
| + | {| align="center" cellspacing="6" width="90%" |
| − | our next task is to examine how the known properties of relations
| + | | |
| − | are modified when an aspect of functionality is spied in the mix.
| + | <math>\begin{matrix} |
| − | | + | (a \,\Leftarrow\, b) \,\Leftarrow\, c & = & a \,\Leftarrow\, b \land c |
| − | Let us then return to our various ways of looking at relational composition,
| + | \\[8pt] |
| − | and see what changes and what stays the same when the relations in question
| + | (a >\!\!\!-~ b) >\!\!\!-~ c & = & a >\!\!\!-~ bc |
| − | happen to be functions of various different kinds at some of their domains.
| + | \\[8pt] |
| − | | + | c ~-\!\!\!< (b ~-\!\!\!< a) & = & cb ~-\!\!\!< a |
| − | Here is one generic picture of relational composition,
| + | \\[8pt] |
| − | cast in a style that hews pretty close to the line of
| + | c \,\Rightarrow\, (b \,\Rightarrow\, a) & = & c \land b \,\Rightarrow\, a |
| − | potentials inherent in Peirce's syntax of this period.
| + | \end{matrix}</math> |
| − | | + | |} |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | P o Q |
| |
| − | | ____________^____________ |
| |
| − | | / \ |
| |
| − | | / P Q \ |
| |
| − | | / @ @ \ |
| |
| − | | / / \ / \ \ |
| |
| − | | / / \ / \ \ |
| |
| − | | o o o o o o |
| |
| − | | X X Y Y Z Z | | |
| − | | 1,__# #'p'__$ $'q'__% %1 |
| |
| − | | o o o o o o |
| |
| − | | \ / \ / \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 16. Anything that is a 'p' of a 'q' of Anything
| |
| − | | |
| − | From this we extract the "hypergraph picture" of relational composition:
| |
| − | | |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | P P o Q Q |
| |
| − | | @ @ @ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | o o o o o o |
| |
| − | | X Y X Z Y Z |
| |
| − | | o o o o o o |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \ / \ / / |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | \ / \___ ___/ \ / |
| |
| − | | @ @ @ |
| |
| − | | !1! !1! !1! |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 17. Relational Composition P o Q
| |
| − | | |
| − | All of the relevant information of these Figures can be compressed
| |
| − | into the form of a "spreadsheet", or constraint satisfaction table:
| |
| − | | |
| − | Table 18. Relational Composition P o Q
| |
| − | o---------o---------o---------o---------o
| |
| − | | # !1! | !1! | !1! |
| |
| − | o=========o=========o=========o=========o
| |
| − | | P # X | Y | |
| |
| − | o---------o---------o---------o---------o
| |
| − | | Q # | Y | Z |
| |
| − | o---------o---------o---------o---------o
| |
| − | | P o Q # X | | Z |
| |
| − | o---------o---------o---------o---------o
| |
| − | | |
| − | So the following presents itself as a reasonable plan of study:
| |
| − | Let's see how much easy mileage we can get in our exploration
| |
| − | of functions by adopting the above templates as a paradigm.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.12
| |
| − | | |
| − | | |
| − | | |
| − | Since functions are special cases of 2-adic relations, and since the space
| |
| − | of 2-adic relations is closed under relational composition, in other words,
| |
| − | the composition of a couple of 2-adic relations is again a 2-adic relation,
| |
| − | we know that the relational composition of a couple of functions has to be
| |
| − | a 2-adic relation. If it is also necessarily a function, then we would be
| |
| − | justified in speaking of "functional composition", and also of saying that
| |
| − | the space of functions is closed under this functional form of composition.
| |
| − | | |
| − | Just for novelty's sake, let's try to prove this
| |
| − | for relations that are functional on correlates.
| |
| − | | |
| − | So our task is this: Given a couple of 2-adic relations,
| |
| − | P c X x Y and Q c Y x Z, that are functional on correlates,
| |
| − | P : X <- Y and Q : Y <- Z, we need to determine whether the
| |
| − | relational composition P o Q c X x Z is also P o Q : X <- Z,
| |
| − | or not.
| |
| − | | |
| − | It always helps to begin by recalling the pertinent definitions.
| |
| − | | |
| − | For a 2-adic relation L c X x Y, we have:
| |
| − | | |
| − | L is a "function" L : X <- Y iff L is 1-regular at Y.
| |
| − | | |
| − | As for the definition of relational composition,
| |
| − | it is enough to consider the coefficient of the
| |
| − | composite on an arbitrary ordered pair like i:j.
| |
| − | | |
| − | (P o Q)_ij = Sum_k (P_ik Q_kj).
| |
| − | | |
| − | So let us begin.
| |
| − | | |
| − | P : X <- Y, or P being 1-regular at Y, means that there
| |
| − | is exactly one ordered pair i:k in P for each k in Y.
| |
| − | | |
| − | Q : Y <- Z, or Q being 1-regular at Z, means that there
| |
| − | is exactly one ordered pair k:j in Q for each j in Z.
| |
| − | | |
| − | Thus, there is exactly one ordered pair i:j in P o Q
| |
| − | for each j in Z, which means that P o Q is 1-regular
| |
| − | at Z, and so we have the function P o Q : X <- Z.
| |
| − | | |
| − | And we are done.
| |
| − | | |
| − | Bur proofs after midnight must be checked the next day.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.13
| |
| − | | |
| − | | |
| − | | |
| − | As we make our way toward the foothills of Peirce's 1870 LOR, there
| |
| − | is one piece of equipment that we dare not leave the plains without --
| |
| − | for there is little hope that "l'or dans les montagnes là" will lie
| |
| − | among our prospects without the ready use of its leverage and lifts --
| |
| − | and that is a facility with the utilities that are variously called
| |
| − | "arrows", "morphisms", "homomorphisms", "structure-preserving maps",
| |
| − | and several other names, in accord with the altitude of abstraction
| |
| − | at which one happens to be working, at the given moment in question.
| |
| − | | |
| − | As a middle but not too beaten track, I will lay out the definition
| |
| − | of a morphism in the forms that we will need right off, in a slight
| |
| − | excess of formality at first, but quickly bringing the bird home to
| |
| − | roost on more familiar perches.
| |
| − | | |
| − | Let's say that we have three functions J, K, L
| |
| − | that have the following types and that satisfy
| |
| − | the equation that follows:
| |
| − | | |
| − | | J : X <- Y
| |
| − | |
| |
| − | | K : X <- X x X
| |
| − | |
| |
| − | | L : Y <- Y x Y
| |
| − | |
| |
| − | | J(L(u, v)) = K(Ju, Jv)
| |
| − | | |
| − | Our sagittarian leitmotif can be rubricized in the following slogan:
| |
| − | | |
| − | >-> The image of the ligature is the compound of the images. <-<
| |
| − | | |
| − | Where J is the "image", K is the "compound", and L is the "ligature".
| |
| − | | |
| − | Figure 19 presents us with a picture of the situation in question.
| |
| − | | |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | K L |
| |
| − | | @ @ |
| |
| − | | /|\ /|\ |
| |
| − | | / | \ / | \ |
| |
| − | | v | \ v | \ |
| |
| − | | o o o o o o |
| |
| − | | X X X Y Y Y |
| |
| − | | o o o o o o |
| |
| − | | ^ ^ ^ / / / |
| |
| − | | \ \ \ / / |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \ \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | @ @ @ |
| |
| − | | J J J |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 19. Structure Preserving Transformation J : K <- L
| |
| − | | |
| − | Here, I have used arrowheads to indicate the relational domains
| |
| − | at which each of the relations J, K, L happens to be functional.
| |
| − | | |
| − | Table 20 gives the constraint matrix version of the same thing.
| |
| − | | |
| − | Table 20. Arrow: J(L(u, v)) = K(Ju, Jv)
| |
| − | o---------o---------o---------o---------o
| |
| − | | # J | J | J |
| |
| − | o=========o=========o=========o=========o
| |
| − | | K # X | X | X |
| |
| − | o---------o---------o---------o---------o
| |
| − | | L # Y | Y | Y |
| |
| − | o---------o---------o---------o---------o
| |
| − | | |
| − | One way to read this Table is in terms of the informational redundancies
| |
| − | that it schematizes. In particular, it can be read to say that when one
| |
| − | satisfies the constraint in the L row, along with all of the constraints
| |
| − | in the J columns, then the constraint in the K row is automatically true.
| |
| − | That is one way of understanding the equation: J(L(u, v)) = K(Ju, Jv).
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.14
| |
| − | | |
| − | | |
| − | | |
| − | First, a correction. Ignore for now the
| |
| − | gloss that I gave in regard to Figure 19:
| |
| − | | |
| − | | Here, I have used arrowheads to indicate the relational domains
| |
| − | | at which each of the relations J, K, L happens to be functional.
| |
| − | | |
| − | It is more like the feathers of the arrows that serve to mark the
| |
| − | relational domains at which the relations J, K, L are functional,
| |
| − | but it would take yet another construction to make this precise,
| |
| − | as the feathers are not uniquely appointed but many splintered.
| |
| − | | |
| − | Now, as promised, let's look at a more homely example of a morphism,
| |
| − | say, any one of the mappings J : R -> R (roughly speaking) that are
| |
| − | commonly known as "logarithm functions", where you get to pick your
| |
| − | favorite base. In this case, K(r, s) = r + s and L(u, v) = u . v,
| |
| − | and the defining formula J(L(u, v)) = K(Ju, Jv) comes out looking
| |
| − | like J(u . v) = J(u) + J(v), writing a dot (.) and a plus sign (+)
| |
| − | for the ordinary 2-ary operations of arithmetical multiplication
| |
| − | and arithmetical summation, respectively.
| |
| − | | |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | {+} {.} |
| |
| − | | @ @ |
| |
| − | | /|\ /|\ |
| |
| − | | / | \ / | \ |
| |
| − | | v | \ v | \ |
| |
| − | | o o o o o o |
| |
| − | | X X X Y Y Y |
| |
| − | | o o o o o o |
| |
| − | | ^ ^ ^ / / / |
| |
| − | | \ \ \ / / |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \ \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | @ @ @ |
| |
| − | | J J J |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 21. Logarithm Arrow J : {+} <- {.}
| |
| − | | |
| − | Thus, where the "image" J is the logarithm map,
| |
| − | the "compound" K is the numerical sum, and the
| |
| − | the "ligature" L is the numerical product, one
| |
| − | obtains the immemorial mnemonic motto:
| |
| − | | |
| − | | The image of the product is the sum of the images.
| |
| − | |
| |
| − | | J(u . v) = J(u) + J(v)
| |
| − | |
| |
| − | | J(L(u, v)) = K(Ju, Jv)
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.15
| |
| − | | |
| − | | |
| − | | |
| − | I'm going to elaborate a little further on the subject
| |
| − | of arrows, morphisms, or structure-preserving maps, as
| |
| − | a modest amount of extra work at this point will repay
| |
| − | ample dividends when it comes time to revisit Peirce's
| |
| − | "number of" function on logical terms.
| |
| − | | |
| − | The "structure" that is being preserved by a structure-preserving map
| |
| − | is just the structure that we all know and love as a 3-adic relation.
| |
| − | Very typically, it will be the type of 3-adic relation that defines
| |
| − | the type of 2-ary operation that obeys the rules of a mathematical
| |
| − | structure that is known as a "group", that is, a structure that
| |
| − | satisfies the axioms for closure, associativity, identities,
| |
| − | and inverses.
| |
| − | | |
| − | For example, in the previous case of the logarithm map J, we have the data:
| |
| − | | |
| − | | J : R <- R (properly restricted)
| |
| − | |
| |
| − | | K : R <- R x R, where K(r, s) = r + s
| |
| − | |
| |
| − | | L : R <- R x R, where L(u, v) = u . v
| |
| − | | |
| − | Real number addition and real number multiplication (suitably restricted)
| |
| − | are examples of group operations. If we write the sign of each operation
| |
| − | in braces as a name for the 3-adic relation that constitutes or defines
| |
| − | the corresponding group, then we have the following set-up:
| |
| − | | |
| − | | J : {+} <- {.}
| |
| − | |
| |
| − | | {+} c R x R x R
| |
| − | |
| |
| − | | {.} c R x R x R
| |
| − | | |
| − | In many cases, one finds that both groups are written with the same
| |
| − | sign of operation, typically ".", "+", "*", or simple concatenation,
| |
| − | but they remain in general distinct whether considered as operations
| |
| − | or as relations, no matter what signs of operation are used. In such
| |
| − | a setting, our chiasmatic theme may run a bit like these two variants:
| |
| − | | |
| − | | The image of the sum is the sum of the images.
| |
| − | |
| |
| − | | The image of the product is the product of the images.
| |
| − | | |
| − | Figure 22 presents a generic picture for groups G and H.
| |
| − | | |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | G H |
| |
| − | | @ @ |
| |
| − | | /|\ /|\ |
| |
| − | | / | \ / | \ |
| |
| − | | v | \ v | \ |
| |
| − | | o o o o o o |
| |
| − | | X X X Y Y Y |
| |
| − | | o o o o o o |
| |
| − | | ^ ^ ^ / / / |
| |
| − | | \ \ \ / / |
| |
| − | | \ \ / \ / / |
| |
| − | | \ \ \ / |
| |
| − | | \ / \ / \ / |
| |
| − | | @ @ @ |
| |
| − | | J J J |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 22. Group Homomorphism J : G <- H
| |
| − | | |
| − | In a setting where both groups are written with a plus sign,
| |
| − | perhaps even constituting the very same group, the defining
| |
| − | formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the
| |
| − | shape J(u + v) = Ju + Jv, which looks very analogous to the
| |
| − | distributive multiplication of a sum (u + v) by a factor J.
| |
| − | Hence another popular name for a morphism: a "linear" map.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.16
| |
| − | | |
| − | | |
| − | | |
| − | I think that we have enough material on morphisms now
| |
| − | to go back and cast a more studied eye on what Peirce
| |
| − | is doing with that "number of" function, the one that
| |
| − | we apply to a logical term 't', absolute or relative
| |
| − | of any number of correlates, by writing it in square
| |
| − | brackets, as ['t']. It is frequently convenient to
| |
| − | have a prefix notation for this function, and since
| |
| − | Peirce reserves 'n' to signify 'not', I will try to
| |
| − | use 'v', personally thinking of it as a Greek 'nu',
| |
| − | which stands for frequency in physics, and which
| |
| − | kind of makes sense if we think of frequency as
| |
| − | it's habitual in statistics. End of mnemonics.
| |
| − | | |
| − | My plan will be nothing less plodding than to work through
| |
| − | all of the principal statements that Peirce has made about
| |
| − | the "number of" function up to our present stopping place
| |
| − | in the paper, namely, those that I collected once before
| |
| − | and placed at this location:
| |
| − | | |
| − | LOR.COM 11.2. http://stderr.org/pipermail/inquiry/2004-November/001814.html
| |
| − | | |
| − | NOF 1.
| |
| − | | |
| − | | I propose to assign to all logical terms, numbers;
| |
| − | | to an absolute term, the number of individuals it denotes;
| |
| − | | to a relative term, the average number of things so related
| |
| − | | to one individual.
| |
| − | |
| |
| − | | Thus in a universe of perfect men ('men'),
| |
| − | | the number of "tooth of" would be 32.
| |
| − | |
| |
| − | | The number of a relative with two correlates would be the
| |
| − | | average number of things so related to a pair of individuals;
| |
| − | | and so on for relatives of higher numbers of correlates.
| |
| − | |
| |
| − | | I propose to denote the number of a logical term by
| |
| − | | enclosing the term in square brackets, thus ['t'].
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.65
| |
| − | | |
| − | We may formalize the role of the "number of" function by assigning it
| |
| − | a local habitation and a name 'v' : S -> R, where S is a suitable set
| |
| − | of signs, called the "syntactic domain", that is ample enough to hold
| |
| − | all of the terms that we might wish to number in a given discussion,
| |
| − | and where R is the real number domain.
| |
| − | | |
| − | Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".
| |
| − | Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect
| |
| − | human dentition, the number of the relative term "tooth of ---" is equal to
| |
| − | the number of teeth of humans divided by the number of humans, that is, 32.
| |
| − | | |
| − | The 2-adic relative term 't' determines a 2-adic relation T c U x V,
| |
| − | where U and V are two universes of discourse, possibly the same one,
| |
| − | that hold among other things all of the teeth and all of the people
| |
| − | that happen to be under discussion, respectively.
| |
| − | | |
| − | A rough indication of the bigraph for T
| |
| − | might be drawn as follows, where I have
| |
| − | tried to sketch in just the toothy part
| |
| − | of U and the peoply part of V.
| |
| − | | |
| − | t_1 t_32 t_33 t_64 t_65 t_96 ... ...
| |
| − | o ... o o ... o o ... o o ... o U
| |
| − | \ | / \ | / \ | / \ | /
| |
| − | \ | / \ | / \ | / \ | / T
| |
| − | \|/ \|/ \|/ \|/
| |
| − | o o o o V
| |
| − | m_1 m_2 m_3 ...
| |
| − | | |
| − | Notice that the "number of" function 'v' : S -> R
| |
| − | needs the data that is represented by this entire
| |
| − | bigraph for T in order to compute the value ['t'].
| |
| − | | |
| − | Finally, one observes that this component of T is a function
| |
| − | in the direction T : U -> V, since we are counting only those
| |
| − | teeth that ideally occupy one and only one mouth of a creature.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.17
| |
| − | | |
| − | | |
| − | | |
| − | I think that the reader is beginning to get an inkling of the crucial importance of
| |
| − | the "number of" map in Peirce's way of looking at logic, for it's one of the plancks
| |
| − | in the bridge from logic to the theories of probability, statistics, and information,
| |
| − | in which logic forms but a limiting case at one scenic turnout on the expanding vista.
| |
| − | It is, as a matter of necessity and a matter of fact, practically speaking, at any rate,
| |
| − | one way that Peirce forges a link between the "eternal", logical, or rational realm and
| |
| − | the "secular", empirical, or real domain.
| |
| − | | |
| − | With that little bit of encouragement and exhortation,
| |
| − | let us return to the nitty gritty details of the text.
| |
| − | | |
| − | NOF 2.
| |
| − | | |
| − | | But not only do the significations of '=' and '<' here adopted fulfill all
| |
| − | | absolute requirements, but they have the supererogatory virtue of being very
| |
| − | | nearly the same as the common significations. Equality is, in fact, nothing
| |
| − | | but the identity of two numbers; numbers that are equal are those which are
| |
| − | | predicable of the same collections, just as terms that are identical are those
| |
| − | | which are predicable of the same classes. So, to write 5 < 7 is to say that 5
| |
| − | | is part of 7, just as to write f < m is to say that Frenchmen are part of men.
| |
| − | | Indeed, if f < m, then the number of Frenchmen is less than the number of men,
| |
| − | | and if v = p, then the number of Vice-Presidents is equal to the number of
| |
| − | | Presidents of the Senate; so that the numbers may always be substituted
| |
| − | | for the terms themselves, in case no signs of operation occur in the
| |
| − | | equations or inequalities.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.66
| |
| − | | |
| − | Peirce is here remarking on the principle that the
| |
| − | measure 'v' on terms "preserves" or "respects" the
| |
| − | prevailing implication, inclusion, or subsumption
| |
| − | relations that impose an ordering on those terms.
| |
| − | | |
| − | In these initiatory passages of the text, Peirce is using a single symbol "<"
| |
| − | to denote the usual linear ordering on numbers, but also what amounts to the
| |
| − | implication ordering on logical terms and the inclusion ordering on classes.
| |
| − | Later, of course, he will introduce distinctive symbols for logical orders.
| |
| − | | |
| − | Now, the links among terms, sets, and numbers can be pursued in all directions,
| |
| − | and Peirce has already indicated in an earlier paper how he would "construct"
| |
| − | the integers from sets, that is, from the aggregate denotations of terms.
| |
| − | | |
| − | We will get back to that at another time.
| |
| − | | |
| − | In the immediate example, we have this sort of statement:
| |
| − | | |
| − | "if f < m, then the number of Frenchmen is less than the number of men"
| |
| − | | |
| − | In symbolic form, this would be written:
| |
| − | | |
| − | f < m => [f] < [m]
| |
| − | | |
| − | Here, the "<" on the left is a logical ordering on syntactic terms
| |
| − | while the "<" on the right is an arithmetic ordering on real numbers.
| |
| − | | |
| − | The type of principle that comes up here is usually discussed
| |
| − | under the question of whether a map between two ordered sets
| |
| − | is "order-preserving" or not. The general type of question
| |
| − | may be formalized in the following way.
| |
| − | | |
| − | Let X_1 be a set with an ordering denoted by "<_1".
| |
| − | Let X_2 be a set with an ordering denoted by "<_2".
| |
| − | | |
| − | What makes an ordering what it is will commonly be
| |
| − | a set of axioms that defines the properties of the
| |
| − | order relation in question. Since one frequently
| |
| − | has occasion to view the same set in the light of
| |
| − | several different order relations, one will often
| |
| − | resort to explicit forms like (X, <_1), (X, <_2),
| |
| − | and so on, to invoke a set with a given ordering.
| |
| − | | |
| − | A map F : (X_1, <_1) -> (X_2, <_2) is "order-preserving"
| |
| − | if and only if a statement of a particular form holds
| |
| − | for all x and y in (X_1, <_1), specifically, this:
| |
| − | | |
| − | x <_1 y => Fx <_2 Fy
| |
| − | | |
| − | The action of the "number of" map 'v' : (S, <_1) -> (R, <_2)
| |
| − | has just this character, as exemplified by its application to
| |
| − | the case where x = f = "frenchman" and y = m = "man", like so:
| |
| − | | |
| − | | f < m => [f] < [m]
| |
| − | |
| |
| − | | f < m => 'v'f < 'v'm
| |
| − | | |
| − | Here, to be more exacting, we may interpret the "<" on the left
| |
| − | as "proper subsumption", that is, excluding the equality case,
| |
| − | while we read the "<" on the right as the usual "less than".
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.18
| |
| − | | |
| − | | |
| − | | |
| − | There is a comment that I ought to make on the concept of
| |
| − | a "structure preserving map", including as a special case
| |
| − | the idea of an "order-preserving map". It seems to be a
| |
| − | peculiarity of mathematical usage in general -- at least,
| |
| − | I don't think it's just me -- that "preserving structure"
| |
| − | always means "preserving 'some', not of necessity 'all',
| |
| − | of the structure in question". People sometimes express
| |
| − | this by speaking of "structure preservation in measure",
| |
| − | the implication being that any property that is amenable
| |
| − | to being qualified in manner is potentially amenable to
| |
| − | being quantified in degree, perhaps in such a way as to
| |
| − | answer questions like "How structure-preserving is it?".
| |
| − | | |
| − | Let's see how this remark applies to the order-preserving property of
| |
| − | the "number of" mapping 'v' : S -> R. For any pair of absolute terms
| |
| − | x and y in the syntactic domain S, we have the following implications,
| |
| − | where "-<" denotes the logical subsumption relation on terms and "=<"
| |
| − | is the "less than or equal to" relation on the real number domain R.
| |
| − | | |
| − | x -< y => 'v'x =< 'v'y
| |
| − | | |
| − | Equivalently:
| |
| − | | |
| − | x -< y => [x] =< [y]
| |
| − | | |
| − | It is easy to see that nowhere near all of the distinctions that make up
| |
| − | the structure of the ordering on the left hand side will be preserved as
| |
| − | one passes to the right hand side of these implication statements, but
| |
| − | that is not required in order to call the map 'v' "order-preserving",
| |
| − | or what is also known as an "order morphism".
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.19
| |
| − | | |
| − | | |
| − | | |
| − | Up to this point in the LOR of 1870, Peirce has introduced the
| |
| − | "number of" measure on logical terms and discussed the extent
| |
| − | to which this measure, 'v' : S -> R such that 'v' : s ~> [s],
| |
| − | exhibits a couple of important measure-theoretic principles:
| |
| − | | |
| − | 1. The "number of" map exhibits a certain type of "uniformity property",
| |
| − | whereby the value of the measure on a uniformly qualified population
| |
| − | is in fact actualized by each member of the population.
| |
| − | | |
| − | 2. The "number of" map satisfies an "order morphism principle", whereby
| |
| − | the illative partial ordering of logical terms is reflected up to a
| |
| − | partial extent by the arithmetical linear ordering of their measures.
| |
| − | | |
| − | Peirce next takes up the action of the "number of" map on the two types of,
| |
| − | loosely speaking, "additive" operations that we normally consider in logic.
| |
| − | | |
| − | NOF 3.
| |
| − | | |
| − | | It is plain that both the regular non-invertible addition and the
| |
| − | | invertible addition satisfy the absolute conditions. (CP 3.67).
| |
| − | | |
| − | The "regular non-invertible addition" is signified by "+,",
| |
| − | corresponding to what we'd call the inclusive disjunction
| |
| − | of logical terms or the union of their extensions as sets.
| |
| − | | |
| − | The "invertible addition" is signified in algebra by "+",
| |
| − | corresponding to what we'd call the exclusive disjunction
| |
| − | of logical terms or the symmetric difference of their sets,
| |
| − | ignoring many details and nuances that are often important,
| |
| − | of course.
| |
| − | | |
| − | | But the notation has other recommendations. The conception of 'taking together'
| |
| − | | involved in these processes is strongly analogous to that of summation, the sum
| |
| − | | of 2 and 5, for example, being the number of a collection which consists of a
| |
| − | | collection of two and a collection of five. (CP 3.67).
| |
| − | | |
| − | A full interpretation of this remark will require us to pick up the precise
| |
| − | technical sense in which Peirce is using the word "collection", and that will
| |
| − | take us back to his logical reconstruction of certain aspects of number theory,
| |
| − | all of which I am putting off to another time, but it is still possible to get
| |
| − | a rough sense of what he's saying relative to the present frame of discussion.
| |
| − | | |
| − | The "number of" map 'v' : S -> R evidently induces
| |
| − | some sort of morphism with respect to logical sums.
| |
| − | If this were straightforwardly true, we could write:
| |
| − | | |
| − | |?| 'v'(x +, y) = 'v'x + 'v'y
| |
| − | |?|
| |
| − | |?| Equivalently:
| |
| − | |?|
| |
| − | |?| [x +, y] = [x] + [y]
| |
| − | | |
| − | Of course, things are just not that simple in the case
| |
| − | of inclusive disjunction and set-theoretic unions, so
| |
| − | we'd "probably" invent a word like "sub-additive" to
| |
| − | describe the principle that does hold here, namely:
| |
| − | | |
| − | | 'v'(x +, y) =< 'v'x + 'v'y
| |
| − | |
| |
| − | | Equivalently:
| |
| − | |
| |
| − | | [x +, y] =< [x] + [y]
| |
| − | | |
| − | This is why Peirce trims his discussion of this point with the following hedge:
| |
| − | | |
| − | | Any logical equation or inequality in which no operation but addition
| |
| − | | is involved may be converted into a numerical equation or inequality by
| |
| − | | substituting the numbers of the several terms for the terms themselves --
| |
| − | | provided all the terms summed are mutually exclusive. (CP 3.67).
| |
| − | | |
| − | Finally, a morphism with respect to addition,
| |
| − | even a contingently qualified one, must do the
| |
| − | right stuff on behalf of the additive identity:
| |
| − | | |
| − | | Addition being taken in this sense,
| |
| − | |'nothing' is to be denoted by 'zero',
| |
| − | | for then:
| |
| − | |
| |
| − | | x +, 0 = x
| |
| − | |
| |
| − | | whatever is denoted by x; and this is the definition
| |
| − | | of 'zero'. This interpretation is given by Boole, and
| |
| − | | is very neat, on account of the resemblance between the
| |
| − | | ordinary conception of 'zero' and that of nothing, and
| |
| − | | because we shall thus have
| |
| − | |
| |
| − | | [0] = 0.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.67
| |
| − | | |
| − | With respect to the nullity 0 in S and the number 0 in R, we have:
| |
| − | | |
| − | 'v'0 = [0] = 0.
| |
| − | | |
| − | In sum, therefor, it also serves that only preserves
| |
| − | a due respect for the function of a vacuum in nature.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.20
| |
| − | | |
| − | | |
| − | | |
| − | We arrive at the last, for the time being, of
| |
| − | Peirce's statements about the "number of" map.
| |
| − | | |
| − | NOF 4.
| |
| − | | |
| − | | The conception of multiplication we have adopted is
| |
| − | | that of the application of one relation to another. ...
| |
| − | |
| |
| − | | Even ordinary numerical multiplication involves the same idea,
| |
| − | | for 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs,
| |
| − | | where "triplet of" and "pair of" are evidently relatives.
| |
| − | |
| |
| − | | If we have an equation of the form:
| |
| − | |
| |
| − | | xy = z
| |
| − | |
| |
| − | | and there are just as many x's per y as there are
| |
| − | |'per' things, things of the universe, then we have
| |
| − | | also the arithmetical equation:
| |
| − | |
| |
| − | | [x][y] = [z].
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| − | | |
| − | Peirce is here observing what we might dub a "contingent morphism"
| |
| − | or a "skeptraphotic arrow", if you will. Provided that a certain
| |
| − | condition, to be named and, what is more hopeful, to be clarified
| |
| − | in short order, happens to be satisfied, we would find it holding
| |
| − | that the "number of" map 'v' : S -> R such that 'v's = [s] serves
| |
| − | to preserve the multiplication of relative terms, that is as much
| |
| − | to say, the composition of relations, in the form: [xy] = [x][y].
| |
| − | | |
| − | So let us try to uncross Peirce's manifestly chiasmatic encryption
| |
| − | of the condition that is called on in support of this preservation.
| |
| − | | |
| − | Proviso for [xy] = [x][y] --
| |
| − | | |
| − | | there are just as many x's per y
| |
| − | | as there are 'per' things<,>
| |
| − | | things of the universe ...
| |
| − | | |
| − | I have placed angle brackets around
| |
| − | a comma that CP shows but CE omits,
| |
| − | not that it helps much either way.
| |
| − | So let us resort to the example:
| |
| − | | |
| − | | For instance, if our universe is perfect men, and there
| |
| − | | are as many teeth to a Frenchman (perfect understood)
| |
| − | | as there are to any one of the universe, then:
| |
| − | |
| |
| − | | ['t'][f] = ['t'f]
| |
| − | |
| |
| − | | holds arithmetically. (CP 3.76).
| |
| − | | |
| − | Now that is something that we can sink our teeth into,
| |
| − | and trace the bigraph representation of the situation.
| |
| − | In order to do this, it will help to recall our first
| |
| − | examination of the "tooth of" relation, and to adjust
| |
| − | the picture that we sketched of it on that occasion.
| |
| − | | |
| − | Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".
| |
| − | Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect
| |
| − | human dentition, the number of the relative term "tooth of ---" is equal to
| |
| − | the number of teeth of humans divided by the number of humans, that is, 32.
| |
| − | | |
| − | The 2-adic relative term 't' determines a 2-adic relation T c U x V,
| |
| − | where U and V are two universes of discourse, possibly the same one,
| |
| − | that hold among other things all of the teeth and all of the people
| |
| − | that happen to be under discussion, respectively. To make the case
| |
| − | as simple as we can and still cover the point, let's say that there
| |
| − | are just four people in our initial universe of discourse, and that
| |
| − | just two of them are French. The bigraphic composition below shows
| |
| − | all of the pertinent facts of the case.
| |
| − | | |
| − | T_1 T_32 T_33 T_64 T_65 T_96 T_97 T_128
| |
| − | o ... o o ... o o ... o o ... o U
| |
| − | \ | / \ | / \ | / \ | /
| |
| − | \ | / \ | / \ | / \ | / 't'
| |
| − | \|/ \|/ \|/ \|/
| |
| − | o o o o V = m = 1
| |
| − | | |
| |
| − | | | 'f'
| |
| − | | |
| |
| − | o o o o V = m = 1
| |
| − | J K L M
| |
| − | | |
| − | Here, the order of relational composition flows up the page.
| |
| − | For convenience, the absolute term f = "frenchman" has been
| |
| − | converted by using the comma functor to give the idempotent
| |
| − | representation 'f' = f, = "frenchman that is ---", and thus
| |
| − | it can be taken as a selective from the universe of mankind.
| |
| − | | |
| − | By way of a legend for the figure, we have the following data:
| |
| − | | |
| − | | m = J +, K +, L +, M = 1
| |
| − | |
| |
| − | | f = K +, M
| |
| − | |
| |
| − | |'f' = K:K +, M:M
| |
| − | |
| |
| − | |'t' = (T_001 +, ... +, T_032):J +,
| |
| − | | (T_033 +, ... +, T_064):K +,
| |
| − | | (T_065 +, ... +, T_096):L +,
| |
| − | | (T_097 +, ... +, T_128):M
| |
| − | | |
| − | Now let's see if we can use this picture
| |
| − | to make sense of the following statement:
| |
| − | | |
| − | | For instance, if our universe is perfect men, and there
| |
| − | | are as many teeth to a Frenchman (perfect understood)
| |
| − | | as there are to any one of the universe, then:
| |
| − | |
| |
| − | | ['t'][f] = ['t'f]
| |
| − | |
| |
| − | | holds arithmetically. (CP 3.76).
| |
| − | | |
| − | In the lingua franca of statistics, Peirce is saying this:
| |
| − | That if the population of Frenchmen is a "fair sample" of
| |
| − | the general population with regard to dentition, then the
| |
| − | morphic equation ['t'f] = ['t'][f], whose transpose gives
| |
| − | ['t'] = ['t'f]/[f], is every bite as true as the defining
| |
| − | equation in this circumstance, namely, ['t'] = ['t'm]/[m].
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.21
| |
| − | | |
| − | | |
| − | | |
| − | One more example and one more general observation, and then we will
| |
| − | be all caught up with our homework on Peirce's "number of" function.
| |
| − | | |
| − | | So if men are just as apt to be black as things in general:
| |
| − | |
| |
| − | | [m,][b] = [m,b]
| |
| − | |
| |
| − | | where the difference between [m] and [m,] must not be overlooked.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| − | | |
| − | The protasis, "men are just as apt to be black as things in general",
| |
| − | is elliptic in structure, and presents us with a potential ambiguity.
| |
| − | If we had no further clue to its meaning, it might be read as either:
| |
| − | | |
| − | 1. Men are just as apt to be black as things in general are apt to be black.
| |
| − | | |
| − | 2. Men are just as apt to be black as men are apt to be things in general.
| |
| − | | |
| − | The second interpretation, if grammatical, is pointless to state,
| |
| − | since it equates a proper contingency with an absolute certainty.
| |
| − | | |
| − | So I think it is safe to assume this paraphrase of what Peirce intends:
| |
| − | | |
| − | 3. Men are just as likely to be black as things in general are likely to be black.
| |
| − | | |
| − | Stated in terms of the conditional probability:
| |
| − | | |
| − | 4. P(b|m) = P(b)
| |
| − | | |
| − | From the definition of conditional probability:
| |
| − | | |
| − | 5. P(b|m) = P(b m)/P(m)
| |
| − | | |
| − | Equivalently:
| |
| − | | |
| − | 6. P(b m) = P(b|m)P(m)
| |
| − | | |
| − | Thus we may derive the equivalent statement:
| |
| − | | |
| − | 7. P(b m) = P(b|m)P(m) = P(b)P(m)
| |
| − | | |
| − | And this, of course, is the definition of independent events, as
| |
| − | applied to the event of being Black and the event of being a Man.
| |
| − | | |
| − | It seems like a likely guess, then, that this is the content of Peirce's
| |
| − | statement about frequencies, [m,b] = [m,][b], in this case normalized to
| |
| − | produce the equivalent statement about probabilities: P(m b) = P(m)P(b).
| |
| − | | |
| − | Let's see if this checks out.
| |
| − | | |
| − | Let n be the number of things in general, in Peirce's lingo, n = [1].
| |
| − | On the assumption that m and b are associated with independent events,
| |
| − | we get [m,b] = P(m b)n = P(m)P(b)n = P(m)[b] = [m,][b], so we have to
| |
| − | interpret [m,] = "the average number of men per things in general" as
| |
| − | P(m) = the probability of a thing in general being a man. Seems okay.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.22
| |
| − | | |
| − | | |
| − | | |
| − | Let's look at that last example from a different angle.
| |
| − | | |
| − | | So if men are just as apt to be black as things in general:
| |
| − | |
| |
| − | | [m,][b] = [m,b]
| |
| − | |
| |
| − | | where the difference between [m] and [m,] must not be overlooked.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| − | | |
| − | In different lights the formula [m,b] = [m,][b] presents itself
| |
| − | as an "aimed arrow", "fair sample", or "independence" condition.
| |
| − | | |
| − | The example apparently assumes a universe of "things in general",
| |
| − | encompassing among other things the denotations of the absolute
| |
| − | terms m = "man" and b = "black". That suggests to me that we
| |
| − | might well illustrate this case in relief, by returning to
| |
| − | our earlier staging of 'Othello' and seeing how well that
| |
| − | universe of dramatic discourse observes the premiss that
| |
| − | "men are just as apt to be black as things in general".
| |
| − | | |
| − | Here is the relevant data:
| |
| − | | |
| − | | 1 = B +, C +, D +, E +, I +, J +, O
| |
| − | |
| |
| − | | b = O
| |
| − | |
| |
| − | | m = C +, I +, J +, O
| |
| − | |
| |
| − | | 1, = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
| |
| − | |
| |
| − | | b, = O:O
| |
| − | |
| |
| − | | m, = C:C +, I:I +, J:J +, O:O
| |
| − | | |
| − | The "fair sampling" or "episkeptral arrow" condition is tantamount to this:
| |
| − | "Men are just as apt to be black as things in general are apt to be black".
| |
| − | In other words, men are a fair sample of things in general with respect to
| |
| − | the factor of being black.
| |
| − | | |
| − | Should this hold, the consequence would be:
| |
| − | | |
| − | [m,b] = [m,][b].
| |
| − | | |
| − | When [b] is not zero, we obtain the result:
| |
| − | | |
| − | [m,] = [m,b]/[b].
| |
| − | | |
| − | Once again, the absolute term b = "black" is most felicitously depicted
| |
| − | by way of its idempotent representation 'b' = b, = "black that is ---",
| |
| − | and thus it can be taken as a selective from the universe of discourse.
| |
| − | | |
| − | Here is the bigraph for the composition:
| |
| − | | |
| − | m,b = "man that is black",
| |
| − | | |
| − | here represented in the equivalent form:
| |
| − | | |
| − | m,b, = "man that is black that is ---".
| |
| − | | |
| − | B C D E I J O
| |
| − | o o o o o o o 1
| |
| − | | | | |
| |
| − | | | | | m,
| |
| − | | | | |
| |
| − | o o o o o o o 1
| |
| − | |
| |
| − | | b,
| |
| − | |
| |
| − | o o o o o o o 1
| |
| − | B C D E I J O
| |
| − | | |
| − | Thus we observe one of the more factitious facts
| |
| − | that hold in this universe of discourse, namely:
| |
| − | | |
| − | m,b = b.
| |
| − | | |
| − | Another way of saying that is:
| |
| − | | |
| − | b -< m.
| |
| − | | |
| − | That in itself is enough to puncture any notion
| |
| − | that b and m are statistically independent, but
| |
| − | let us continue to develop the plot a bit more.
| |
| − | | |
| − | Putting all of the general formulas and particular facts together,
| |
| − | we arrive at following summation of situation in the Othello case:
| |
| − | | |
| − | If the fair sampling condition holds:
| |
| − | | |
| − | [m,] = [m,b]/[b] = [b]/[b] = `1`,
| |
| − | | |
| − | In fact, however, it is the case that:
| |
| − | | |
| − | [m,] = [m,1]/[1] = [m]/[1] = 4/7.
| |
| − | | |
| − | In sum, it is not the case in the Othello example that
| |
| − | "men are just as apt to be black as things in general".
| |
| − | | |
| − | Expressed in terms of probabilities: P(m) = 4/7 and P(b) = 1/7.
| |
| − | | |
| − | If these were independent we'd have: P(mb) = 4/49.
| |
| − | | |
| − | On the contrary, P(mb) = P(b) = 1/7.
| |
| − | | |
| − | Another way to see it is as follows: P(b|m) = 1/4 while P(b) = 1/7.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.23
| |
| − | | |
| − | | |
| − | | |
| − | Let me try to sum up as succinctly as possible the lesson
| |
| − | that we ought to take away from Peirce's last "number of"
| |
| − | example, since I know that the account that I have given
| |
| − | of it so far may appear to have wandered rather widely.
| |
| − | | |
| − | | So if men are just as apt to be black as things in general:
| |
| − | |
| |
| − | | [m,][b] = [m,b]
| |
| − | |
| |
| − | | where the difference between [m] and [m,] must not be overlooked.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| − | | |
| − | In different lights the formula [m,b] = [m,][b] presents itself
| |
| − | as an "aimed arrow", "fair sample", or "independence" condition.
| |
| − | I had taken the tack of illustrating this polymorphous theme in
| |
| − | bas relief, that is, via detour through a universe of discourse
| |
| − | where it fails. Here's a brief reminder of the Othello example:
| |
| − | | |
| − | B C D E I J O
| |
| − | o o o o o o o 1
| |
| − | | | | |
| |
| − | | | | | m,
| |
| − | | | | |
| |
| − | o o o o o o o 1
| |
| − | |
| |
| − | | b,
| |
| − | |
| |
| − | o o o o o o o 1
| |
| − | B C D E I J O
| |
| − | | |
| − | The condition, "men are just as apt to be black as things in general",
| |
| − | is expressible in terms of conditional probabilities as P(b|m) = P(b),
| |
| − | written out, the probability of the event Black given the event Male
| |
| − | is exactly equal to the unconditional probability of the event Black.
| |
| − | | |
| − | Thus, for example, it is sufficient to observe in the Othello setting
| |
| − | that P(b|m) = 1/4 while P(b) = 1/7 in order to cognize the dependency,
| |
| − | and thereby to tell that the ostensible arrow is anaclinically biased.
| |
| − | | |
| − | This reduction of a conditional probability to an absolute probability,
| |
| − | in the form P(A|Z) = P(A), is a familiar disguise, and yet in practice
| |
| − | one of the ways that we most commonly come to recognize the condition
| |
| − | of independence P(AZ) = P(A)P(Z), via the definition of a conditional
| |
| − | probability according to the rule P(A|Z) = P(AZ)/P(Z). To recall the
| |
| − | familiar consequences, the definition of conditional probability plus
| |
| − | the independence condition yields P(A|Z) = P(AZ)/P(Z) = P(A)P(Z)/P(Z),
| |
| − | to wit, P(A|Z) = P(A).
| |
| − | | |
| − | As Hamlet discovered, there's a lot to be learned from turning a crank.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 11.24
| |
| − | | |
| − | | |
| − | | |
| − | And so we come to the end of the "number of" examples
| |
| − | that we found on our agenda at this point in the text:
| |
| − | | |
| − | | It is to be observed that:
| |
| − | |
| |
| − | | [!1!] = `1`.
| |
| − | |
| |
| − | | Boole was the first to show this connection between logic and
| |
| − | | probabilities. He was restricted, however, to absolute terms.
| |
| − | | I do not remember having seen any extension of probability to
| |
| − | | relatives, except the ordinary theory of 'expectation'.
| |
| − | |
| |
| − | | Our logical multiplication, then, satisfies the essential conditions
| |
| − | | of multiplication, has a unity, has a conception similar to that of
| |
| − | | admitted multiplications, and contains numerical multiplication as
| |
| − | | a case under it.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.76
| |
| − | | |
| − | There appears to be a problem with the printing of the text at this point.
| |
| − | Let us first recall the conventions that I am using in this transcription:
| |
| − | `1` for the "antique 1" that Peirce defines as !1!_oo = "something", and
| |
| − | !1! for the "bold 1" that signifies the ordinary 2-identity relation.
| |
| − | | |
| − | CP 3 gives [!1!] = `1`, which I cannot make any sense of.
| |
| − | CE 2 gives [!1!] = 1 , which makes sense on the reading
| |
| − | of "1" as denoting the natural number 1, and not as the
| |
| − | absolute term "1" that denotes the universe of discourse.
| |
| − | On this reading, [!1!] is the average number of things
| |
| − | related by the identity relation !1! to one individual,
| |
| − | and so it makes sense that [!1!] = 1 : N, where "N" is
| |
| − | the set or type of the natural numbers {0, 1, 2, ...}.
| |
| − | | |
| − | With respect to the 2-identity !1! in the syntactic domain S
| |
| − | and the number 1 in the non-negative integers N c R, we have:
| |
| − | | |
| − | 'v'!1! = [!1!] = 1.
| |
| − | | |
| − | And so the "number of" mapping 'v' : S -> R has another one
| |
| − | of the properties that would be required of an arrow S -> R.
| |
| − | | |
| − | The manner in which these arrows and qualified arrows help us
| |
| − | to construct a suspension bridge that unifies logic, semiotics,
| |
| − | statistics, stochastics, and information theory will be one of
| |
| − | the main themes that I aim to elaborate throughout the rest of
| |
| − | this inquiry.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Note 12
| |
| − | | |
| − | | |
| − | | |
| − | | The Sign of Involution
| |
| − | |
| |
| − | | I shall take involution in such a sense that x^y
| |
| − | | will denote everything which is an x for every
| |
| − | | individual of y.
| |
| − | |
| |
| − | | Thus
| |
| − | |
| |
| − | | 'l'^w
| |
| − | |
| |
| − | | will be a lover of every woman.
| |
| − | |
| |
| − | | Then
| |
| − | |
| |
| − | | ('s'^'l')^w
| |
| − | |
| |
| − | | will denote whatever stands to every woman in
| |
| − | | the relation of servant of every lover of hers;
| |
| − | |
| |
| − | | and
| |
| − | |
| |
| − | | 's'^('l'w)
| |
| − | |
| |
| − | | will denote whatever is a servant of
| |
| − | | everything that is lover of a woman.
| |
| − | |
| |
| − | | So that
| |
| − | |
| |
| − | | ('s'^'l')^w = 's'^('l'w).
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.77
| |
| − | |
| |
| − | | Charles Sanders Peirce,
| |
| − | |"Description of a Notation for the Logic of Relatives,
| |
| − | | Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
| |
| − | |'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
| |
| − | |'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Commentary Note 12
| |
| − | | |
| − | | |
| − | | |
| − | Let us make a few preliminary observations about the
| |
| − | "logical sign of involution", as Peirce uses it here:
| |
| − | | |
| − | | The Sign of Involution
| |
| − | |
| |
| − | | I shall take involution in such a sense that x^y
| |
| − | | will denote everything which is an x for every
| |
| − | | individual of y.
| |
| − | |
| |
| − | | Thus
| |
| − | |
| |
| − | | 'l'^w
| |
| − | |
| |
| − | | will be a lover of every woman.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.77
| |
| − | | |
| − | In arithmetic, the "involution" x^y, or the "exponentiation" of x
| |
| − | to the power of y, is the iterated multiplication of the factor x,
| |
| − | repeated as many times as there are ones making up the exponent y.
| |
| − | | |
| − | In analogous fashion, 'l'^w is the iterated multiplication of 'l',
| |
| − | repeated as many times as there are individuals under the term w.
| |
| − | | |
| − | For example, suppose that the universe of discourse has,
| |
| − | among other things, just the three women, W_1, W_2, W_3.
| |
| − | This could be expressed in Peirce's notation by writing:
| |
| − | | |
| − | w = W_1 +, W_2 +, W_3.
| |
| − | | |
| − | In this setting, we would have:
| |
| − | | |
| − | 'l'^w = 'l'^(W_1 +, W_2 +, W_3) = 'l'W_1 , 'l'W_2 , 'l'W_3.
| |
| − | | |
| − | That is, a lover of every woman in the universe of discourse
| |
| − | would be a lover of W_1 and a lover of W_2 and lover of W_3.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Note 13
| |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | LOR. Work Area
| |
| − | | |
| − | | |
| − | | |
| − | Up to this point in the discussion, we have observed that
| |
| − | the "number of" map 'v' : S -> R such that 'v's = [s] has
| |
| − | the following morphic properties:
| |
| − | | |
| − | 0. [0] = 0
| |
| − | | |
| − | 1. 'v'
| |
| − | | |
| − | 2. x -< y => [x] =< [y]
| |
| − | | |
| − | 3. [x +, y] =< [x] + [y]
| |
| − | | |
| − | contingent:
| |
| − | | |
| − | 4. [xy] = [x][y]
| |
| − | | |
| − | view relation P c X x Y x Z as related to three functions:
| |
| − | | |
| − | `p_1` c
| |
| − | `p_3` c X x Y x Pow(Z)
| |
| − | | |
| − | | |
| − | f(x)
| |
| − | | |
| − | f(x+y) = f(x) + f(y)
| |
| − | | |
| − | f(p(x, y)) = q(f(x), f(y))
| |
| − | | |
| − | P(x, y, z)
| |
| − | | |
| − | (f^-1)(y)
| |
| − | | |
| − | f(z(x, y)) = z'(f(x), f(y))
| |
| − | | |
| − | Definition. f(x:y:z) = (fx:fy:fz).
| |
| − | | |
| − | f(x:y:z) = (fx:fy:
| |
| − | | |
| − | x:y:z in R => fx:fy:fz in fR
| |
| − | | |
| − | R(x, y, z) => (fR)(fx, fy, fz)
| |
| − | | |
| − | (L, x, y, z) => (fL, fx, fy, fz)
| |
| − | | |
| − | (x, y, z, L) => (xf, yf, zf, Lf)
| |
| − | | |
| − | (x, y, z, b) => (xf, yf, zf, bf)
| |
| − | | |
| − | | |
| − | fzxy = z'(fx)(fy)
| |
| − | | |
| − | | |
| − | F
| |
| − | o
| |
| − | |
| |
| − | o
| |
| − | / \
| |
| − | o o
| |
| − | o
| |
| − | . | .
| |
| − | . | .
| |
| − | . | .
| |
| − | . o .
| |
| − | . / \ .
| |
| − | . / \ .
| |
| − | . / \ .
| |
| − | . o o .
| |
| − | . . .
| |
| − | . . .
| |
| − | .
| |
| − | | |
| − |
| |
| − | C o . / \ . o
| |
| − | | . / \ . | CF
| |
| − | | . o o . |
| |
| − | f o . . . o fF
| |
| − | / \ . . . / \
| |
| − | / . \ . o o
| |
| − | X o o Y XF YF
| |
| − | | |
| − | <u, v, w> in P ->
| |
| − | | |
| − | o---------o---------o---------o---------o
| |
| − | | # h | h | f |
| |
| − | o=========o=========o=========o=========o
| |
| − | | P # X | Y | Z |
| |
| − | o---------o---------o---------o---------o
| |
| − | | Q # U | V | W |
| |
| − | o---------o---------o---------o---------o
| |
| − | | |
| − | Products of diagonal extensions:
| |
| − | | |
| − | 1,1, = !1!!1!
| |
| − | | |
| − | = "anything that is anything that is ---"
| |
| − | | |
| − | = "anything that is ---"
| |
| − | | |
| − | = !1!
| |
| − | | |
| − | m,n = "man that is noble"
| |
| − | | |
| − | = (C:C +, I:I +, J:J +, O:O)(C +, D +, O)
| |
| − | | |
| − | = C +, O
| |
| − | | |
| − | n,m = "noble that is man"
| |
| − | | |
| − | = (C:C +, D:D +, O:O)(C +, I +, J +, O)
| |
| − | | |
| − | = C +, O
| |
| − | | |
| − | n,w = "noble that is woman"
| |
| − | | |
| − | = (C:C +, D:D +, O:O)(B +, D +, E)
| |
| − | | |
| − | = D
| |
| − | | |
| − | w,n = "woman that is noble"
| |
| − | | |
| − | = (B:B +, D:D +, E:E)(C +, D +, O)
| |
| − | | |
| − | = D
| |
| − | | |
| − | Given a set X and a subset M c X, define e_M,
| |
| − | the "idempotent representation" of M over X,
| |
| − | as the 2-adic relation e_M c X x X which is
| |
| − | the identity relation on M. In other words,
| |
| − | e_M = {<x, x> : x in M}.
| |
| − | | |
| − | Transposing this by steps into Peirce's notation:
| |
| − | | |
| − | e_M = {<x, x> : x in M}
| |
| − | | |
| − | = {x:x : x in M}
| |
| − | | |
| − | = Sum_X |x in M| x:x
| |
| − | | |
| − | 'l' = "lover of ---"
| |
| − | | |
| − | 's' = "servant of ---"
| |
| − | | |
| − | 'l', = "lover that is --- of ---"
| |
| − | | |
| − | 's', = "servant that is --- of ---"
| |
| − | | |
| − | | But not only may any absolute term be thus regarded as a relative term,
| |
| − | | but any relative term may in the same way be regarded as a relative with
| |
| − | | one correlate more. It is convenient to take this additional correlate
| |
| − | | as the first one.
| |
| − | |
| |
| − | | Then:
| |
| − | |
| |
| − | | 'l','s'w
| |
| − | |
| |
| − | | will denote a lover of a woman that is a servant of that woman.
| |
| − | |
| |
| − | | C.S. Peirce, CP 3.73
| |
| − | | |
| − | o~~~~~~~~~o~~~~+~~~~o~~~~~~~~~o~~~~~~~~~o~~~~+~~~~o~~~~~~~~~o
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | Objective Framework (OF) | Interpretive Framework (IF) |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | Objects | Signs |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | |
| |
| − | | C o--------------- |
| |
| − | | |
| |
| − | | F o--------------- |
| |
| − | | |
| |
| − | | I o--------------- |
| |
| − | | |
| |
| − | | O o--------------- |
| |
| − | | |
| |
| − | | B o--------------- |
| |
| − | | |
| |
| − | | D o--------------- |
| |
| − | | |
| |
| − | | E o--------------- |
| |
| − | | o "m" |
| |
| − | | / |
| |
| − | | / |
| |
| − | | / |
| |
| − | | o o o-----------@ |
| |
| − | | \ |
| |
| − | | \ |
| |
| − | | \ |
| |
| − | | o |
| |
| − | | |
| |
| − | o-----------------------------o-----------------------------o
| |
| − | | |
| − | †‡||§¶
| |
| − | @#||$%
| |
| − | | |
| − | quality, reflection, synecdoche
| |
| − | | |
| − | 1. neglect of
| |
| − | 2. neglect of
| |
| − | 3. neglect of nil?
| |
| − | | |
| − | Now, it's not the end of the story, of course, but it's a start.
| |
| − | The significant thing is what is usually the significant thing
| |
| − | in mathematics, at least, that two distinct descriptions refer
| |
| − | to the same things. Incidentally, Peirce is not really being
| |
| − | as indifferent to the distinctions between signs and things
| |
| − | as this ascii text makes him look, but uses a host of other
| |
| − | type-faces to distinguish the types and the uses of signs.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 1
| |
| − | | |
| − | | |
| − | | |
| − | GR = Gary Richmond
| |
| − | | |
| − | GR: I wonder if the necessary "elementary triad" spoken of
| |
| − | below isn't somehow implicated in those discussions
| |
| − | "invoking a 'closure principle'".
| |
| − | | |
| − | GR, quoting CSP:
| |
| − | | |
| − | | CP 1.292. It can further be said in advance, not, indeed,
| |
| − | | purely a priori but with the degree of apriority that is
| |
| − | | proper to logic, namely, as a necessary deduction from
| |
| − | | the fact that there are signs, that there must be an
| |
| − | | elementary triad. For were every element of the
| |
| − | | phaneron a monad or a dyad, without the relative
| |
| − | | of teridentity (which is, of course, a triad),
| |
| − | | it is evident that no triad could ever be
| |
| − | | built up. Now the relation of every sign
| |
| − | | to its object and interpretant is plainly
| |
| − | | a triad. A triad might be built up of
| |
| − | | pentads or of any higher perissad
| |
| − | | elements in many ways. But it
| |
| − | | can be proved -- and really
| |
| − | | with extreme simplicity,
| |
| − | | though the statement of
| |
| − | | the general proof is
| |
| − | | confusing -- that no
| |
| − | | element can have
| |
| − | | a higher valency
| |
| − | | than three.
| |
| − | | |
| − | GR: (Of course this passage also directly relates
| |
| − | to the recent thread on Identity and Teridentity.)
| |
| − | | |
| − | Yes, generally speaking, I think that there are deep formal principles here
| |
| − | that manifest themselves in these various guises: the levels of intention
| |
| − | or the orders of reflection, the sign relation, pragmatic conceivability,
| |
| − | the generative sufficiency of 3-adic relations for all practical intents,
| |
| − | and the irreducibility of continuous relations. I have run into themes
| |
| − | in combinatorics, group theory, and Lie algebras that are tantalizingly
| |
| − | reminiscent of the things that Peirce says here, but it will take me
| |
| − | some time to investigate them far enough to see what's going on.
| |
| − | | |
| − | GR: PS. I came upon the above passage last night reading through
| |
| − | the Peirce selections in John J. Stuhr's 'Classical American
| |
| − | Philosophy: Essential Readings and Interpretive Essays',
| |
| − | Oxford University, 1987 (the passage above is found on
| |
| − | pp 61-62), readily available in paperback in a new
| |
| − | edition, I believe.
| |
| − | | |
| − | GR: An aside: These excerpts in Sturh include versions of a fascinating
| |
| − | "Intellectual Autobiography", Peirce's summary of his scientific,
| |
| − | especially, philosophic accomplishments. I've seen them published
| |
| − | nowhere else.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 2
| |
| − | | |
| − | | |
| − | | |
| − | BU = Ben Udell
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | BU: I'm in the process of moving back to NYC and have had little opportunity
| |
| − | to do more than glance through posts during the past few weeks, but this
| |
| − | struck me because it sounds something I really would like to know about,
| |
| − | but I didn't understand it:
| |
| − | | |
| − | JA: Notice that Peirce follows the mathematician's usual practice,
| |
| − | then and now, of making the status of being an "individual" or
| |
| − | a "universal" relative to a discourse in progress. I have come
| |
| − | to appreciate more and more of late how radically different this
| |
| − | "patchwork" or "piecewise" approach to things is from the way of
| |
| − | some philosophers who seem to be content with nothing less than
| |
| − | many worlds domination, which means that they are never content
| |
| − | and rarely get started toward the solution of any real problem.
| |
| − | Just my observation, I hope you understand.
| |
| − | | |
| − | BU: "Many worlds domination", "nothing less than many worlds domination" --
| |
| − | as opposed to the patchwork or piecewise approach. What is many worlds
| |
| − | domination? When I hear "many worlds" I think of Everett's Many Worlds
| |
| − | interpretation of quantum mechanics.
| |
| − | | |
| − | Yes, it is a resonance of Edward, Everett, and All the Other Whos in Whoville,
| |
| − | but that whole microcosm is itself but the frumious reverberation of Leibniz's
| |
| − | Maenadolatry.
| |
| − | | |
| − | More sequitur, though, this is an issue that has simmered beneath
| |
| − | the surface of my consciousness for several decades now and only
| |
| − | periodically percolates itself over the hyper-critical thrashold
| |
| − | of expression. Let me see if I can a better job of it this time.
| |
| − | | |
| − | The topic is itself a patchwork of infernally recurrent patterns.
| |
| − | Here are a few pieces of it that I can remember arising recently:
| |
| − | | |
| − | | Zeroth Law Of Semantics
| |
| − | |
| |
| − | | Meaning is a privilege not a right.
| |
| − | | Not all pictures depict.
| |
| − | | Not all signs denote.
| |
| − | |
| |
| − | | Never confuse a property of a sign,
| |
| − | | for instance, existence,
| |
| − | | with a sign of a property,
| |
| − | | for instance, existence.
| |
| − | |
| |
| − | | Taking a property of a sign,
| |
| − | | for a sign of a property,
| |
| − | | is the zeroth sign of
| |
| − | | nominal thinking,
| |
| − | | and the first
| |
| − | | mistake.
| |
| − | |
| |
| − | | Also Sprach Zero*
| |
| − | | |
| − | A less catchy way of saying "meaning is a privilege not a right"
| |
| − | would most likely be "meaning is a contingency not a necessity".
| |
| − | But if I reflect on that phrase, it does not quite satisfy me,
| |
| − | since a deeper lying truth is that contingency and necessity,
| |
| − | connections in fact and connections beyond the reach of fact,
| |
| − | depend on a line of distinction that is itself drawn on the
| |
| − | scene of observation from the embodied, material, physical,
| |
| − | non-point massive, non-purely-spectrelative point of view
| |
| − | of an agent or community of interpretation, a discursive
| |
| − | universe, an engauged interpretant, a frame of at least
| |
| − | partial self-reverence, a hermeneutics in progress, or
| |
| − | a participant observer. In short, this distinction
| |
| − | between the contingent and the necessary is itself
| |
| − | contingent, which means, among other things, that
| |
| − | signs are always indexical at some least quantum.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 3
| |
| − | | |
| − | | |
| − | | |
| − | JR = Joe Ransdell
| |
| − | | |
| − | JR: Would the Kripke conception of the "rigid designator" be an instance
| |
| − | of the "many worlds domination"? I was struck by your speaking of
| |
| − | the "patchwork or piecewise" approach as well in that it seemed to
| |
| − | me you might be expressing the same general idea that I have usually
| |
| − | thought of in terms of contextualism instead: I mean the limits it
| |
| − | puts upon what you can say a priori if you really take contextualism
| |
| − | seriously, which is the same as recognizing indexicality as incapable
| |
| − | of elimination, I think.
| |
| − | | |
| − | Yes, I think this is the same ballpark of topics.
| |
| − | I can't really speak for what Kripke had in mind,
| |
| − | but I have a practical acquaintance with the way
| |
| − | that some people have been trying to put notions
| |
| − | like this to work on the applied ontology scene,
| |
| − | and it strikes me as a lot of nonsense. I love
| |
| − | a good parallel worlds story as much as anybody,
| |
| − | but it strikes me that many worlds philosophers
| |
| − | have the least imagination of anybody as to what
| |
| − | an alternative universe might really be like and
| |
| − | so I prefer to read more creative writers when it
| |
| − | comes to that. But serially, folks, I think that
| |
| − | the reason why some people evidently feel the need
| |
| − | for such outlandish schemes -- and the vast majority
| |
| − | of the literature on counterfactual conditionals falls
| |
| − | into the same spaceboat as this -- is simply that they
| |
| − | have failed to absorb, through the fault of Principian
| |
| − | filters, a quality that Peirce's logic is thoroughly
| |
| − | steeped in, namely, the functional interpretation
| |
| − | of logical terms, that is, as signs referring to
| |
| − | patterns of contingencies. It is why he speaks
| |
| − | more often, and certainly more sensibly and to
| |
| − | greater effect, of "conditional generals" than
| |
| − | of "modal subjunctives". This is also bound up
| |
| − | with that element of sensibility that got lost in
| |
| − | the transition from Peircean to Fregean quantifiers.
| |
| − | Peirce's apriorities are always hedged with risky bets.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 4
| |
| − | | |
| − | | |
| − | | |
| − | BU = Benjamin Udell
| |
| − | | |
| − | BU: I wish I had more time to ponder the "many-worlds" issue (& that my books
| |
| − | were not currently disappearing into heavily taped boxes). I had thought
| |
| − | of the piecemeal approach's opposite as the attempt to build a kind of
| |
| − | monolithic picture, e.g., to worry that there is not an infinite number
| |
| − | of particles in the physical universe for the infinity integers. But
| |
| − | maybe the business with rigid designators & domination of many worlds
| |
| − | has somehow to do with monolithism.
| |
| − | | |
| − | Yes, that's another way of saying it. When I look to my own priorities,
| |
| − | my big worry is that logic as a discipline is not fulfilling its promise.
| |
| − | I have worked in too many settings where the qualitative researchers and
| |
| − | the quantitative researchers could barely even talk to one an Other with
| |
| − | any understanding, and this I recognized as a big block to inquiry since
| |
| − | our first notice of salient facts and significant phenomena is usually
| |
| − | in logical, natural language, or qualitative forms, while our eventual
| |
| − | success in resolving anomalies and solving practical problems depends
| |
| − | on our ability to formalize, operationalize, and quantify the issues,
| |
| − | even if only to a very partial degree, as it generally turns out.
| |
| − | | |
| − | When I look to the history of how logic has been deployed in mathematics,
| |
| − | and through those media in science generally, it seems to me that the
| |
| − | Piece Train started to go off track with the 'Principia Mathematica'.
| |
| − | All pokes in the rib aside, however, I tend to regard this event
| |
| − | more as the symptom of a localized cultural phenomenon than as
| |
| − | the root cause of the broader malaise.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 5
| |
| − | | |
| − | | |
| − | | |
| − | CG = Clark Goble
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | JA, quoting CSP:
| |
| − | | |
| − | | For example,
| |
| − | |
| |
| − | | f + u
| |
| − | |
| |
| − | | means all Frenchmen besides all violinists, and,
| |
| − | | therefore, considered as a logical term, implies
| |
| − | | that all French violinists are 'besides themselves'.
| |
| − | | |
| − | CG: Could you clarify your use of "besides"?
| |
| − | | |
| − | CG: I think I am following your thinking in that you
| |
| − | don't want the logical terms to be considered
| |
| − | to have any necessary identity between them.
| |
| − | Is that right?
| |
| − | | |
| − | I use vertical sidebars "|" for long quotations, so this
| |
| − | is me quoting Peirce at CP 3.67 who is explaining in an
| |
| − | idiomatic way Boole's use of the plus sign for a logical
| |
| − | operation that is strictly speaking limited to terms for
| |
| − | mutually exclusive classes. The operation would normally
| |
| − | be extended to signify the "symmetric difference" operator.
| |
| − | But Peirce is saying that he prefers to use the sign "+,"
| |
| − | for inclusive disjunction, corresponding to the union of
| |
| − | the associated classes. Peirce calls Boole's operation
| |
| − | "invertible" because it amounts to the sum operation in
| |
| − | a field, whereas the inclusive disjunction or union is
| |
| − | "non-invertible", since knowing that A |_| B = C does
| |
| − | not allow one to say determinately that A = C - B.
| |
| − | I can't recall if Boole uses this 'besides' idiom,
| |
| − | but will check later.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 6
| |
| − | | |
| − | | |
| − | | |
| − | CG = Clark Goble
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | JA: I use vertical sidebars "|" for long quotations, so this
| |
| − | is me quoting Peirce at CP 3.67 who is explaining in an
| |
| − | idiomatic way Boole's use of the plus sign for a logical
| |
| − | operation that is strictly speaking limited to terms for
| |
| − | mutually exclusive classes.
| |
| − | | |
| − | CG: Is that essay related to any of the essays
| |
| − | in the two volume 'Essential Peirce'? I'm
| |
| − | rather interested in how he speaks there.
| |
| − | | |
| − | No, the EP volumes are extremely weak on logical selections.
| |
| − | I see nothing there that deals with the logic of relatives.
| |
| − | | |
| − | JA: But Peirce is saying that he prefers to use the sign "+,"
| |
| − | for inclusive disjunction, corresponding to the union of
| |
| − | the associated classes.
| |
| − | | |
| − | CG: The reason I asked was more because it seemed
| |
| − | somewhat interesting in light of the logic of
| |
| − | operators in quantum mechanics. I was curious
| |
| − | if the use of "beside" might relate to that.
| |
| − | But from what you say it probably was just me
| |
| − | reading too much into the quote. The issue of
| |
| − | significance was whether the operation entailed
| |
| − | the necessity of mutual exclusivity or whether
| |
| − | some relationship between the classes might be
| |
| − | possible. I kind of latched on to Peirce's
| |
| − | odd statement about "all French violinists
| |
| − | are 'beside themselves'".
| |
| − | | |
| − | CG: Did Peirce have anything to say about
| |
| − | what we'd call non-commuting operators?
| |
| − | | |
| − | In general, 2-adic relative terms are non-commutative.
| |
| − | For example, a brother of a mother is not identical to
| |
| − | a mother of a brother.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 7
| |
| − | | |
| − | | |
| − | | |
| − | GR = Gary Richmond
| |
| − | | |
| − | GR: I am very much enjoying, which is to say,
| |
| − | learning from your interlacing commentary
| |
| − | on Peirce's 1870 "Logic of Relatives" paper.
| |
| − | | |
| − | GR: What an extraordinary paper the 1870 "LOG" is! Your notes helped
| |
| − | me appreciate the importance of the unanticipated proposal of P's
| |
| − | to "assign to all logical terms, numbers". On the other hand,
| |
| − | the excerpts suggested to we why Peirce finally framed his
| |
| − | Logic of Relatives into graphical form. Still, I think
| |
| − | that a thorough examination of the 1970 paper might
| |
| − | serve as propaedeutic (and of course, much more)
| |
| − | for the study of the alpha and beta graphs.
| |
| − | | |
| − | Yes, there's gold in them thar early logic papers that has been "panned"
| |
| − | but nowhere near mined in depth yet. The whole quiver of arrows between
| |
| − | terms and numbers harks back to the 'numeri characteristici' of Leibniz,
| |
| − | of course, but Leibniz attended more on the intensional chains of being
| |
| − | while Peirce will here start to "escavate" the extensional hierarchies.
| |
| − | | |
| − | I consider myself rewarded that you see the incipient impulse toward
| |
| − | logical graphs, as one of the most striking things to me about this
| |
| − | paper is to see these precursory seeds already planted here within
| |
| − | it and yet to know how long it will take them to sprout and bloom.
| |
| − | | |
| − | Peirce is obviously struggling to stay within the linotyper's art --
| |
| − | a thing that we, for all our exorbitant hype about markable text,
| |
| − | are still curiously saddled with -- but I do not believe that it
| |
| − | is possible for any mind equipped with a geometrical imagination
| |
| − | to entertain these schemes for connecting up terminological hubs
| |
| − | with their terminological terminals without perforce stretching
| |
| − | imaginary strings between the imaginary gumdrops.
| |
| − | | |
| − | GR: I must say though that the pace at which you've been throwing this at us
| |
| − | is not to be kept up with by anyone I know "in person or by reputation".
| |
| − | I took notes on the first 5 or 6 Notes, but can now just barely find
| |
| − | time to read through your posts.
| |
| − | | |
| − | Oh, I was trying to burrow as fast as I could toward the more untapped veins --
| |
| − | I am guessing that things will probably "descalate" a bit over the next week,
| |
| − | but then, so will our attention spans ...
| |
| − | | |
| − | Speaking of which, I will have to break here, and pick up the rest later ...
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 8
| |
| − | | |
| − | | |
| − | | |
| − | GR = Gary Richmond
| |
| − | | |
| − | GR: In any event, I wish that you'd comment on Note 5 more directly (though
| |
| − | you do obliquely in your own diagramming of "every [US] Vice-President(s) ...
| |
| − | [who is] every President(s) of the US Senate".
| |
| − | | |
| − | There are several layers of things to say about that,
| |
| − | and I think that it would be better to illustrate the
| |
| − | issues by way of the examples that Peirce will soon be
| |
| − | getting to, but I will see what I can speak to for now.
| |
| − | | |
| − | GR: But what interested me even more in LOR, Note 5, was the sign < ("less than"
| |
| − | joined to the sign of identity = to yield P's famous sign -< (or more clearly,
| |
| − | =<) of inference, which combines the two (so that -< (literally, "as small as")
| |
| − | means "is". I must say I both "get" this and don't quite (Peirce's example(s) of
| |
| − | the frenchman helped a little). Perhaps your considerably more mathematical mind
| |
| − | can help clarify this for a non-mathematician such as myself. (My sense is that
| |
| − | "as small as" narrows the terms so that "everything that occurs in the conclusion
| |
| − | is already contained in the premise.) I hope I'm not being obtuse here. I'm sure
| |
| − | it's "all too simple for words".
| |
| − | | |
| − | Then let us draw a picture.
| |
| − | | |
| − | "(F (G))", read "not F without G", means that F (G), that is, F and not G,
| |
| − | is the only region exempted from the occupation of being in this universe:
| |
| − | | |
| − | o-----------------------------------------------------------o
| |
| − | |`X`````````````````````````````````````````````````````````|
| |
| − | |```````````````````````````````````````````````````````````|
| |
| − | |`````````````o-------------o```o-------------o`````````````|
| |
| − | |````````````/ \`/```````````````\````````````|
| |
| − | |```````````/ o`````````````````\```````````|
| |
| − | |``````````/ /`\`````````````````\``````````|
| |
| − | |`````````/ /```\`````````````````\`````````|
| |
| − | |````````/ /`````\`````````````````\````````|
| |
| − | |```````o o```````o`````````````````o```````|
| |
| − | |```````| |```````|`````````````````|```````|
| |
| − | |```````| |```````|`````````````````|```````|
| |
| − | |```````| F |```````|````````G````````|```````|
| |
| − | |```````| |```````|`````````````````|```````|
| |
| − | |```````| |```````|`````````````````|```````|
| |
| − | |```````o o```````o`````````````````o```````|
| |
| − | |````````\ \`````/`````````````````/````````|
| |
| − | |`````````\ \```/`````````````````/`````````|
| |
| − | |``````````\ \`/`````````````````/``````````|
| |
| − | |```````````\ o`````````````````/```````````|
| |
| − | |````````````\ /`\```````````````/````````````|
| |
| − | |`````````````o-------------o```o-------------o`````````````|
| |
| − | |```````````````````````````````````````````````````````````|
| |
| − | |```````````````````````````````````````````````````````````|
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| − | Collapsing the vacuous region like soapfilm popping on a wire frame,
| |
| − | we draw the constraint (F (G)) in the following alternative fashion:
| |
| − | | |
| − | o-----------------------------------------------------------o
| |
| − | |`X`````````````````````````````````````````````````````````|
| |
| − | |```````````````````````````````````````````````````````````|
| |
| − | |```````````````````````````````o-------------o`````````````|
| |
| − | |``````````````````````````````/```````````````\````````````|
| |
| − | |`````````````````````````````o`````````````````\```````````|
| |
| − | |````````````````````````````/`\`````````````````\``````````|
| |
| − | |```````````````````````````/```\`````````````````\`````````|
| |
| − | |``````````````````````````/`````\`````````````````\````````|
| |
| − | |`````````````````````````o```````o`````````````````o```````|
| |
| − | |`````````````````````````|```````|`````````````````|```````|
| |
| − | |`````````````````````````|```````|`````````````````|```````|
| |
| − | |`````````````````````````|```F```|````````G````````|```````|
| |
| − | |`````````````````````````|```````|`````````````````|```````|
| |
| − | |`````````````````````````|```````|`````````````````|```````|
| |
| − | |`````````````````````````o```````o`````````````````o```````|
| |
| − | |``````````````````````````\`````/`````````````````/````````|
| |
| − | |```````````````````````````\```/`````````````````/`````````|
| |
| − | |````````````````````````````\`/`````````````````/``````````|
| |
| − | |`````````````````````````````o`````````````````/```````````|
| |
| − | |``````````````````````````````\```````````````/````````````|
| |
| − | |```````````````````````````````o-------------o`````````````|
| |
| − | |```````````````````````````````````````````````````````````|
| |
| − | |```````````````````````````````````````````````````````````|
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| − | So, "(F (G))", "F => G", "F =< G", "F -< G", "F c G",
| |
| − | under suitable mutations of interpretation, are just
| |
| − | so many ways of saying that the denotation of "F" is
| |
| − | contained within the denotation of "G".
| |
| − | | |
| − | Now, let us look to the "characteristic functions" or "indicator functions"
| |
| − | of the various regions of being. It is frequently convenient to ab-use the
| |
| − | same letters for them and merely keep a variant interpretation "en thy meme",
| |
| − | but let us be more meticulous here, and reserve the corresponding lower case
| |
| − | letters "f" and "g" to denote the indicator functions of the regions F and G,
| |
| − | respectively.
| |
| − | | |
| − | Taking B = {0, 1} as the boolean domain, we have:
| |
| − | | |
| − | f, g : X -> B
| |
| − | | |
| − | (f^(-1))(1) = F
| |
| − | | |
| − | (g^(-1))(1) = G
| |
| − | | |
| − | In general, for h : X -> B, an expression like "(h^(-1))(1)"
| |
| − | can be read as "the inverse of h evaluated at 1", in effect,
| |
| − | denoting the set of points in X where h evaluates to "true".
| |
| − | This is called the "fiber of truth" in h, and I have gotten
| |
| − | where I like to abbreviate it as "[|h|]".
| |
| − | | |
| − | Accordingly, we have:
| |
| − | | |
| − | F = [|f|] = (f^(-1))(1) c X
| |
| − | | |
| − | G = [|g|] = (g^(-1))(1) c X
| |
| − | | |
| − | This brings us to the question, what sort
| |
| − | of "functional equation" between f and g
| |
| − | goes with the regional constraint (F (G))?
| |
| − | | |
| − | Just this, that f(x) =< g(x) for all x in X,
| |
| − | where the '=<' relation on the values in B
| |
| − | has the following operational table for
| |
| − | the pairing "row head =< column head".
| |
| − | | |
| − | o---------o---------o---------o
| |
| − | | =< # 0 | 1 |
| |
| − | o=========o=========o=========o
| |
| − | | 0 # 1 | 1 |
| |
| − | o---------o---------o---------o
| |
| − | | 1 # 0 | 1 |
| |
| − | o---------o---------o---------o
| |
| − | | |
| − | And this, of course, is the same thing as the truth table
| |
| − | for the conditional connective or the implication relation.
| |
| − | | |
| − | GR: By the way, in the semiosis implied by the modal gamma graphs,
| |
| − | could -< (were it used there, which of course it is not) ever
| |
| − | be taken to mean,"leads to" or "becomes" or "evolves into"?
| |
| − | I informally use it that way myself, using the ordinary
| |
| − | arrow for implication.
| |
| − | | |
| − | I am a bit insensitive to the need for modal logic,
| |
| − | since necessity in mathematics always seems to come
| |
| − | down to being a matter of truth for all actual cases,
| |
| − | if under an expanded sense of actuality that makes it
| |
| − | indiscernible from possibility, so I must beg off here.
| |
| − | But there are places where Peirce makes a big deal about
| |
| − | the advisability of drawing the '-<' symbol in one fell
| |
| − | stroke of the pen, kind of like a "lazy gamma" -- an old
| |
| − | texican cattle brand -- and I have seen another place where
| |
| − | he reads "A -< B" as "A, in every way that it can be, is B",
| |
| − | as if this '-<' fork in the road led into a veritable garden
| |
| − | of branching paths.
| |
| − | | |
| − | And out again ...
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 9
| |
| − | | |
| − | | |
| − | | |
| − | GR = Gary Richmond
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | JA: I am a bit insensitive to the need for modal logic,
| |
| − | since necessity in mathematics always seems to come
| |
| − | down to being a matter of truth for all actual cases,
| |
| − | if under an expanded sense of actuality that makes it
| |
| − | indiscernible from possibility, so I must beg off here.
| |
| − | | |
| − | GR: I cannot agree with you regarding modal logic. Personally
| |
| − | I feel that the gamma part of the EG's is of the greatest
| |
| − | interest and potential importance, and as Jay Zeman has
| |
| − | made clear in his dissertation, Peirce certainly thought
| |
| − | this as well.
| |
| − | | |
| − | You disagree that I am insensitive? Well, certainly nobody has ever done that before!
| |
| − | No, I phrased it that way to emphasize the circumstance that it ever hardly comes up
| |
| − | as an issue within the limited purview of my experience, and when it does -- as in
| |
| − | topo-logical boundary situations -- it seems to require a sort of analysis that
| |
| − | doesn't comport all that well with the classical modes and natural figures of
| |
| − | speech about it. Then again, I spent thirty years trying to motorize Alpha,
| |
| − | have only a few good clues how I would go about Beta, and so Gamma doesn't
| |
| − | look like one of those items on my plate.
| |
| − | | |
| − | Speeching Of Which ---
| |
| − | Best Of The Season ...
| |
| − | And Happy Trailing ...
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 10
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | BM: Thanks for your very informative talk. There
| |
| − | is a point that I did not understand in note 35:
| |
| − | | |
| − | JA: If we operate in accordance with Peirce's example of `g`'o'h
| |
| − | as the "giver of a horse to an owner of that horse", then we
| |
| − | may assume that the associative law and the distributive law
| |
| − | are by default in force, allowing us to derive this equation:
| |
| − | | |
| − | JA: 'l','s'w = 'l','s'(B +, D +, E) = 'l','s'B +, 'l','s'D +, 'l','s'E
| |
| − | | |
| − | BM: May be because language or more probably my lack of training in logic, what
| |
| − | does mean that "associative law and distributive law are by default in force"?
| |
| − | | |
| − | Those were some tricky Peirces,
| |
| − | and I was trying to dodge them
| |
| − | as artful as could be, but now
| |
| − | you have fastly apprehended me!
| |
| − | | |
| − | It may be partly that I left out the initial sections of this paper where Peirce
| |
| − | discusses how he will regard the ordinarily applicable principles in the process
| |
| − | of trying to extend and generalize them (CP 3.45-62), but there may be also an
| |
| − | ambiguity in Peirce's use of the phrase "absolute conditions" (CP 3.62-68).
| |
| − | Does he mean "absolutely necessary", "indispensable", "inviolate", or
| |
| − | does he mean "the conditions applying to the logic of absolute terms",
| |
| − | in which latter case we would expect to alter them sooner or later?
| |
| − | | |
| − | We lose the commutative law, xy = yx, as soon as we extend to 2-adic relations,
| |
| − | but keep the associative law, x(yz) = (xy)z, as the multiplication of 2-adics
| |
| − | is the logical analogue of ordinary matrix multiplication, and Peirce like
| |
| − | most mathematicians treats the double distributive law, x(y + z) = xy + xz
| |
| − | and (x + y)z = xz + yz, and as something that must be striven to preserve
| |
| − | as far as possible.
| |
| − | | |
| − | Strictly speaking, Peirce is already using a principle that goes beyond
| |
| − | the ordinary associative law, but that is recognizably analogous to it,
| |
| − | for example, in the modified Othello case, where (J:J:D)(J:D)(D) = J.
| |
| − | If it were strictly associative, then we would have the following:
| |
| − | | |
| − | 1. (J:J:D)((J:D)(D)) = (J:J:D)(J) = 0?
| |
| − | | |
| − | 2. ((J:J:D)(J:D))(D) = (J)(D) = 0?
| |
| − | | |
| − | In other words, the intended relational linkage would be broken.
| |
| − | However, the type of product that Peirce is taking for granted
| |
| − | in this situation often occurs in mathematics in just this way.
| |
| − | There is another location where he comments more fully on this,
| |
| − | but I have the sense that it was a late retrospective remark,
| |
| − | and I do not recall if it was in CP or in the microfilm MS's
| |
| − | that I read it.
| |
| − | | |
| − | By "default" conditions I am referring more or less to what
| |
| − | Peirce says at the end of CP 3.69, where he use an argument
| |
| − | based on the distributive principle to rationalize the idea
| |
| − | that 'A term multiplied by two relatives shows that the same
| |
| − | individual is in the two relations'. This means, for example,
| |
| − | that one can let "`g`'o'h", without subjacent marks or numbers,
| |
| − | be interpreted on the default convention of "overlapping scopes",
| |
| − | where the two correlates of `g` are given by the next two terms
| |
| − | in line, namely, 'o' and h, and the single correlate of 'o' is
| |
| − | given by the very next term in line, namely, h. Thus, it is
| |
| − | only when this natural scoping cannot convey the intended
| |
| − | sense that we have to use more explicit mark-up devices.
| |
| − | | |
| − | BM: About another point: do you think that the LOR could be of some help to solve
| |
| − | the puzzle of the "second way of dividing signs" where CSP concludes that 66
| |
| − | classes could be made out of the 10 divisions (Letters to lady Welby)?
| |
| − | (As I see them, the ten divisions involve a mix of relative terms,
| |
| − | dyadic relations and a triadic one. In order to make 66 classes
| |
| − | it is clear that these 10 divisions have to be stated under some
| |
| − | linear order. The nature of this order is at the bottom of the
| |
| − | disagreements on the subject).
| |
| − | | |
| − | This topic requires a longer excuse from me
| |
| − | than I am able to make right now, but maybe
| |
| − | I'll get back to it later today or tomorrow.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 11
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | | |
| − | BM: About another point: do you think that the LOR could be of some help
| |
| − | to solve the puzzle of the "second way of dividing signs" where CSP
| |
| − | concludes that 66 classes could be made out of the 10 divisions
| |
| − | (Letters to lady Welby)? (As I see them, the ten divisions
| |
| − | involve a mix of relative terms, dyadic relations and
| |
| − | a triadic one. In order to make 66 classes it is
| |
| − | clear that these 10 divisions have to be stated
| |
| − | under some linear order. The nature of this
| |
| − | order is at the bottom of the disagreements
| |
| − | on the subject).
| |
| − | | |
| − | Yes. At any rate, I have a pretty clear sense from reading Peirce's work
| |
| − | in the period 1865-1870 that the need to understand the function of signs
| |
| − | in scientific inquiry is one of the main reasons he found himself forced
| |
| − | to develop both the theory of information and the logic of relatives.
| |
| − | | |
| − | Peirce's work of this period is evenly distributed across the extensional
| |
| − | and intensional pans of the balance in a way that is very difficult for us
| |
| − | to follow anymore. I remember when I started looking into this I thought of
| |
| − | myself as more of an "intensional, synthetic" than an "extensional, analytic"
| |
| − | type of thinker, but that seems like a long time ago, as it soon became clear
| |
| − | that much less work had been done in the Peirce community on the extensional
| |
| − | side of things, while that was the very facet that needed to be polished up
| |
| − | in order to reconnect logic with empirical research and mathematical models.
| |
| − | So I fear that I must be content that other able people are working on the
| |
| − | intensional classification of sign relations.
| |
| − | | |
| − | Still, the way that you pose the question is very enticing,
| |
| − | so maybe it is time for me to start thinking about this
| |
| − | aspect of sign relations again, if you could say more
| |
| − | about it.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 12
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | | |
| − | BM: The pairing "intensional, synthetic" against the other "extensional, analytic"
| |
| − | is not one that I would have thought so. I would have paired synthetic with
| |
| − | extensional because synthesis consists in adding new facts to an already made
| |
| − | conception. On the other side analysis looks to be the determination of
| |
| − | features while neglecting facts. But may be there is something like
| |
| − | a symmetry effect leading to the same view from two different points.
| |
| − | | |
| − | Oh, it's not too important, as I don't put a lot of faith in such divisions,
| |
| − | and the problem for me is always how to integrate the facets of the object,
| |
| − | or the faculties of the mind -- but there I go being synthetic again!
| |
| − | | |
| − | I was only thinking of a conventional contrast that used to be drawn
| |
| − | between different styles of thinking in mathematics, typically one
| |
| − | points to Descartes, and the extensionality of analytic geometry,
| |
| − | versus Desargues, and the intensionality of synthetic geometry.
| |
| − | | |
| − | It may appear that one has side-stepped the issue of empiricism
| |
| − | that way, but then all that stuff about the synthetic a priori
| |
| − | raises its head, and we have Peirce's insight that mathematics
| |
| − | is observational and even experimental, and so I must trail off
| |
| − | into uncoordinated elliptical thoughts ...
| |
| − | | |
| − | The rest I have to work at a while, and maybe go back to the Welby letters.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 13
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | | |
| − | BM: I will try to make clear the matter, at least as far as I understand it
| |
| − | for now. We can summarize in a table the 10 divisions with their number
| |
| − | in a first column, their title in current (peircean) language in the second
| |
| − | and some kind of logical notation in the third. The sources come mainly from
| |
| − | the letters to Lady Welby. While the titles come from CP 8.344, the third column
| |
| − | comes from my own interpretation.
| |
| − | | |
| − | BM: So we get:
| |
| − | | |
| − | I - According to the Mode of Apprehension of the Sign itself - S
| |
| − | II - According to the Mode of Presentation of the Immediate Object - Oi
| |
| − | III - According to the Mode of Being of the Dynamical Object - Od
| |
| − | IV - According to the Relation of the Sign to its Dynamical Object - S-Od
| |
| − | V - According to the Mode of Presentation of the Immediate Interpretant - Ii
| |
| − | VI - According to the Mode of Being of the Dynamical Interpretant - Id
| |
| − | VII - According to the relation of the Sign to the Dynamical Interpretant - S-Id
| |
| − | VIII - According to the Nature of the Normal Interpretant - If
| |
| − | IX - According to the the relation of the Sign to the Normal Interpretant - S-If
| |
| − | X - According to the Triadic Relation of the Sign to its Dynamical Object
| |
| − | and to its Normal Interpretant - S-Od-If
| |
| − | | |
| − | For my future study, I will reformat the table in a way that I can muse upon.
| |
| − | I hope the roman numerals have not become canonical, as I cannot abide them.
| |
| − | | |
| − | Table. Ten Divisions of Signs (Peirce, Morand)
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | | According To: | Of: | To: | |
| |
| − | o===o===============o==================o==================o===============o
| |
| − | | 1 | Apprehension | Sign Itself | | S |
| |
| − | | 2 | Presentation | Immediate Object | | O_i |
| |
| − | | 3 | Being | Dynamical Object | | O_d |
| |
| − | | 4 | Relation | Sign | Dynamical Object | S : O_d |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | 5 | Presentation | Immediate Interp | | I_i |
| |
| − | | 6 | Being | Dynamical Interp | | I_d |
| |
| − | | 7 | Relation | Sign | Dynamical Interp | S : I_d |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | 8 | Nature | Normal Interp | | I_f |
| |
| − | | 9 | Relation | Sign | Normal Interp | S : I_f |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | A | Relation | Sign | Dynamical Object | |
| |
| − | | | | | & Normal Interp | S : O_d : I_f |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | |
| − | Just as I have always feared, this classification mania
| |
| − | appears to be communicable! But now I must definitely
| |
| − | review the Welby correspondence, as all this stuff was
| |
| − | a blur to my sensibilities the last 10 times I read it.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 14
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | | |
| − | [Table. Ten Divisions of Signs (Peirce, Morand)]
| |
| − | | |
| − | BM: Yes this is clearer (in particular in expressing relations with :)
| |
| − | | |
| − | This is what Peirce used to form elementary relatives, for example,
| |
| − | o:s:i = <o, s, i>, and I find it utterly ubertous in a wide variety
| |
| − | of syntactic circumstances.
| |
| − | | |
| − | BM: I suggest making a correction to myself if
| |
| − | the table is destinate to become canonic.
| |
| − | | |
| − | Hah! Good one!
| |
| − | | |
| − | BM: I probably made a too quick jump from Normal Interpretant to Final Interpretant.
| |
| − | As we know, the final interpretant, the ultimate one is not a sign for Peirce
| |
| − | but a habit. So for the sake of things to come it would be more careful to
| |
| − | retain I_n in place of I_f for now.
| |
| − | | |
| − | This accords with my understanding of how the word is used in mathematics.
| |
| − | In my own work it has been necessary to distinguish many different species
| |
| − | of expressions along somewhat similar lines, for example: arbitrary, basic,
| |
| − | canonical, decidable, normal, periodic, persistent, prototypical, recurrent,
| |
| − | representative, stable, typical, and so on. So I will make the changes below:
| |
| − | | |
| − | Table. Ten Divisions of Signs (Peirce, Morand)
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | | According To: | Of: | To: | |
| |
| − | o===o===============o==================o==================o===============o
| |
| − | | 1 | Apprehension | Sign Itself | | S |
| |
| − | | 2 | Presentation | Immediate Object | | O_i |
| |
| − | | 3 | Being | Dynamical Object | | O_d |
| |
| − | | 4 | Relation | Sign | Dynamical Object | S : O_d |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | 5 | Presentation | Immediate Interp | | I_i |
| |
| − | | 6 | Being | Dynamical Interp | | I_d |
| |
| − | | 7 | Relation | Sign | Dynamical Interp | S : I_d |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | 8 | Nature | Normal Interp | | I_n |
| |
| − | | 9 | Relation | Sign | Normal Interp | S : I_n |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | A | Tri. Relation | Sign | Dynamical Object | |
| |
| − | | | | | & Normal Interp | S : O_d : I_n |
| |
| − | o---o---------------o------------------o------------------o---------------o
| |
| − | | |
| − | BM: Peirce gives the following definition (CP 8.343):
| |
| − | | |
| − | BM, quoting CSP:
| |
| − | | |
| − | | It is likewise requisite to distinguish
| |
| − | | the 'Immediate Interpretant', i.e. the
| |
| − | | Interpretant represented or signified in
| |
| − | | the Sign, from the 'Dynamic Interpretant',
| |
| − | | or effect actually produced on the mind
| |
| − | | by the Sign; and both of these from
| |
| − | | the 'Normal Interpretant', or effect
| |
| − | | that would be produced on the mind by
| |
| − | | the Sign after sufficient development
| |
| − | | of thought.
| |
| − | |
| |
| − | | C.S. Peirce, 'Collected Papers', CP 8.343.
| |
| − | | |
| − | Well, you've really tossed me in the middle of the briar patch now!
| |
| − | I must continue with my reading from the 1870 LOR, but now I have
| |
| − | to add to my do-list the problems of comparing the whole variorum
| |
| − | of letters and drafts of letters to Lady Welby. I only have the
| |
| − | CP 8 and Wiener versions here, so I will depend on you for ample
| |
| − | excerpts from the Lieb volume.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 15
| |
| − | | |
| − | | |
| − | | |
| − | I will need to go back and pick up the broader contexts of your quotes.
| |
| − | For ease of study I break Peirce's long paragraphs into smaller pieces.
| |
| − | | |
| − | | It seems to me that one of the first useful steps toward a science
| |
| − | | of 'semeiotic' ([Greek 'semeiootike']), or the cenoscopic science
| |
| − | | of signs, must be the accurate definition, or logical analysis,
| |
| − | | of the concepts of the science.
| |
| − | |
| |
| − | | I define a 'Sign' as anything which on the one hand
| |
| − | | is so determined by an Object and on the other hand
| |
| − | | so determines an idea in a person's mind, that this
| |
| − | | latter determination, which I term the 'Interpretant'
| |
| − | | of the sign, is thereby mediately determined by that
| |
| − | | Object.
| |
| − | |
| |
| − | | A sign, therefore, has a triadic relation to
| |
| − | | its Object and to its Interpretant. But it is
| |
| − | | necessary to distinguish the 'Immediate Object',
| |
| − | | or the Object as the Sign represents it, from
| |
| − | | the 'Dynamical Object', or really efficient
| |
| − | | but not immediately present Object.
| |
| − | |
| |
| − | | It is likewise requisite to distinguish
| |
| − | | the 'Immediate Interpretant', i.e. the
| |
| − | | Interpretant represented or signified in
| |
| − | | the Sign, from the 'Dynamic Interpretant',
| |
| − | | or effect actually produced on the mind
| |
| − | | by the Sign; and both of these from
| |
| − | | the 'Normal Interpretant', or effect
| |
| − | | that would be produced on the mind by
| |
| − | | the Sign after sufficient development
| |
| − | | of thought.
| |
| − | |
| |
| − | | On these considerations I base a recognition of ten respects in which Signs
| |
| − | | may be divided. I do not say that these divisions are enough. But since
| |
| − | | every one of them turns out to be a trichotomy, it follows that in order
| |
| − | | to decide what classes of signs result from them, I have 3^10, or 59049,
| |
| − | | difficult questions to carefully consider; and therefore I will not
| |
| − | | undertake to carry my systematical division of signs any further,
| |
| − | | but will leave that for future explorers.
| |
| − | |
| |
| − | | C.S. Peirce, 'Collected Papers', CP 8.343.
| |
| − | | |
| − | You never know when the future explorer will be yourself.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 16
| |
| − | | |
| − | | |
| − | | |
| − | Burks, the editor of CP 8, attaches this footnote
| |
| − | to CP 8.342-379, "On the Classification of Signs":
| |
| − | | |
| − | | From a partial draft of a letter to Lady Welby, bearing
| |
| − | | the dates of 24, 25, and 28 December 1908, Widener IB3a,
| |
| − | | with an added quotation in 368n23. ...
| |
| − | | |
| − | There is a passage roughly comparable to CP 8.343 in a letter
| |
| − | to Lady Welby dated 23 December 1908, pages 397-409 in Wiener,
| |
| − | which is incidentally the notorious "sop to Cerberus" letter:
| |
| − | | |
| − | | It is usual and proper to distinguish two Objects of a Sign,
| |
| − | | the Mediate without, and the Immediate within the Sign. Its
| |
| − | | Interpretant is all that the Sign conveys: acquaintance with
| |
| − | | its Object must be gained by collateral experience.
| |
| − | |
| |
| − | | The Mediate Object is the Object outside of the Sign; I call
| |
| − | | it the 'Dynamoid' Object. The Sign must indicate it by a hint;
| |
| − | | and this hint, or its substance, is the 'Immediate' Object.
| |
| − | |
| |
| − | | Each of these two Objects may be said to be capable of either of
| |
| − | | the three Modalities, though in the case of the Immediate Object,
| |
| − | | this is not quite literally true.
| |
| − | |
| |
| − | | Accordingly, the Dynamoid Object may be a Possible; when I term
| |
| − | | the Sign an 'Abstractive'; such as the word Beauty; and it will be
| |
| − | | none the less an Abstractive if I speak of "the Beautiful", since it is
| |
| − | | the ultimate reference, and not the grammatical form, that makes the sign
| |
| − | | an 'Abstractive'.
| |
| − | |
| |
| − | | When the Dynamoid Object is an Occurrence (Existent thing or Actual fact
| |
| − | | of past or future), I term the Sign a 'Concretive'; any one barometer
| |
| − | | is an example; and so is a written narrative of any series of events.
| |
| − | |
| |
| − | | For a 'Sign' whose Dynamoid Object is a Necessitant, I have at present
| |
| − | | no better designation than a 'Collective', which is not quite so bad a
| |
| − | | name as it sounds to be until one studies the matter: but for a person,
| |
| − | | like me, who thinks in quite a different system of symbols to words, it
| |
| − | | is so awkward and often puzzling to translate one's thought into words!
| |
| − | |
| |
| − | | If the Immediate Object is a "Possible", that is, if the Dynamoid Object
| |
| − | | is indicated (always more or less vaguely) by means of its Qualities, etc.,
| |
| − | | I call the Sign a 'Descriptive';
| |
| − | |
| |
| − | | if the Immediate is an Occurrence, I call the Sign a 'Designative';
| |
| − | |
| |
| − | | and if the Immediate Object is a Necessitant, I call the Sign a
| |
| − | | 'Copulant'; for in that case the Object has to be so identified
| |
| − | | by the Interpreter that the Sign may represent a necessitation.
| |
| − | | My name is certainly a temporary expedient.
| |
| − | |
| |
| − | | It is evident that a possible can determine nothing but a Possible,
| |
| − | | it is equally so that a Necessitant can be determined by nothing but
| |
| − | | a Necessitant. Hence it follows from the Definition of a Sign that
| |
| − | | since the Dynamoid Object determines the Immediate Object,
| |
| − | |
| |
| − | | Which determines the Sign itself,
| |
| − | | which determines the Destinate Interpretant
| |
| − | | which determines the Effective Interpretant
| |
| − | | which determines the Explicit Interpretant
| |
| − | |
| |
| − | | the six trichotomies, instead of determining 729 classes of signs,
| |
| − | | as they would if they were independent, only yield 28 classes;
| |
| − | | and if, as I strongly opine (not to say almost prove), there
| |
| − | | are four other trichotomies of signs of the same order of
| |
| − | | importance, instead of making 59,049 classes, these will
| |
| − | | only come to 66.
| |
| − | |
| |
| − | | The additional 4 trichotomies are undoubtedly, first:
| |
| − | |
| |
| − | | Icons*, Symbols, Indices,
| |
| − | |
| |
| − | |*(or Simulacra, Aristotle's 'homoiomata'), caught from Plato, who I guess took it
| |
| − | | from the Mathematical school of logic, for it earliest appears in the 'Phaedrus'
| |
| − | | which marks the beginning of Plato's being decisively influenced by that school.
| |
| − | | Lutoslowski is right in saying that the 'Phaedrus' is later than the 'Republic'
| |
| − | | but his date 379 B.C. is about eight years too early.
| |
| − | |
| |
| − | | and then 3 referring to the Interpretants. One of these I am pretty confident
| |
| − | | is into: 'Suggestives', 'Imperatives', 'Indicatives', where the Imperatives
| |
| − | | include the Interrogatives. Of the other two I 'think' that one must be
| |
| − | | into Signs assuring their Interpretants by:
| |
| − | |
| |
| − | | Instinct, Experience, Form.
| |
| − | |
| |
| − | | The other I suppose to be what, in my 'Monist'
| |
| − | | exposition of Existential Graphs, I called:
| |
| − | |
| |
| − | | Semes, Phemes, Delomes.
| |
| − | |
| |
| − | | CSP, 'Selected Writings', pp. 406-408.
| |
| − | |
| |
| − | |'Charles S. Peirce: Selected Writings (Values in a Universe of Chance)',
| |
| − | | edited with an introduction and notes by Philip P. Wiener, Dover,
| |
| − | | New York, NY, 1966. Originally published under the subtitle
| |
| − | | in parentheses above, Doubleday & Company, 1958.
| |
| − | | |
| − | But see CP 4.549-550 for a significant distinction between
| |
| − | the categories (or modalities) and the orders of intention.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 17
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | JA: In closing, observe that the teridentity relation has turned up again
| |
| − | in this context, as the second comma-ing of the universal term itself:
| |
| − | | |
| − | 1,, = B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O.
| |
| − | | |
| − | HC: I see that you've come around to a mention of teridentity again, Jon.
| |
| − | Still, if I recall the prior discussions, then no one doubts that we
| |
| − | can have a system of notation in which teridentity appears (I don't
| |
| − | actually see it here).
| |
| − | | |
| − | Perhaps we could get at the root of the misunderstanding
| |
| − | if you tell me why you don't actually see the concept of
| |
| − | teridentity being exemplified here.
| |
| − | | |
| − | If it's only a matter of having lost the context of the
| |
| − | present discussion over the break, then you may find the
| |
| − | previous notes archived at the distal ends of the ur-links
| |
| − | that I append below (except for the first nine discussion
| |
| − | notes that got lost in a disk crash at the Arisbe Dev site).
| |
| − | | |
| − | HC: Also, I think we can have a system of notation in which
| |
| − | teridentity is needed. Those points seem reasonably clear.
| |
| − | | |
| − | The advantage of a concept is the integration of a species of manifold.
| |
| − | The necessity of a concept is the incapacity to integrate it otherwise.
| |
| − | | |
| − | Of course, no one should be too impressed with a concept that
| |
| − | is only the artifact of a particular system of representation.
| |
| − | So before we accord a concept the status of addressing reality,
| |
| − | and declare it a term of some tenured office in our intellects,
| |
| − | we would want to see some evidence that it helps us to manage
| |
| − | a reality that we cannot see a way to manage any other way.
| |
| − | | |
| − | Granted.
| |
| − | | |
| − | Now how in general do we go about an investiture of this sort?
| |
| − | That is the big question that would serve us well to consider
| |
| − | in the process of the more limited investigation of identity.
| |
| − | Indeed, I do not see how it is possible to answer the small
| |
| − | question if no understanding is reached on the big question.
| |
| − | | |
| − | HC: What remains relatively unclear is why we should need a system of notation
| |
| − | in which teridentity appears or is needed as against one in which it seems
| |
| − | not to be needed -- since assertion of identity can be made for any number
| |
| − | of terms in the standard predicate calculus.
| |
| − | | |
| − | This sort of statement totally non-plusses me.
| |
| − | It seems like a complete non-sequitur or even
| |
| − | a contradiction in terms to me.
| |
| − | | |
| − | The question is about the minimal adequate resource base for
| |
| − | defining, deriving, or generating all of the concepts that we
| |
| − | need for a given but very general type of application that we
| |
| − | conventionally but equivocally refer to as "logic". You seem
| |
| − | to be saying something like this: We don't need 3-identity
| |
| − | because we have 4-identity, 5-identity, 6-identity, ..., in
| |
| − | the "standard predicate calculus". The question is not what
| |
| − | concepts are generated in all the generations that follow the
| |
| − | establishment of the conceptual resource base (axiom system),
| |
| − | but what is the minimal set of concepts that we can use to
| |
| − | generate the needed collection of concepts. And there the
| |
| − | answer is, in a way that is subject to the usual sorts of
| |
| − | mathematical proof, that 3-identity is the minimum while
| |
| − | 2-identity is not big enough to do the job we want to do.
| |
| − | | |
| − | Logic Of Relatives 01-41, LOR Discussion Notes 10-17.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 18
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | JA: but now I have to add to my do-list the problems of comparing the
| |
| − | whole variorum of letters and drafts of letters to Lady Welby.
| |
| − | I only have the CP 8 and Wiener versions here, so I will
| |
| − | depend on you for ample excerpts from the Lieb volume.
| |
| − | | |
| − | BM: I made such a kind of comparison some time ago. I selected
| |
| − | the following 3 cases on the criterium of alternate "grounds".
| |
| − | Hoping it could save some labor. The first rank expressions
| |
| − | come from the MS 339 written in Oct. 1904 and I label them
| |
| − | with an (a). I think that it is interesting to note that
| |
| − | they were written four years before the letters to Welby
| |
| − | and just one or two years after the Syllabus which is the
| |
| − | usual reference for the classification in 3 trichotomies
| |
| − | and 10 classes. The second (b) is our initial table (from
| |
| − | a draft to Lady Welby, Dec. 1908, CP 8.344) and the third
| |
| − | (c) comes from a letter sent in Dec. 1908 (CP 8.345-8.376).
| |
| − | A tabular presentation would be better but I can't do it.
| |
| − | Comparing (c) against (a) and (b) is informative, I think.
| |
| − | | |
| − | Is this anywhere that it can be linked to from Arisbe?
| |
| − | I've seen many pretty pictures of these things over the
| |
| − | years, but may have to follow my own gnosis for a while.
| |
| − | | |
| − | Pages I have bookmarked just recently,
| |
| − | but not really had the chance to study:
| |
| − | | |
| − | http://www.digitalpeirce.org/hoffmann/p-sighof.htm
| |
| − | http://www.csd.uwo.ca/~merkle/thesis/Introduction.html
| |
| − | http://members.door.net/arisbe/menu/library/aboutcsp/merkle/hci-abstract.htm
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 19
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | I now have three partially answered messages on the table,
| |
| − | so I will just grab this fragment off the top of the deck.
| |
| − | | |
| − | BM: Peirce gives the following definition (CP 8.343):
| |
| − | | |
| − | BM, quoting CSP:
| |
| − | | |
| − | | It is likewise requisite to distinguish
| |
| − | | the 'Immediate Interpretant', i.e. the
| |
| − | | Interpretant represented or signified in
| |
| − | | the Sign, from the 'Dynamic Interpretant',
| |
| − | | or effect actually produced on the mind
| |
| − | | by the Sign; and both of these from
| |
| − | | the 'Normal Interpretant', or effect
| |
| − | | that would be produced on the mind by
| |
| − | | the Sign after sufficient development
| |
| − | | of thought.
| |
| − | |
| |
| − | | C.S. Peirce, 'Collected Papers', CP 8.343.
| |
| − | | |
| − | JA: Well, you've really tossed me in the middle of the briar patch now!
| |
| − | I must continue with my reading from the 1870 LOR, ...
| |
| − | | |
| − | BM: Yes indeed! I am irritated by having not the necessary
| |
| − | turn of mind to fully grasp it. But it seems to be a
| |
| − | prerequisite in order to understand the very meaning
| |
| − | of the above table. It could be the same for:
| |
| − | | |
| − | BM, quoting CSP:
| |
| − | | |
| − | | I define a 'Sign' as anything which on the one hand
| |
| − | | is so determined by an Object and on the other hand
| |
| − | | so determines an idea in a person's mind, that this
| |
| − | | latter determination, which I term the 'Interpretant'
| |
| − | | of the sign, is thereby mediately determined by that
| |
| − | | Object.
| |
| − | | |
| − | BM: The so-called "latter determination" would make the 'Interpretant'
| |
| − | a tri-relative term into a teridentity involving Sign and Object.
| |
| − | Isn't it?
| |
| − | | |
| − | BM: I thought previously that the Peirce's phrasing was just applying the
| |
| − | principle of transitivity. From O determines S and S determines I,
| |
| − | it follows: O determines I. But this is not the same as teridentity.
| |
| − | Do you think so or otherwise?
| |
| − | | |
| − | My answers are "No" and "Otherwise".
| |
| − | | |
| − | Continuing to discourse about definite universes thereof,
| |
| − | the 3-identity term over the universe 1 = {A, B, C, D, ...} --
| |
| − | I only said it was definite, I didn't say it wasn't vague! --
| |
| − | designates, roughly speaking, the 3-adic relation that may
| |
| − | be hinted at by way of the following series:
| |
| − | | |
| − | 1,, = A:A:A +, B:B:B +, C:C:C +, D:D:D +, ...
| |
| − | | |
| − | I did a study on Peirce's notion of "determination".
| |
| − | As I understand it so far, we need to keep in mind
| |
| − | that it is more fundamental than causation, can be
| |
| − | a form of "partial determination", and is roughly
| |
| − | formal, mathematical, or "information-theoretic",
| |
| − | not of necessity invoking any temporal order.
| |
| − | | |
| − | For example, when we say "The points A and B determine the line AB",
| |
| − | this invokes the concept of a 3-adic relation of determination that
| |
| − | does not identify A, B, AB, is not transitive, as transitivity has
| |
| − | to do with the composition of 2-adic relations and would amount to
| |
| − | the consideration of a degenerate 3-adic relation in this context.
| |
| − | | |
| − | Now, it is possible to have a sign relation q whose sum enlists
| |
| − | an elementary sign relation O:S:I where O = S = I. For example,
| |
| − | it makes perfect sense to me to say that the whole universe may
| |
| − | be a sign of itself to itself, so the conception is admissable.
| |
| − | But this amounts to a very special case, by no means general.
| |
| − | More generally, we are contemplating sums like the following:
| |
| − | | |
| − | q = O1:S1:I1 +, O2:S2:I2 +, O3:S3:I3 +, ...
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 20
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | JR = Joe Ransdell
| |
| − | | |
| − | HC: Though I certainly hesitate to think that we are separated
| |
| − | from the world by a veil of signs, it seems clear, too, on
| |
| − | Peircean grounds, that no sign can ever capture its object
| |
| − | completely.
| |
| − | | |
| − | JR: Any case of self-representation is a case of sign-object identity,
| |
| − | in some sense of "identity". I have argued in various places that
| |
| − | this is the key to the doctrine of immediate perception as it occurs
| |
| − | in Peirce's theory.
| |
| − | | |
| − | To put the phrase back on the lathe:
| |
| − | | |
| − | | We are not separated from the world by a veil of signs --
| |
| − | | we are the veil of signs.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 21
| |
| − | | |
| − | | |
| − | | |
| − | AS = Armando Sercovich
| |
| − | | |
| − | AS: We are not separated from the world by a veil of signs nor we are a veil of signs.
| |
| − | Simply we are signs.
| |
| − | | |
| − | AS, quoting CSP:
| |
| − | | |
| − | | The *man-sign* acquires information, and comes to mean more than he did before.
| |
| − | | But so do words. Does not electricity mean more now than it did in the days
| |
| − | | of Franklin? Man makes the word, and the word means nothing which the man
| |
| − | | has not made it mean, and that only to some man. But since man can think
| |
| − | | only by means of words or other external symbols, these might turn round
| |
| − | | and say: "You mean nothing which we have not taught you, and then only
| |
| − | | so far as you address some word as the interpretant of your thought".
| |
| − | | In fact, therefore, men and words reciprocally educate each other;
| |
| − | | each increase of a man's information involves, and is involved by,
| |
| − | | a corresponding increase of a word's information.
| |
| − | |
| |
| − | | Without fatiguing the reader by stretching this parallelism too far, it is
| |
| − | | sufficient to say that there is no element whatever of man's consciousness
| |
| − | | which has not something corresponding to it in the word; and the reason is
| |
| − | | obvious. It is that the word or sign which man uses *is* the man itself.
| |
| − | | For, as the fact that every thought is a sign, taken in conjunction with
| |
| − | | the fact that life is a train of thought, proves that man is a sign; so,
| |
| − | | that every thought is an *external* sign proves that man is an external
| |
| − | | sign. That is to say, the man and the external sign are identical, in
| |
| − | | the same sense in which the words 'homo' and 'man' are identical. Thus
| |
| − | | my language is the sum total of myself; for the man is the thought ...
| |
| − | |
| |
| − | |'Charles S. Peirce: Selected Writings (Values in a Universe of Chance)',
| |
| − | | edited with an introduction and notes by Philip P. Wiener, Dover,
| |
| − | | New York, NY, 1966. Originally published under the subtitle
| |
| − | | in parentheses above, Doubleday & Company, 1958.
| |
| − | | |
| − | I read you loud and clear.
| |
| − | Every manifold must have
| |
| − | its catalytic converter.
| |
| − | | |
| − | <Innumerate Continuation:>
| |
| − | | |
| − | TUC = The Usual CISPEC
| |
| − | | |
| − | TUC Alert:
| |
| − | | |
| − | | E.P.A. Says Catalytic Converter Is
| |
| − | | Growing Cause of Global Warming
| |
| − | | By Matthew L. Wald
| |
| − | | Copyright 1998 The New York Times
| |
| − | | May 29, 1998
| |
| − | | -----------------------------------------------------------------------
| |
| − | | WASHINGTON -- The catalytic converter, an invention that has sharply
| |
| − | | reduced smog from cars, has now become a significant and growing cause
| |
| − | | of global warming, according to the Environmental Protection Agency
| |
| − | | |
| − | Much as I would like to speculate ad libitum on these exciting new prospects for the
| |
| − | application of Peirce's chemico-algebraic theory of logic to the theorem-o-dynamics
| |
| − | of auto-semeiosis, I must get back to "business as usual" (BAU) ...
| |
| − | | |
| − | And now a word from our sponsor ...
| |
| − | | |
| − | http://www2.naias.com/
| |
| − | | |
| − | Reporting from Motown ---
| |
| − | | |
| − | Jon Awbrey
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 22
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | | |
| − | HC: You quote the following passage from a prior posting of mine:
| |
| − | | |
| − | HC: What remains relatively unclear is why we should need a system of notation
| |
| − | in which teridentity appears or is needed as against one in which it seems
| |
| − | not to be needed -- since assertion of identity can be made for any number
| |
| − | of terms in the standard predicate calculus.
| |
| − | | |
| − | HC: You comment as follows:
| |
| − | | |
| − | JA: This sort of statement totally non-plusses me.
| |
| − | It seems like a complete non-sequitur or even
| |
| − | a contradiction in terms to me.
| |
| − | | |
| − | JA: The question is about the minimal adequate resource base for
| |
| − | defining, deriving, or generating all of the concepts that we
| |
| − | need for a given but very general type of application that we
| |
| − | conventionally but equivocally refer to as "logic". You seem
| |
| − | to be saying something like this: We don't need 3-identity
| |
| − | because we have 4-identity, 5-identity, 6-identity, ..., in
| |
| − | the "standard predicate calculus". The question is not what
| |
| − | concepts are generated in all the generations that follow the
| |
| − | establishment of the conceptual resource base (axiom system),
| |
| − | but what is the minimal set of concepts that we can use to
| |
| − | generate the needed collection of concepts. And there the
| |
| − | answer is, in a way that is subject to the usual sorts of
| |
| − | mathematical proof, that 3-identity is the minimum while
| |
| − | 2-identity is not big enough to do the job we want to do.
| |
| − | | |
| − | HC: I have fallen a bit behind on this thread while attending to some other
| |
| − | matters, but in this reply, you do seem to me to be coming around to an
| |
| − | understanding of the issues involved, as I see them. You put the matter
| |
| − | this way, "We don't need 3-identity because we have 4-identity, 5-identity,
| |
| − | 6-identity, ..., in the 'standard predicate calculus'". Actually, as I think
| |
| − | you must know, there is no such thing as "4-identity", "5-identity", etc., in
| |
| − | the standard predicate calculus. It is more that such concepts are not needed,
| |
| − | just as teridentity is not needed, since the general apparatus of the predicate
| |
| − | calculus allows us to express identity among any number of terms without special
| |
| − | provision beyond "=".
| |
| − | | |
| − | No, that is not the case. Standard predicate calculus allows the expression
| |
| − | of predicates I_k, for k = 2, 3, 4, ..., such that I_k (x_1, ..., x_k) holds
| |
| − | if and only if all x_j, for j = 1 to k, are identical. So predicate calculus
| |
| − | contains a k-identity predicate for all such k. So whether "they're in there"
| |
| − | is not an issue. The question is whether these or any other predicates can be
| |
| − | constructed or defined in terms of 2-adic relations alone. And the answer is
| |
| − | no, they cannot. The vector of the misconception counterwise appears to be
| |
| − | as various a virus as the common cold, and every bit as resistant to cure.
| |
| − | I have taken the trouble to enumerate some of the more prevalent strains,
| |
| − | but most of them appear to go back to the 'Principia Mathematica', and
| |
| − | the variety of nominalism called "syntacticism" -- Ges-und-heit! --
| |
| − | that was spread by it, however unwittedly by some of its carriers.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 23
| |
| − | | |
| − | | |
| − | | |
| − | In trying to answer the rest of your last note,
| |
| − | it seems that we cannot go any further without
| |
| − | achieving some concrete clarity as to what is
| |
| − | denominated by "standard predicate calculus",
| |
| − | that is, "first order logic", or whatever.
| |
| − | | |
| − | There is a "canonical" presentation of the subject, as I remember it, anyway,
| |
| − | in the following sample of materials from Chang & Keisler's 'Model Theory'.
| |
| − | (There's a newer edition of the book, but this part of the subject hasn't
| |
| − | really changed all that much in ages.)
| |
| − | | |
| − | Model Theory 01-39
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 24
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | | |
| − | HC: I might object that "teridentity" seems to come
| |
| − | to a matter of "a=b & b=c", so that a specific
| |
| − | predicate of teridentity seems unnecessary.
| |
| − | | |
| − | I am presently concerned with expositing and interpreting
| |
| − | the logical system that Peirce laid out in the LOR of 1870.
| |
| − | It is my considered opinion after thirty years of study that
| |
| − | there are untapped resources remaining in this work that have
| |
| − | yet to make it through the filters of that ilk of syntacticism
| |
| − | that was all the rage in the late great 1900's. I find there
| |
| − | to be an appreciably different point of view on logic that is
| |
| − | embodied in Peirce's work, and until we have made the minimal
| |
| − | effort to read what he wrote it is just plain futile to keep
| |
| − | on pretending that we have already assimilated it, or that
| |
| − | we are qualified to evaluate its cogency.
| |
| − | | |
| − | The symbol "&" that you employ above denotes a mathematical object that
| |
| − | qualifies as a 3-adic relation. Independently of my own views, there
| |
| − | is an abundance of statements in evidence that mathematical thinkers
| |
| − | from Peirce to Goedel consider the appreciation of facts like this
| |
| − | to mark the boundary between realism and nominalism in regard to
| |
| − | mathematical objects.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 25
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | HC: I might object that "teridentity" seems to come
| |
| − | to a matter of "a=b & b=c", so that a specific
| |
| − | predicate of teridentity seems unnecessary.
| |
| − | | |
| − | JA: I am presently concerned with expositing and interpreting
| |
| − | the logical system that Peirce laid out in the LOR of 1870.
| |
| − | It is my considered opinion after thirty years of study that
| |
| − | there are untapped resources remaining in this work that have
| |
| − | yet to make it through the filters of that ilk of syntacticism
| |
| − | that was all the rage in the late great 1900's. I find there
| |
| − | to be an appreciably different point of view on logic that is
| |
| − | embodied in Peirce's work, and until we have made the minimal
| |
| − | effort to read what he wrote it is just plain futile to keep
| |
| − | on pretending that we have already assimilated it, or that
| |
| − | we are qualified to evaluate its cogency.
| |
| − | | |
| − | JA: The symbol "&" that you employ above denotes a mathematical object that
| |
| − | qualifies as a 3-adic relation. Independently of my own views, there
| |
| − | is an abundance of statements in evidence that mathematical thinkers
| |
| − | from Peirce to Goedel consider the appreciation of facts like this
| |
| − | to mark the boundary between realism and nominalism in regard to
| |
| − | mathematical objects.
| |
| − | | |
| − | HC: I would agree, I think, that "&" may be thought of
| |
| − | as a function mapping pairs of statements onto the
| |
| − | conjunction of that pair.
| |
| − | | |
| − | Yes, indeed, in the immortal words of my very first college algebra book:
| |
| − | "A binary operation is a ternary relation". As it happens, the symbol "&"
| |
| − | is equivocal in its interpretation -- computerese today steals a Freudian
| |
| − | line and dubs it "polymorphous" -- it can be regarded in various contexts
| |
| − | as a 3-adic relation on syntactic elements called "sentences", on logical
| |
| − | elements called "propositions", or on truth values collated in the boolean
| |
| − | domain B = {false, true} = {0, 1}. But the mappings and relations between
| |
| − | all of these interpretive choices are moderately well understood. Still,
| |
| − | no matter how many ways you enumerate for looking at a B-bird, the "&" is
| |
| − | always 3-adic. And that is sufficient to meet your objection, so I think
| |
| − | I will just leave it there until next time.
| |
| − | | |
| − | On a related note, that I must postpone until later:
| |
| − | We seem to congrue that there is a skewness between
| |
| − | the way that most mathematicians use logic and some
| |
| − | philosophers talk about logic, but I may not be the
| |
| − | one to set it adjoint, much as I am inclined to try.
| |
| − | At the moment I have this long-post-poned exponency
| |
| − | to carry out. I will simply recommend for your due
| |
| − | consideration Peirce's 1870 Logic Of Relatives, and
| |
| − | leave it at that. There's a cornucopiousness to it
| |
| − | that's yet to be dreamt of in the philosophy of the
| |
| − | 1900's. I am doing what I can to infotain you with
| |
| − | the Gardens of Mathematical Recreations that I find
| |
| − | within Peirce's work, and that's in direct response
| |
| − | to many, okay, a couple of requests. Perhaps I can
| |
| − | not hope to attain the degree of horticultural arts
| |
| − | that Gardners before me have exhibited in this work,
| |
| − | but then again, who could? Everybody's a critic --
| |
| − | but the better ones read first, and criticize later.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 26
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | | |
| − | HC: But on the other hand, it is not customary to think of "&" as
| |
| − | a relation among statements or sentences -- as, for instance,
| |
| − | logical implication is considered a logical relation between
| |
| − | statements or sentences.
| |
| − | | |
| − | Actually, it is the custom in many quarters to treat all of the
| |
| − | boolean operations, logical connectives, propositional relations,
| |
| − | or whatever you want to call them, as "equal citizens", having each
| |
| − | their "functional" (f : B^k -> B) and their "relational" (L c B^(k+1))
| |
| − | interpretations and applications. From this vantage, the interpretive
| |
| − | distinction that is commonly regarded as that between "assertion" and
| |
| − | mere "contemplation" is tantamount to a "pragmatic" difference between
| |
| − | computing the values of a function on a given domain of arguments and
| |
| − | computing the inverse of a function vis-a-vis a prospective true value.
| |
| − | This is the logical analogue of the way that our mathematical models
| |
| − | of reality have long been working, unsuspected and undisturbed by
| |
| − | most philosophers of science, I might add. If only the logical
| |
| − | side of the ledger were to be developed rather more fully than
| |
| − | it is at present, we might wake one of these days to find our
| |
| − | logical accounts of reality, finally, at long last, after an
| |
| − | overweaningly longish adolescence, beginning to come of age.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 27
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | | |
| − | HC: For, if I make an assertion A&B, then I am not asserting
| |
| − | that the statement A stands in a relation to a statement B.
| |
| − | Instead, I am asserting the conjunction A&B (which logically
| |
| − | implies both the conjuncts in view of the definition of "&").
| |
| − | | |
| − | Please try to remember where we came in. This whole play of
| |
| − | animadversions about 3-adicity and 3-identity is set against
| |
| − | the backdrop of a single point, over the issue as to whether
| |
| − | 3-adic relations are wholly dispensable or somehow essential
| |
| − | to logic, mathematics, and indeed to argument, communication,
| |
| − | and reasoning in general. Some folks clamor "Off with their
| |
| − | unnecessary heads!" -- other people, who are forced by their
| |
| − | occupations to pay close attention to the ongoing complexity
| |
| − | of the processes at stake, know that, far from finding 3-ads
| |
| − | in this or that isolated corner of the realm, one can hardly
| |
| − | do anything at all in the ways of logging or mathing without
| |
| − | running smack dab into veritable hosts of them.
| |
| − | | |
| − | I have just shown that "a=b & b=c" involves a 3-adic relation.
| |
| − | Some people would consider this particular 3-adic relation to
| |
| − | be more complex than the 3-identity relation, but that may be
| |
| − | a question of taste. At any rate, the 3-adic aspect persists.
| |
| − | | |
| − | HC: If "&" counts as a triadic relation, simply because it serves
| |
| − | to conjoin two statements into a third, then it would seem that
| |
| − | any binary relation 'R' will count as triadic, simply because
| |
| − | it places two things into a relation, which is a "third" thing.
| |
| − | By the same kind of reasoning a triadic relation, as ordinarily
| |
| − | understood would be really 4-adic.
| |
| − | | |
| − | The rest of your comments are just confused,
| |
| − | and do not use the terms as they are defined.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 28
| |
| − | | |
| − | | |
| − | | |
| − | JA = Jon Awbrey
| |
| − | JR = Joseph Ransdell
| |
| − | | |
| − | JA: Notice that Peirce follows the mathematician's usual practice,
| |
| − | then and now, of making the status of being an "individual" or
| |
| − | a "universal" relative to a discourse in progress. I have come
| |
| − | to appreciate more and more of late how radically different this
| |
| − | "patchwork" or "piecewise" approach to things is from the way of
| |
| − | some philosophers who seem to be content with nothing less than
| |
| − | many worlds domination, which means that they are never content
| |
| − | and rarely get started toward the solution of any real problem.
| |
| − | Just my observation, I hope you understand.
| |
| − | | |
| − | JR: Yes, I take this as underscoring and explicating the import of
| |
| − | making logic prior to rather than dependent upon metaphysics.
| |
| − | | |
| − | I think that Peirce, and of course many math folks, would take math
| |
| − | as prior, on a par, or even identical with logic. Myself I've been
| |
| − | of many minds about this over the years. The succinctest picture
| |
| − | that I get from Peirce is always this one:
| |
| − | | |
| − | | [Riddle of the Sphynx]
| |
| − | |
| |
| − | | Normative science rests largely on phenomenology and on mathematics;
| |
| − | | Metaphysics on phenomenology and on normative science.
| |
| − | |
| |
| − | | C.S. Peirce, CP 1.186 (1903)
| |
| − | |
| |
| − | |
| |
| − | | o Metaphysics
| |
| − | | /|
| |
| − | | / |
| |
| − | | / |
| |
| − | | Normative Science o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \|
| |
| − | | Mathematics o o Phenomenology
| |
| − | |
| |
| − | |
| |
| − | | ROTS. http://stderr.org/pipermail/inquiry/2004-March/001262.html
| |
| − | | |
| − | Logic being a normative science must depend on math and phenomenology.
| |
| − | | |
| − | Of course, it all depends on what a person means by "logic" ...
| |
| − | | |
| − | JA: I also observe that Peirce takes the individual objects of
| |
| − | a particular universe of discourse in a "generative" way,
| |
| − | not a "totalizing" way, and thus they afford us with the
| |
| − | basis for talking freely about collections, constructions,
| |
| − | properties, qualities, subsets, and "higher types", as
| |
| − | the phrase is mint.
| |
| − | | |
| − | JR: Would this be essentially the same as regarding quantification as
| |
| − | distributive rather than collective, i.e. we take the individuals
| |
| − | of a class one-by-one as selectable rather than as somehow given
| |
| − | all at once, collectively?
| |
| − | | |
| − | Gosh, that's a harder question. Your suggestion reminds me
| |
| − | of the way that some intuitionist and even some finitist
| |
| − | mathematicians talk when they reflect on math practice.
| |
| − | I have leanings that way, but when I have tried to
| |
| − | give up the classical logic axioms, I have found
| |
| − | them too built in to my way of thinking to quit.
| |
| − | Still, a healthy circumspection about about our
| |
| − | often-wrongly vaunted capacties to conceive of
| |
| − | totalities is a habitual part of current math.
| |
| − | Again, I think individuals are made not born,
| |
| − | that is, to some degree factitious and mere
| |
| − | compromises of this or that conveniency.
| |
| − | This is one of the reasons that I have
| |
| − | been trying to work out the details
| |
| − | of a functional approach to logic,
| |
| − | propostional, quantificational,
| |
| − | and relational.
| |
| − | | |
| − | Cf: INTRO 30. http://stderr.org/pipermail/inquiry/2004-November/001765.html
| |
| − | In: INTRO. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1720
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 29
| |
| − | | |
| − | | |
| − | | |
| − | JA = Jon Awbrey
| |
| − | GR = Gary Richmond
| |
| − | | |
| − | Re: LOR.COM 11.24. http://stderr.org/pipermail/inquiry/2004-November/001836.html
| |
| − | In: LOR.COM. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1755
| |
| − | | |
| − | JA: The manner in which these arrows and qualified arrows help us
| |
| − | to construct a suspension bridge that unifies logic, semiotics,
| |
| − | statistics, stochastics, and information theory will be one of
| |
| − | the main themes that I aim to elaborate throughout the rest of
| |
| − | this inquiry.
| |
| − | | |
| − | GR: Pretty ambitious, Jon. I'm sure you're up to it.
| |
| − | | |
| − | GR: I'd like to anticipate 3 versions: The mathematical (cactus diagrams, etc.),
| |
| − | the poetic, and the commonsensical -- ordinary language for those who are
| |
| − | NEITHER logicians NOR poets.
| |
| − | | |
| − | GR: Are you up to THAT?
| |
| − | | |
| − | Riddle A Body: "Time Enough, And Space, Excalibrate Co-Arthurs Should Apply"
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 30
| |
| − | | |
| − | | |
| − | | |
| − | JA = Jon Awbrey
| |
| − | GR = Gary Richmond
| |
| − | | |
| − | Re: LOR.DIS 29. http://stderr.org/pipermail/inquiry/2004-November/001838.html
| |
| − | In: LOR.DIS. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1768
| |
| − | | |
| − | JA: Riddle A Body: "Time Enough, And Space, Excalibrate Co-Arthurs Should Apply"
| |
| − | | |
| − | GR: Well said, and truly!
| |
| − | | |
| − | Body A Riddle: TEASE CASA = Fun House.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 31
| |
| − | | |
| − | | |
| − | | |
| − | Many illusions of selective reading -- like the myth that Peirce did not
| |
| − | discover quantification over indices until 1885 -- can be dispelled by
| |
| − | looking into his 1870 "Logic of Relatives". I started a web study of
| |
| − | this in 2002, reworked again in 2003 and 2004, the current version
| |
| − | of which can be found here:
| |
| − | | |
| − | LOR. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1750
| |
| − | LOR-COM. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1755
| |
| − | LOR-DIS. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1768
| |
| − | | |
| − | I've only gotten as far as the bare infrastructure of Peirce's 1870 LOR,
| |
| − | but an interesting feature of the study is that, if one draws the pictures
| |
| − | that seem almost demanded by his way of linking up indices over expressions,
| |
| − | then one can see a prototype of his much later logical graphs developing in
| |
| − | the text.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Discussion Note 32
| |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 1
| |
| − | | |
| − | | |
| − | | |
| − | BM: Several discussions could take place there,
| |
| − | as to the reasons for the number of divisions,
| |
| − | the reasons of the titles themselves. Another
| |
| − | one is my translation from "normal interpretant"
| |
| − | into "final interpretant" (which one is called
| |
| − | elsewhere "Eventual Interpretant" or "Destinate
| |
| − | Interpretant" by CSP). I let all this aside
| |
| − | to focus on the following remark:
| |
| − | | |
| − | BM: 6 divisions correspond to individual correlates:
| |
| − | | |
| − | (S, O_i, O_d, I_i, I_d, I_n),
| |
| − | | |
| − | 3 divisions correspond to dyads:
| |
| − | | |
| − | (S : O_d, S : I_d, S : I_n),
| |
| − | | |
| − | and the tenth to a triad:
| |
| − | | |
| − | (S : O_d : I_n).
| |
| − | | |
| − | This remark would itself deserve
| |
| − | a lot of explanations but one
| |
| − | more time I let this aside.
| |
| − | | |
| − | BM: Then we have the following very clear statement from Peirce:
| |
| − | | |
| − | | It follows from the Definition of a Sign
| |
| − | | that since the Dynamoid Object determines
| |
| − | | the Immediate Object,
| |
| − | | which determines the Sign,
| |
| − | | which determines the Destinate Interpretant
| |
| − | | which determines the Effective Interpretant
| |
| − | | which determines the Explicit Interpretant
| |
| − | |
| |
| − | | the six trichotomies, instead of determining 729 classes of signs,
| |
| − | | as they would if they were independent, only yield 28 classes; and
| |
| − | | if, as I strongly opine (not to say almost prove) there are four other
| |
| − | | trichotomies of signs of the same order of importance, instead of making
| |
| − | | 59049 classes, these will only come to 66.
| |
| − | |
| |
| − | | CSP, "Letter to Lady Welby", 14 Dec 1908, LW, p. 84.
| |
| − | | |
| − | BM: The separation made by CSP between 6 divisions and four others
| |
| − | seems to rely upon the suggested difference between individual
| |
| − | correlates and relations. We get the idea that the 10 divisions
| |
| − | are ordered on the whole and will end into 66 classes (by means of
| |
| − | three ordered modal values on each division: maybe, canbe, wouldbe).
| |
| − | Finally we have too the ordering for the divisions relative to the
| |
| − | correlates that I write in my notation:
| |
| − | | |
| − | Od -> Oi -> S -> If -> Id -> Ii.
| |
| − | | |
| − | BM: This order of "determinations" has bothered many people
| |
| − | but if we think of it as operative in semiosis, it seems
| |
| − | to be correct (at least to my eyes). Thus the question is:
| |
| − | where, how, and why the "four other trichotomies" fit in this
| |
| − | schema to obtain a linear ordering on the whole 10 divisions?
| |
| − | May be the question can be rephrased as: how intensional
| |
| − | relationships fit into an extensional one? Possibly the
| |
| − | question could be asked the other way. R. Marty responds
| |
| − | that in a certain sense the four trichotomies give nothing
| |
| − | more than the previous six ones but I strongly doubt of this.
| |
| − | | |
| − | BM: I put the problem in graphical form in an attached file
| |
| − | because my message editor will probably make some mistakes.
| |
| − | I make a distinction between arrow types drawing because I am
| |
| − | not sure that the sequence of correlates determinations is of
| |
| − | the same nature than correlates determination inside relations.
| |
| − | | |
| − | BM: It looks as if the problem amounts to some kind of projection
| |
| − | of relations on the horizontal axis made of correlates.
| |
| − | | |
| − | BM: If we consider some kind of equivalence (and this seems necessary to
| |
| − | obtain a linear ordering), by means of Agent -> Patient reductions on
| |
| − | relations, then erasing transitive determinations leads to:
| |
| − | | |
| − | Od -> Oi -> S -> S-Od -> If -> S-If -> S-Od-If -> Id -> S-Id -> Ii
| |
| − | | |
| − | BM: While it is interesting to compare the subsequence
| |
| − | S-Od -> If -> S-If -> S-Od-If with the pragmatic maxim,
| |
| − | I have no clear idea of the (in-) validity of such a result.
| |
| − | But I am convinced that the clarity has to come from the
| |
| − | Logic Of Relatives.
| |
| − | | |
| − | BM: I will be very grateful if you can make something with all that stuff.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 2
| |
| − | | |
| − | | |
| − | | |
| − | BM: I also found this passage which may be of some interest
| |
| − | (CP 4.540, Prolegomena to an Apology of Pragmatism):
| |
| − | | |
| − | | But though an Interpretant is not necessarily a Conclusion, yet a
| |
| − | | Conclusion is necessarily an Interpretant. So that if an Interpretant is
| |
| − | | not subject to the rules of Conclusions there is nothing monstrous in my
| |
| − | | thinking it is subject to some generalization of such rules. For any
| |
| − | | evolution of thought, whether it leads to a Conclusion or not, there is a
| |
| − | | certain normal course, which is to be determined by considerations not in
| |
| − | | the least psychological, and which I wish to expound in my next
| |
| − | | article;†1 and while I entirely agree, in opposition to distinguished
| |
| − | | logicians, that normality can be no criterion for what I call
| |
| − | | rationalistic reasoning, such as alone is admissible in science, yet it
| |
| − | | is precisely the criterion of instinctive or common-sense reasoning,
| |
| − | | which, within its own field, is much more trustworthy than rationalistic
| |
| − | | reasoning. In my opinion, it is self-control which makes any other than
| |
| − | | the normal course of thought possible, just as nothing else makes any
| |
| − | | other than the normal course of action possible; and just as it is
| |
| − | | precisely that that gives room for an ought-to-be of conduct, I mean
| |
| − | | Morality, so it equally gives room for an ought-to-be of thought, which
| |
| − | | is Right Reason; and where there is no self-control, nothing but the
| |
| − | | normal is possible. If your reflections have led you to a different
| |
| − | | conclusion from mine, I can still hope that when you come to read my next
| |
| − | | article, in which I shall endeavor to show what the forms of thought are,
| |
| − | | in general and in some detail, you may yet find that I have not missed
| |
| − | | the truth.
| |
| − | | |
| − | JA: Just as I have always feared, this classification mania
| |
| − | appears to be communicable! But now I must definitely
| |
| − | review the Welby correspondence, as all this stuff was
| |
| − | a blur to my sensibilities the last 10 times I read it.
| |
| − | | |
| − | BM: I think that I understand your reticence. I wonder if:
| |
| − | | |
| − | a. the fact that the letters to Lady Welby have been published as such,
| |
| − | has not lead to approach the matter in a certain way.
| |
| − | | |
| − | b. other sources, eventually unpublished, would give another lighting on
| |
| − | the subject, namely a logical one. I think of MS 339 for example that
| |
| − | seems to be part of the Logic Notebook. I have had access to some pages
| |
| − | of it, but not to the whole MS.
| |
| − | | |
| − | BM: A last remark. I don't think that classification is a mania for CSP but I
| |
| − | know that you know that! It is an instrument of thought and I think that
| |
| − | it is in this case much more a plan for experimenting than the exposition
| |
| − | of a conclusion. Experimenting what ? There is a strange statement in a
| |
| − | letter to W. James where CSP says that what is in question in his "second
| |
| − | way of dividing signs" is the logical theory of numbers. I give this from
| |
| − | memory. I have not the quote at hand now but I will search for it if needed.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 3
| |
| − | | |
| − | | |
| − | | |
| − | BM = Bernard Morand
| |
| − | JA = Jon Awbrey
| |
| − | | |
| − | JA: ... but now I have to add to my do-list the problems of comparing
| |
| − | the whole variorum of letters and drafts of letters to Lady Welby.
| |
| − | I only have the CP 8 and Wiener versions here, so I will depend
| |
| − | on you for ample excerpts from the Lieb volume.
| |
| − | | |
| − | BM: I made such a kind of comparison some time ago. I selected the following
| |
| − | 3 cases on the criterium of alternate "grounds". Hoping it could save
| |
| − | some labor. The first rank expressions come from the MS 339 written in
| |
| − | Oct. 1904 and I label them with an (a). I think that it is interesting to
| |
| − | note that they were written four years before the letters to Welby and
| |
| − | just one or two years after the Syllabus which is the usual reference for
| |
| − | the classification in 3 trichotomies and 10 classes. The second (b) is
| |
| − | our initial table (from a draft to Lady Welby, Dec. 1908, CP 8.344) and
| |
| − | the third (c) comes from a letter sent in Dec. 1908 (CP 8.345-8.376). A
| |
| − | tabular presentation would be better but I can't do it. Comparing (c)
| |
| − | against (a) and (b) is informative, I think.
| |
| − | | |
| − | Division 1
| |
| − | | |
| − | (a) According to the matter of the Sign
| |
| − | | |
| − | (b) According to the Mode of Apprehension of the Sign itself
| |
| − | | |
| − | (c) Signs in respect to their Modes of possible Presentation
| |
| − | | |
| − | Division 2
| |
| − | | |
| − | (a) According to the Immediate Object
| |
| − | | |
| − | (b) According to the Mode of Presentation of the Immediate Object
| |
| − | | |
| − | (c) Objects, as they may be presented
| |
| − | | |
| − | Division 3
| |
| − | | |
| − | (a) According to the Matter of the Dynamic Object
| |
| − | | |
| − | (b) According to the Mode of Being of the Dynamical Object
| |
| − | | |
| − | (c) In respect to the Nature of the Dynamical Objects of Signs
| |
| − | | |
| − | Division 4
| |
| − | | |
| − | (a) According to the mode of representing object by the Dynamic Object
| |
| − | | |
| − | (b) According to the Relation of the Sign to its Dynamical Object
| |
| − | | |
| − | (c) The fourth Trichotomy
| |
| − | | |
| − | Division 5
| |
| − | | |
| − | (a) According to the Immédiate Interpretant
| |
| − | | |
| − | (b) According to the Mode of Presentation of the Immediate Interpretant
| |
| − | | |
| − | (c) As to the nature of the Immediate (or Felt ?) Interpretant
| |
| − | | |
| − | Division 6
| |
| − | | |
| − | (a) According to the Matter of Dynamic Interpretant
| |
| − | | |
| − | (b) According to the Mode of Being of the Dynamical Interpretant
| |
| − | | |
| − | (c) As to the Nature of the Dynamical Interpretant
| |
| − | | |
| − | Division 7
| |
| − | | |
| − | (a) According to the Mode of Affecting Dynamic Interpretant
| |
| − | | |
| − | (b) According to the relation of the Sign to the Dynamical Interpretant
| |
| − | | |
| − | (c) As to the Manner of Appeal to the Dynamic Interpretant
| |
| − | | |
| − | Division 8
| |
| − | | |
| − | (a) According to the Matter of Representative Interpretant
| |
| − | | |
| − | (b) According to the Nature of the Normal Interpretant
| |
| − | | |
| − | (c) According to the Purpose of the Eventual Interpretant
| |
| − | | |
| − | Division 9
| |
| − | | |
| − | (a) According to the Mode of being represented by Representative Interpretant
| |
| − | | |
| − | (b) According to the the relation of the Sign to the Normal Interpretant
| |
| − | | |
| − | (c) As to the Nature of the Influence of the Sign
| |
| − | | |
| − | Division 10
| |
| − | | |
| − | (a) According to the Mode of being represented to represent object by Sign, Truly
| |
| − | | |
| − | (b) According to the Triadic Relation of the Sign to its Dynamical Object and to
| |
| − | its Normal Interpretant
| |
| − | | |
| − | (c) As to the Nature of the Assurance of the Utterance
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 4
| |
| − | | |
| − | | |
| − | | |
| − | JA: It may appear that one has side-stepped the issue of empiricism
| |
| − | that way, but then all that stuff about the synthetic a priori
| |
| − | raises its head, and we have Peirce's insight that mathematics
| |
| − | is observational and even experimental, and so I must trail off
| |
| − | into uncoordinated elliptical thoughts ...
| |
| − | | |
| − | HC: In contrast with this it strikes me that not all meanings of "analytic"
| |
| − | and "synthetic" have much, if anything, to do with the "analytic and the
| |
| − | synthetic", say, as in Quine's criticism of the "dualism" of empiricism.
| |
| − | Surely no one thinks that a plausible analysis must be analytic or that
| |
| − | synthetic materials tell us much about epistemology. So, it is not
| |
| − | clear that anything connected with analyticity or a priori knowledge
| |
| − | will plausibly or immediately arise from a discussion of analytical
| |
| − | geometry. Prevalent mathematical assumptions or postulates, yes --
| |
| − | but who says these are a prior? Can't non-Euclidean geometry also
| |
| − | be treated in the style of analytic geometry?
| |
| − | | |
| − | HC: I can imagine the a discussion might be forced in
| |
| − | that direction, but the connections don't strike me
| |
| − | as at all obvious or pressing. Perhaps Jon would just
| |
| − | like to bring up the notion of the synthetic apriori?
| |
| − | But why?
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 5
| |
| − | | |
| − | | |
| − | | |
| − | HC = Howard Callaway
| |
| − | | |
| − | HC: But I see you as closer to my theme or challenge, when you say
| |
| − | "The question is about the minimal adequate resource base for
| |
| − | defining, deriving, or generating all of the concepts that we
| |
| − | need for a given but very general type of application that we
| |
| − | conventinally but equivocally refer to as 'logic'".
| |
| − | | |
| − | HC: I think it is accepted on all sides of the discussion that there
| |
| − | is some sort of "equivalence" between the standard predicate logic
| |
| − | and Peirce's graphs.
| |
| − | | |
| − | There you would be mistaken, except perhaps for the fact that
| |
| − | "some sort of equivalence" is vague to the depths of vacuity.
| |
| − | It most particularly does not mean "all sorts of equivalence"
| |
| − | or even "all important sorts of equivalence". It is usually
| |
| − | interpreted to mean an extremely abstract type of syntactic
| |
| − | equivalence, and that is undoubtedly one important type of
| |
| − | equivalence that it is worth examining whether two formal
| |
| − | systems have or not. But it precisely here that we find
| |
| − | another symptom of syntacticism, namely, the deprecation
| |
| − | of all other important qualities of formal systems, most
| |
| − | pointedly their "analystic, "semantic", and "pragmatic"
| |
| − | qualities, which make all the difference in how well the
| |
| − | system actually serves its users in a real world practice.
| |
| − | You can almost hear the whining and poohing coming from the
| |
| − | syntactic day camp, but those are the hard facts of the case.
| |
| − | | |
| − | HC: But we find this difference in relation to the vocabulary used to express
| |
| − | identity. From the point of view of starting with the predicate calculus,
| |
| − | we don't need "teridentity". So, this seems to suggest there is something
| |
| − | of interesting contrast in Peirce's logic, which brings in this concept.
| |
| − | The obvious question may be expressed by asking why we need teridentity
| |
| − | in Peirce's system and how Peirce's system may recommend itself in contrast
| |
| − | to the standard way with related concepts. This does seem to call for
| |
| − | a comparative evaluation of distinctive systems. That is not an easy task,
| |
| − | as I think we all understand. But I do think that if it is a goal to have
| |
| − | Peirce's system better appreciated, then that kind of question must be
| |
| − | addressed. If "=" is sufficient in the standard predicate calculus,
| |
| − | to say whatever we may need to say about the identity of terms, then
| |
| − | what is the advantage of an alternative system which insists on always
| |
| − | expressing identity of triples?
| |
| − | | |
| − | HC: The questions may look quite different, depending on where we start.
| |
| − | But in any case, I thought I saw some better appreciation of the
| |
| − | questions in your comments above.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 6
| |
| − | | |
| − | | |
| − | | |
| − | It's been that way for about as long as anybody can remember, and
| |
| − | it will remain so, in spite of the spate of history rewriting and
| |
| − | image re-engineering that has become the new rage in self-styled
| |
| − | "analytic" circles.
| |
| − | | |
| − | | |
| − | | |
| − | LOR. Work 7
| |
| − | | |
| − | | |
| − | | |
| − | The brands of objection that you continue to make, with no evidence
| |
| − | of reflection on the many explanations that I and others have taken
| |
| − | the time to write out for you, lead me to believe that you are just
| |
| − | not interested in making that effort. That's okay, life is short,
| |
| − | the arts are long and many, there is always something else to do.
| |
| − | | |
| − | HC: For, if I make an assertion A&B, then I am not asserting
| |
| − | that the statement A stands in a relation to a statement B.
| |
| − | Instead, I am asserting the conjunction A&B (which logically
| |
| − | implies both the conjuncts in view of the definition of "&").
| |
| − | If "&" counts as a triadic relation, simply because it serves
| |
| − | to conjoin two statements into a third, then it would seem that
| |
| − | any binary relation 'R' will count as triadic, simply because
| |
| − | it places two things into a relation, which is a "third" thing.
| |
| − | By the same kind of reasoning a triadic relation, as ordinarily
| |
| − | understood would be really 4-adic.
| |
| − | | |
| − | HC: Now, I think this is the kind of argument you are making, ...
| |
| − | | |
| − | No, it's the kind of argument that you are making.
| |
| − | I am not making that kind of argument, and Peirce
| |
| − | did not make that kind of argument. Peirce used
| |
| − | his terms subject to definitions that would have
| |
| − | been understandable, and remain understandable,
| |
| − | to those of his readers who understand these
| |
| − | elementary definitions, either though their
| |
| − | prior acquaintance with standard concepts
| |
| − | or through their basic capacity to read
| |
| − | a well-formed, if novel definition.
| |
| − | | |
| − | Peirce made certain observations about the structure of logical concepts
| |
| − | and the structure of their referents. Those observations are accurate
| |
| − | and important. He expressed those observations in a form that is clear
| |
| − | to anybody who knows the meanings of the technical terms that he used,
| |
| − | and he is not responsible for the interpretations of those who don't.
| |
| − | | |
| − | HC: ... and it seems to both trivialize the claimed argument
| |
| − | for teridentity, by trivializing the conception of what
| |
| − | is to count as a triadic, as contrasted with a binary
| |
| − | relation, and it also seems to introduce a confusion
| |
| − | about what is is count as a binary, vs. a triadic
| |
| − | relation.
| |
| − | | |
| − | Yes, the argument that you are making trivializes
| |
| − | just about everything in sight, but that is the
| |
| − | common and well-known property of any argument
| |
| − | that fails to base itself on a grasp of the
| |
| − | first elements of the subject matter.
| |
| − | | |
| − | HC: If this is mathematical realism, then so much the worse for
| |
| − | mathematical realism. I am content to think that we do not
| |
| − | have a free hand in making up mathematical truth.
| |
| | | | |
| − | No, it's not mathematical realism. It is your reasoning,
| + | ===Commentary Note 12.6=== |
| − | and it exhibits all of the symptoms of syntacticism that
| |
| − | I have already diagnosed. It's a whole other culture
| |
| − | from what is pandemic in the practice of mathematics,
| |
| − | and it never fails to surprise me that people who
| |
| − | would never call themselves "relativists" in any
| |
| − | other matter of culture suddenly turn into just
| |
| − | that in matters of simple mathematical fact.
| |
| − | </pre>
| |
| | | | |
| − | ==Logic Of Relatives : Old Series== | + | ==References== |
| | | | |
| − | <pre>
| + | * Boole, George (1854), ''An Investigation of the Laws of Thought, On Which are Founded the Mathematical Theories of Logic and Probabilities'', Macmillan, 1854. Reprinted, Dover Publications, New York, NY, 1958. |
| − | 00. http://suo.ieee.org/ontology/thrd20.html#04416
| |
| − | 01. http://suo.ieee.org/ontology/msg04416.html
| |
| − | 02. http://suo.ieee.org/ontology/msg04417.html
| |
| − | 03. http://suo.ieee.org/ontology/msg04418.html
| |
| − | 04. http://suo.ieee.org/ontology/msg04419.html
| |
| − | 05. http://suo.ieee.org/ontology/msg04421.html
| |
| − | 06. http://suo.ieee.org/ontology/msg04422.html
| |
| − | 07. http://suo.ieee.org/ontology/msg04423.html
| |
| − | 08. http://suo.ieee.org/ontology/msg04424.html
| |
| − | 09. http://suo.ieee.org/ontology/msg04425.html
| |
| − | 10. http://suo.ieee.org/ontology/msg04426.html
| |
| − | 11. http://suo.ieee.org/ontology/msg04427.html
| |
| − | 12. http://suo.ieee.org/ontology/msg04431.html
| |
| − | 13. http://suo.ieee.org/ontology/msg04432.html
| |
| − | 14. http://suo.ieee.org/ontology/msg04435.html
| |
| − | 15. http://suo.ieee.org/ontology/msg04436.html
| |
| − | 16. http://suo.ieee.org/ontology/msg04437.html
| |
| − | 17. http://suo.ieee.org/ontology/msg04438.html
| |
| − | 18. http://suo.ieee.org/ontology/msg04439.html
| |
| − | 19. http://suo.ieee.org/ontology/msg04440.html
| |
| − | 20. http://suo.ieee.org/ontology/msg04441.html
| |
| − | 21. http://suo.ieee.org/ontology/msg04442.html
| |
| − | 22. http://suo.ieee.org/ontology/msg04443.html
| |
| − | 23. http://suo.ieee.org/ontology/msg04444.html
| |
| − | 24. http://suo.ieee.org/ontology/msg04445.html
| |
| − | 25. http://suo.ieee.org/ontology/msg04446.html
| |
| − | 26. http://suo.ieee.org/ontology/msg04447.html
| |
| − | 27. http://suo.ieee.org/ontology/msg04448.html
| |
| − | 28. http://suo.ieee.org/ontology/msg04449.html
| |
| − | 29. http://suo.ieee.org/ontology/msg04450.html
| |
| − | 30. http://suo.ieee.org/ontology/msg04451.html
| |
| − | 31. http://suo.ieee.org/ontology/msg04452.html
| |
| − | 32. http://suo.ieee.org/ontology/msg04453.html
| |
| − | 33. http://suo.ieee.org/ontology/msg04454.html
| |
| − | 34. http://suo.ieee.org/ontology/msg04456.html
| |
| − | 35. http://suo.ieee.org/ontology/msg04457.html
| |
| − | 36. http://suo.ieee.org/ontology/msg04458.html
| |
| − | 37. http://suo.ieee.org/ontology/msg04459.html
| |
| − | 38. http://suo.ieee.org/ontology/msg04462.html
| |
| − | 39. http://suo.ieee.org/ontology/msg04464.html
| |
| − | 40. http://suo.ieee.org/ontology/msg04473.html
| |
| − | 41. http://suo.ieee.org/ontology/msg04478.html
| |
| − | 42. http://suo.ieee.org/ontology/msg04484.html
| |
| − | 43. http://suo.ieee.org/ontology/msg04487.html
| |
| − | 44. http://suo.ieee.org/ontology/msg04488.html
| |
| − | 45. http://suo.ieee.org/ontology/msg04492.html
| |
| − | 46. http://suo.ieee.org/ontology/msg04497.html
| |
| − | 47. http://suo.ieee.org/ontology/msg04498.html
| |
| − | 48. http://suo.ieee.org/ontology/msg04499.html
| |
| − | 49. http://suo.ieee.org/ontology/msg04500.html
| |
| − | 50. http://suo.ieee.org/ontology/msg04501.html
| |
| − | 51. http://suo.ieee.org/ontology/msg04502.html
| |
| − | 52. http://suo.ieee.org/ontology/msg04503.html
| |
| − | 53. http://suo.ieee.org/ontology/msg04504.html
| |
| − | 54. http://suo.ieee.org/ontology/msg04506.html
| |
| − | 55. http://suo.ieee.org/ontology/msg04508.html
| |
| − | 56. http://suo.ieee.org/ontology/msg04509.html
| |
| − | 57. http://suo.ieee.org/ontology/msg04510.html
| |
| − | 58. http://suo.ieee.org/ontology/msg04511.html
| |
| − | 59. http://suo.ieee.org/ontology/msg04512.html
| |
| − | 60. http://suo.ieee.org/ontology/msg04513.html
| |
| − | 61. http://suo.ieee.org/ontology/msg04516.html
| |
| − | 62. http://suo.ieee.org/ontology/msg04517.html
| |
| − | 63. http://suo.ieee.org/ontology/msg04518.html
| |
| − | 64. http://suo.ieee.org/ontology/msg04521.html
| |
| − | 65. http://suo.ieee.org/ontology/msg04539.html
| |
| − | 66. http://suo.ieee.org/ontology/msg04541.html
| |
| − | 67. http://suo.ieee.org/ontology/msg04542.html
| |
| − | 68. http://suo.ieee.org/ontology/msg04543.html
| |
| − | </pre>
| |
| | | | |
| − | ==Logic of Relatives : Discussion Notes==
| + | * Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, ''Memoirs of the American Academy of Arts and Sciences'' 9, 317–378, 26 January 1870. Reprinted, ''Collected Papers'' (CP 3.45–149), ''Chronological Edition'' (CE 2, 359–429). Online [http://www.jstor.org/stable/25058006 (1)] [https://archive.org/details/jstor-25058006 (2)] [http://books.google.com/books?id=fFnWmf5oLaoC (3)]. |
| | | | |
| − | <pre>
| + | * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as (CP volume.paragraph). |
| − | 00. http://suo.ieee.org/ontology/thrd19.html#04460
| |
| − | 10. http://suo.ieee.org/ontology/msg04460.html
| |
| − | 11. http://suo.ieee.org/ontology/msg04461.html
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| − | 12. http://suo.ieee.org/ontology/msg04471.html
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| − | 13. http://suo.ieee.org/ontology/msg04472.html
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| − | 14. http://suo.ieee.org/ontology/msg04475.html
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| − | 15. http://suo.ieee.org/ontology/msg04476.html
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| − | 16. http://suo.ieee.org/ontology/msg04477.html
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| − | 17. http://suo.ieee.org/ontology/msg04479.html
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| − | 18. http://suo.ieee.org/ontology/msg04480.html
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| − | 19. http://suo.ieee.org/ontology/msg04481.html
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| − | 20. http://suo.ieee.org/ontology/msg04482.html
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| − | 21. http://suo.ieee.org/ontology/msg04483.html
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| − | 22. http://suo.ieee.org/ontology/msg04485.html
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| − | 23. http://suo.ieee.org/ontology/msg04486.html
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| − | 24. http://suo.ieee.org/ontology/msg04493.html
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| − | 25. http://suo.ieee.org/ontology/msg04494.html
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| − | 26. http://suo.ieee.org/ontology/msg04495.html
| |
| − | 27. http://suo.ieee.org/ontology/msg04496.html
| |
| − | </pre>
| |
| | | | |
| − | ==Logic Of Relatives : 2003==
| + | * Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN, 1981–. Cited as (CE volume, page). |
| | | | |
| − | <pre>
| + | ==Further Reading== |
| − | LOR. http://stderr.org/pipermail/inquiry/2003-March/thread.html#186
| |
| − | LOR. http://stderr.org/pipermail/inquiry/2003-April/thread.html#245
| |
| | | | |
| − | 01. http://stderr.org/pipermail/inquiry/2003-March/000186.html
| + | * [[Charles Sanders Peirce (Bibliography)|Bibliography : Charles Sanders Peirce]]. |
| − | 02. http://stderr.org/pipermail/inquiry/2003-March/000187.html
| |
| − | 03. http://stderr.org/pipermail/inquiry/2003-March/000188.html
| |
| − | 04. http://stderr.org/pipermail/inquiry/2003-March/000189.html
| |
| − | 05. http://stderr.org/pipermail/inquiry/2003-March/000190.html
| |
| − | 06. http://stderr.org/pipermail/inquiry/2003-March/000191.html
| |
| − | 07. http://stderr.org/pipermail/inquiry/2003-March/000194.html
| |
| − | 08. http://stderr.org/pipermail/inquiry/2003-March/000195.html
| |
| − | 09. http://stderr.org/pipermail/inquiry/2003-April/000245.html
| |
| − | 10. http://stderr.org/pipermail/inquiry/2003-April/000246.html
| |
| − | 11. http://stderr.org/pipermail/inquiry/2003-April/000247.html
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| − | 12. http://stderr.org/pipermail/inquiry/2003-April/000248.html
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| − | 13. http://stderr.org/pipermail/inquiry/2003-April/000249.html
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| − | 14. http://stderr.org/pipermail/inquiry/2003-April/000250.html
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| − | 15. http://stderr.org/pipermail/inquiry/2003-April/000251.html
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| − | 16. http://stderr.org/pipermail/inquiry/2003-April/000252.html
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| − | 17. http://stderr.org/pipermail/inquiry/2003-April/000253.html
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| − | 18. http://stderr.org/pipermail/inquiry/2003-April/000254.html
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| − | 19. http://stderr.org/pipermail/inquiry/2003-April/000255.html
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| − | 20. http://stderr.org/pipermail/inquiry/2003-April/000256.html
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| − | 21. http://stderr.org/pipermail/inquiry/2003-April/000257.html
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| − | 22. http://stderr.org/pipermail/inquiry/2003-April/000258.html
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| − | 23. http://stderr.org/pipermail/inquiry/2003-April/000259.html
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| − | 24. http://stderr.org/pipermail/inquiry/2003-April/000260.html
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| − | 25. http://stderr.org/pipermail/inquiry/2003-April/000261.html
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| − | 26. http://stderr.org/pipermail/inquiry/2003-April/000262.html
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| − | 27. http://stderr.org/pipermail/inquiry/2003-April/000263.html
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| − | 28. http://stderr.org/pipermail/inquiry/2003-April/000264.html
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| − | 29. http://stderr.org/pipermail/inquiry/2003-April/000265.html
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| − | 30. http://stderr.org/pipermail/inquiry/2003-April/000267.html
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| − | 31. http://stderr.org/pipermail/inquiry/2003-April/000268.html
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| − | 32. http://stderr.org/pipermail/inquiry/2003-April/000269.html
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| − | 33. http://stderr.org/pipermail/inquiry/2003-April/000270.html
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| − | 34. http://stderr.org/pipermail/inquiry/2003-April/000271.html
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| − | 35. http://stderr.org/pipermail/inquiry/2003-April/000273.html
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| − | 36. http://stderr.org/pipermail/inquiry/2003-April/000274.html
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| − | 37. http://stderr.org/pipermail/inquiry/2003-April/000275.html
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| − | 38. http://stderr.org/pipermail/inquiry/2003-April/000276.html
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| − | 39. http://stderr.org/pipermail/inquiry/2003-April/000277.html
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| − | 40. http://stderr.org/pipermail/inquiry/2003-April/000278.html
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| − | 41. http://stderr.org/pipermail/inquiry/2003-April/000279.html
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| − | 42. http://stderr.org/pipermail/inquiry/2003-April/000280.html
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| − | 43. http://stderr.org/pipermail/inquiry/2003-April/000281.html
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| − | 44. http://stderr.org/pipermail/inquiry/2003-April/000282.html
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| − | 45. http://stderr.org/pipermail/inquiry/2003-April/000283.html
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| − | 46. http://stderr.org/pipermail/inquiry/2003-April/000284.html
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| − | 47. http://stderr.org/pipermail/inquiry/2003-April/000285.html
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| − | 48. http://stderr.org/pipermail/inquiry/2003-April/000286.html
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| − | 49. http://stderr.org/pipermail/inquiry/2003-April/000287.html
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| − | 50. http://stderr.org/pipermail/inquiry/2003-April/000288.html
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| − | 51. http://stderr.org/pipermail/inquiry/2003-April/000289.html
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| − | 52. http://stderr.org/pipermail/inquiry/2003-April/000290.html
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| − | 53. http://stderr.org/pipermail/inquiry/2003-April/000291.html
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| − | 54. http://stderr.org/pipermail/inquiry/2003-April/000294.html
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| − | 55. http://stderr.org/pipermail/inquiry/2003-April/000295.html
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| − | 56. http://stderr.org/pipermail/inquiry/2003-April/000296.html
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| − | 57. http://stderr.org/pipermail/inquiry/2003-April/000297.html
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| − | 58. http://stderr.org/pipermail/inquiry/2003-April/000298.html
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| − | 59. http://stderr.org/pipermail/inquiry/2003-April/000299.html
| |
| − | 60. http://stderr.org/pipermail/inquiry/2003-April/000300.html
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| − | 61. http://stderr.org/pipermail/inquiry/2003-April/000301.html
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| − | 62. http://stderr.org/pipermail/inquiry/2003-April/000302.html
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| − | 63. http://stderr.org/pipermail/inquiry/2003-April/000303.html
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| − | 64. http://stderr.org/pipermail/inquiry/2003-April/000305.html
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| − | 65. http://stderr.org/pipermail/inquiry/2003-April/000306.html
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| − | 66. http://stderr.org/pipermail/inquiry/2003-April/000307.html
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| − | 67. http://stderr.org/pipermail/inquiry/2003-April/000308.html
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| − | 68. http://stderr.org/pipermail/inquiry/2003-April/000309.html
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| − | </pre>
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| | | | |
| − | ==Logic Of Relatives : 2004==
| + | * Brady, G. (2000), ''From Peirce to Skolem : A Neglected Chapter in the History of Logic'', Elsevier, Amsterdam. [http://books.google.com/books?id=ahoH-tLm2S0C Online Preview]. |
| | | | |
| − | <pre>
| + | * Lambek, J., and Scott, P.J. (1986), ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK. |
| − | 00. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1750
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| − | 01. http://stderr.org/pipermail/inquiry/2004-November/001750.html
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| − | 02. http://stderr.org/pipermail/inquiry/2004-November/001751.html
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| − | 03. http://stderr.org/pipermail/inquiry/2004-November/001752.html
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| − | 04. http://stderr.org/pipermail/inquiry/2004-November/001753.html
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| − | 05. http://stderr.org/pipermail/inquiry/2004-November/001754.html
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| − | 06. http://stderr.org/pipermail/inquiry/2004-November/001760.html
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| − | 07. http://stderr.org/pipermail/inquiry/2004-November/001769.html
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| − | 08. http://stderr.org/pipermail/inquiry/2004-November/001774.html
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| − | 09. http://stderr.org/pipermail/inquiry/2004-November/001783.html
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| − | 10. http://stderr.org/pipermail/inquiry/2004-November/001794.html
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| − | 11. http://stderr.org/pipermail/inquiry/2004-November/001812.html
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| − | 12. http://stderr.org/pipermail/inquiry/2004-November/001842.html
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| − | </pre>
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| | | | |
| − | ==Logic Of Relatives : Commentary==
| + | * Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY. |
| | | | |
| − | <pre>
| + | * Walsh, A. (2012), ''Relations Between Logic and Mathematics in the Work of Benjamin and Charles S. Peirce'', Docent Press, Boston, MA. |
| − | 00. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1755
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| − | 01. http://stderr.org/pipermail/inquiry/2004-November/001755.html
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| − | 02. http://stderr.org/pipermail/inquiry/2004-November/001756.html
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| − | 03. http://stderr.org/pipermail/inquiry/2004-November/001757.html
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| − | 04. http://stderr.org/pipermail/inquiry/2004-November/001758.html
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| − | 05. http://stderr.org/pipermail/inquiry/2004-November/001759.html
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| − | 06. http://stderr.org/pipermail/inquiry/2004-November/001761.html
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| − | 07. http://stderr.org/pipermail/inquiry/2004-November/001770.html
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| − | 08.1. http://stderr.org/pipermail/inquiry/2004-November/001775.html
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| − | 08.2. http://stderr.org/pipermail/inquiry/2004-November/001776.html
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| − | 08.3. http://stderr.org/pipermail/inquiry/2004-November/001777.html
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| − | 08.4. http://stderr.org/pipermail/inquiry/2004-November/001778.html
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| − | 08.5. http://stderr.org/pipermail/inquiry/2004-November/001781.html
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| − | 08.6. http://stderr.org/pipermail/inquiry/2004-November/001782.html
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| − | 09.1. http://stderr.org/pipermail/inquiry/2004-November/001787.html
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| − | 09.2. http://stderr.org/pipermail/inquiry/2004-November/001788.html
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| − | 09.3. http://stderr.org/pipermail/inquiry/2004-November/001789.html
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| − | 09.4. http://stderr.org/pipermail/inquiry/2004-November/001790.html
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| − | 09.5. http://stderr.org/pipermail/inquiry/2004-November/001791.html
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| − | 09.6. http://stderr.org/pipermail/inquiry/2004-November/001792.html
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| − | 09.7. http://stderr.org/pipermail/inquiry/2004-November/001793.html
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| − | 10.01. http://stderr.org/pipermail/inquiry/2004-November/001795.html
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| − | 10.02. http://stderr.org/pipermail/inquiry/2004-November/001796.html
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| − | 10.03. http://stderr.org/pipermail/inquiry/2004-November/001797.html
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| − | 10.04. http://stderr.org/pipermail/inquiry/2004-November/001798.html
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| − | 10.05. http://stderr.org/pipermail/inquiry/2004-November/001799.html
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| − | 10.06. http://stderr.org/pipermail/inquiry/2004-November/001800.html
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| − | 10.07. http://stderr.org/pipermail/inquiry/2004-November/001801.html
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| − | 10.08. http://stderr.org/pipermail/inquiry/2004-November/001802.html
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| − | 10.09. http://stderr.org/pipermail/inquiry/2004-November/001803.html
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| − | 10.10. http://stderr.org/pipermail/inquiry/2004-November/001804.html
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| − | 10.11. http://stderr.org/pipermail/inquiry/2004-November/001805.html
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| − | 11.01. http://stderr.org/pipermail/inquiry/2004-November/001813.html
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| − | 11.02. http://stderr.org/pipermail/inquiry/2004-November/001814.html
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| − | 11.03. http://stderr.org/pipermail/inquiry/2004-November/001815.html
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| − | 11.04. http://stderr.org/pipermail/inquiry/2004-November/001816.html
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| − | 11.05. http://stderr.org/pipermail/inquiry/2004-November/001817.html
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| − | 11.06. http://stderr.org/pipermail/inquiry/2004-November/001818.html
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| − | 11.07. http://stderr.org/pipermail/inquiry/2004-November/001819.html
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| − | 11.08. http://stderr.org/pipermail/inquiry/2004-November/001820.html
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| − | 11.09. http://stderr.org/pipermail/inquiry/2004-November/001821.html
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| − | 11.10. http://stderr.org/pipermail/inquiry/2004-November/001822.html
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| − | 11.11. http://stderr.org/pipermail/inquiry/2004-November/001823.html
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| − | 11.12. http://stderr.org/pipermail/inquiry/2004-November/001824.html
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| − | 11.13. http://stderr.org/pipermail/inquiry/2004-November/001825.html
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| − | 11.14. http://stderr.org/pipermail/inquiry/2004-November/001826.html
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| − | 11.15. http://stderr.org/pipermail/inquiry/2004-November/001827.html
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| − | 11.16. http://stderr.org/pipermail/inquiry/2004-November/001828.html
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| − | 11.17. http://stderr.org/pipermail/inquiry/2004-November/001829.html
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| − | 11.18. http://stderr.org/pipermail/inquiry/2004-November/001830.html
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| − | 11.19. http://stderr.org/pipermail/inquiry/2004-November/001831.html
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| − | 11.20. http://stderr.org/pipermail/inquiry/2004-November/001832.html
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| − | 11.21. http://stderr.org/pipermail/inquiry/2004-November/001833.html
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| − | 11.22. http://stderr.org/pipermail/inquiry/2004-November/001834.html
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| − | 11.23. http://stderr.org/pipermail/inquiry/2004-November/001835.html
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| − | 11.24. http://stderr.org/pipermail/inquiry/2004-November/001836.html
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| − | 12. http://stderr.org/pipermail/inquiry/2004-November/001843.html
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| − | </pre>
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| | | | |
| − | ==Logic Of Relatives : Discussion== | + | ==See Also== |
| | | | |
| − | <pre>
| + | {{col-begin}} |
| − | 00. http://suo.ieee.org/ontology/thrd20.html#04460
| + | {{col-break}} |
| − | 00. http://stderr.org/pipermail/inquiry/2004-November/thread.html#1768
| + | * [[Charles Sanders Peirce]] |
| − | 00. http://stderr.org/pipermail/inquiry/2005-January/thread.html#2301
| + | * [[Logic of relatives]] |
| | + | * [[Logic of Relatives (1870)]] |
| | + | * [[Logic of Relatives (1883)]] |
| | + | {{col-break}} |
| | + | * [[Relation (mathematics)|Relation]] |
| | + | * [[Relation theory]] |
| | + | * [[Sign relation]] |
| | + | * [[Triadic relation]] |
| | + | {{col-end}} |
| | | | |
| − | 10. http://suo.ieee.org/ontology/msg04460.html
| + | [[Category:Artificial Intelligence]] |
| − | 11. http://suo.ieee.org/ontology/msg04461.html
| + | [[Category:Charles Sanders Peirce]] |
| − | 12. http://suo.ieee.org/ontology/msg04471.html
| + | [[Category:Critical Thinking]] |
| − | 13. http://suo.ieee.org/ontology/msg04472.html
| + | [[Category:Cybernetics]] |
| − | 14. http://suo.ieee.org/ontology/msg04475.html
| + | [[Category:Education]] |
| − | 15. http://suo.ieee.org/ontology/msg04476.html
| + | [[Category:Hermeneutics]] |
| − | 16. http://suo.ieee.org/ontology/msg04477.html
| + | [[Category:Information Systems]] |
| − | 17. http://suo.ieee.org/ontology/msg04479.html
| + | [[Category:Inquiry]] |
| − | 18. http://suo.ieee.org/ontology/msg04480.html
| + | [[Category:Intelligence Amplification]] |
| − | 19. http://suo.ieee.org/ontology/msg04481.html
| + | [[Category:Learning Organizations]] |
| − | 20. http://suo.ieee.org/ontology/msg04482.html
| + | [[Category:Knowledge Representation]] |
| − | 21. http://suo.ieee.org/ontology/msg04483.html
| + | [[Category:Logic]] |
| − | 22. http://suo.ieee.org/ontology/msg04485.html
| + | [[Category:Logic Of Relatives]] |
| − | 23. http://suo.ieee.org/ontology/msg04486.html
| + | [[Category:Logical Graphs]] |
| − | 24. http://suo.ieee.org/ontology/msg04493.html
| + | [[Category:Mathematics]] |
| − | 25. http://suo.ieee.org/ontology/msg04494.html
| + | [[Category:Normative Sciences]] |
| − | 26. http://suo.ieee.org/ontology/msg04495.html
| + | [[Category:Philosophy]] |
| − | 27. http://suo.ieee.org/ontology/msg04496.html
| + | [[Category:Pragmatics]] |
| − | 28. http://stderr.org/pipermail/inquiry/2004-November/001768.html
| + | [[Category:Pragmatism]] |
| − | 29. http://stderr.org/pipermail/inquiry/2004-November/001838.html
| + | [[Category:Relation Theory]] |
| − | 30. http://stderr.org/pipermail/inquiry/2004-November/001840.html
| + | [[Category:Science]] |
| − | 31. http://stderr.org/pipermail/inquiry/2005-January/002301.html
| + | [[Category:Semantics]] |
| − | </pre>
| + | [[Category:Semiotics]] |
| | + | [[Category:Systems Science]] |
| | + | [[Category:Visualization]] |
