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{{DISPLAYTITLE:Peirce's 1870 Logic Of Relatives}}
{{DISPLAYTITLE:Peirce's 1870 Logic Of Relatives}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
'''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]].
<div class="nonumtoc">__TOC__</div>
==Preliminaries==
===Application of the Algebraic Signs to Logic===
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, for example, absolute terms, dyadic relative terms, or 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.
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, for example, absolute terms, dyadic relative terms, or 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.
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''.
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.
The path to understanding Peirce's system and its wider implications for logic can be 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 does, 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.
==Selection 1==
==Selection 1==
<p>Now logical terms are of three grand classes.</p>
<p>Now logical terms are of three grand classes.</p>
<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>
<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>
<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>
<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>
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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.
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 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.
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 predicate]]s", his definition of [[sign relation]]s, and in the [[pragmatic maxim]] itself.
One might also call attention to the following two statements:
One might also call attention to the following two statements:
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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<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>
<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>
<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>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>
<p>(Peirce, CP 3.65).</p>
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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<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>
<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>
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| align="center" | <math>\mathrm{v} = \mathrm{p}\!</math>
| align="center" | <math>\mathrm{v} = \mathrm{p}\!</math>
<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>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>
<p>The sign "less than" is to be so taken that</p>
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| align="center" | <math>\mathrm{f} < \mathrm{m}~\!</math>
| align="center" | <math>\mathrm{f} < \mathrm{m}\!</math>
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<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>
<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>
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| align="center" | <math>\mathrm{f} ~-\!\!\!< \mathrm{m}</math>
| align="center" | <math>\mathrm{f} ~-\!\!\!< \mathrm{m}</math>
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<p>means “every Frenchman is a man”, without saying whether there are any other men or not. So,</p>
<p>means "every Frenchman is a man", without saying whether there are any other men or not. So,</p>
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| align="center" | <math>\mathit{m} ~-\!\!\!< \mathit{l}</math>
| align="center" | <math>\mathit{m} ~-\!\!\!< \mathit{l}</math>
<p>that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.</p>
<p>that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.</p>
<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>
<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>
<p>(Peirce, CP 3.66).</p>
<p>(Peirce, CP 3.66).</p>
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.
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.
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.
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.
==Selection 4==
==Selection 4==
<p>Thus</p>
<p>Thus</p>
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| align="center" | <math>\mathrm{m} + \mathrm{w}~\!</math>
| align="center" | <math>\mathrm{m} + \mathrm{w}\!</math>
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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.
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 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:
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:
:* [http://suo.ieee.org/ontology/msg04350.html]
:* [http://suo.ieee.org/ontology/msg04351.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.
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.
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.
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.
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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<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>
<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 apears not to have had multiplication in his mind.</p>
<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>
<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>
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| align="center" | <math>\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w}) ~=~ \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}.</math>
| align="center" | <math>\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w}) ~=~ \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}</math>.
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<p><math>(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w}</math> will denote whatever is lover or servant to a woman, and:</p>
<p><math>(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w}</math> will denote whatever is lover or servant to a woman, and:</p>
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| align="center" | <math>(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w} ~=~ \mathit{l}\mathrm{w} ~+\!\!,~ \mathit{s}\mathrm{w}.</math>
| align="center" | <math>(\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w} ~=~ \mathit{l}\mathrm{w} ~+\!\!,~ \mathit{s}\mathrm{w}</math>.
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<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>
<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>
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| align="center" | <math>(\mathit{s}\mathit{l})\mathrm{w} ~=~ \mathit{s}(\mathit{l}\mathrm{w}).</math>
| align="center" | <math>(\mathit{s}\mathit{l})\mathrm{w} ~=~ \mathit{s}(\mathit{l}\mathrm{w})</math>.
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<p>Thus all the absolute conditions of multiplication are satisfied.</p>
<p>Thus all the absolute conditions of multiplication are satisfied.</p>
<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>
<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>
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| align="center" | <math>x \mathit{1} ~=~ x ~ ,</math>
| align="center" | <math>x \mathit{1} ~=~ x</math>,
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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 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.
[[Sign relation]]s, like any non-trivial brand of [[3-adic relation]]s, 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.
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.
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.
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.
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.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 1.jpg]]
| align="center" |
|-
<pre>
| height="20px" valign="top" | <math>\text{Figure 1}~\!</math>
o-----------------------------o-----------------------------o
| Objective Framework (OF) | Interpretive Framework (IF) |
o-----------------------------o-----------------------------o
| Objects | Signs |
o-----------------------------o-----------------------------o
| |
| o "v" |
| / |
| / |
| / |
| o ... o-----------@ |
| \ |
| \ |
| \ |
| o "p" |
| |
o-----------------------------o-----------------------------o
</pre>
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|}
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.
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.
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.
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.
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.
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.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 2.jpg]]
| align="center" |
|-
<pre>
| height="20px" valign="top" | <math>\text{Figure 2}\!</math>
o-----------------------------o-----------------------------o
| Objective Framework (OF) | Interpretive Framework (IF) |
o-----------------------------o-----------------------------o
| Objects | Signs |
o-----------------------------o-----------------------------o
| |
| o "'s'(m +, w)" |
| / |
| / |
| / |
| o ... o-----------@ |
| \ |
| \ |
| \ |
| o "'s'm +, 's'w" |
| |
o-----------------------------o-----------------------------o
</pre>
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|}
<p>A conjugative term like ''giver'' naturally requires two correlates, one denoting the thing given, the other the recipient of the gift.</p>
<p>A conjugative term like ''giver'' naturally requires two correlates, one denoting the thing given, the other the recipient of the gift.</p>
<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>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>
<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>
<p>we abandon the associative principle of multiplication.</p>
<p>we abandon the associative principle of multiplication.</p>
<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>
<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>
<p>Then, the giver of a horse to a lover of a woman may be written:</p>
<p>Then, the giver of a horse to a lover of a woman may be written:</p>
|-
|-
|
|
<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>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>
<p>(Peirce, CP 3.69–70).</p>
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.
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.
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:
We can represent this problem setup as the problem of computing the following product:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>(1 + a)(1 + b)(1 + c).\!</math>
| <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.
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:
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:
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>
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:
Multiplying out the product <math>(1 + a)(1 + b)(1 + c)\!</math>, one obtains:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{*{15}{c}}
<math>\begin{array}{*{15}{c}}
1 & + & a & + & b & + & c & + & ab & + & ac & + & bc & + & abc.
1 & + & a & + & b & + & c & + & ab & + & ac & + & bc & + & abc
\end{array}</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\varnothing, & \{ a \}, & \{ b \}, & \{ c \}, & \{ a, b \}, & \{ a, c \}, & \{ b, c \}, & \{ a, b, c \}.
\varnothing, & \{ a \}, & \{ b \}, & \{ c \}, & \{ a, b \}, & \{ a, c \}, & \{ b, c \}, & \{ a, b, c \}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
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.
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:
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"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 3.jpg]] || (3)
| align="center" |
<pre>
o---------------------------------------o
| |
| ! # |
| / \ / \ |
| o o o o |
| `g`_!@ !'l'_# #w @h |
| o o |
| \_____________/ |
| @ |
| |
o---------------------------------------o
Giver of a Horse to a Lover of a Woman
</pre>
|}
|}
|}
|}
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).
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).
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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
===Commentary Note 8.2===
===Commentary Note 8.2===
I continue with my commentary on CP 3.73, developing the ''Othello'' example as a way of illustrating Peirce's concepts.
I will continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating its concepts.
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:
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:
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}
^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}
\\[6pt]
\\[6pt]
^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}
^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}
\\[6pt]
\\[6pt]
^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}
^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}
\end{array}</math>
\end{array}</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<p>The relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>The relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute term <math>^{\backprime\backprime}\, \text{Emilia}\, ^{\prime\prime}</math></p>
<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>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><math>\operatorname{Iago}</math> is a lover of <math>\operatorname{Emilia},</math> so the relate-correlate pair <math>\operatorname{I}:\operatorname{E}</math></p>
<p>lies in the 2-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math></p>
<p>lies in the 2-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math></p>
|-
|-
|
|
<p>The relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>The relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<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>
<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>
<p>from the absolute term <math>^{\backprime\backprime}\, \text{betrayer to Othello of Desdemona}\, ^{\prime\prime}.</math></p>
<p>from the absolute term <math>^{\backprime\backprime}\, \text{betrayer to Othello of Desdemona}\, ^{\prime\prime}.</math></p>
<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>
<p><math>\operatorname{Iago}</math> is a betrayer to <math>\operatorname{Othello}</math> of <math>\operatorname{Desdemona},</math> so the relate-correlate-correlate triple <math>\operatorname{I}:\operatorname{O}:\operatorname{D}</math></p>
<p>lies in the 3-adic relation assciated with the relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.\!</math></p>
<p>lies in the 3-adic relation assciated with the relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}.\!</math></p>
|-
|-
|
|
<p>The relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>The relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<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>
<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>
<p>from the absolute term <math>^{\backprime\backprime}\, \text{winner over of Othello to Iago from Cassio}\, ^{\prime\prime}.</math></p>
<p>from the absolute term <math>^{\backprime\backprime}\, \text{winner over of Othello to Iago from Cassio}\, ^{\prime\prime}.</math></p>
<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><math>\operatorname{Iago}</math> is a winner over of <math>\operatorname{Othello}</math> to <math>\operatorname{Iago}</math> from <math>\operatorname{Cassio},\!</math> so the elementary relative term <math>\operatorname{I}:\operatorname{O}:\operatorname{I}:\operatorname{C}</math></p>
<p>lies in the 4-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math></p>
<p>lies in the 4-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math></p>
|}
|}
|}
|}
Returning to the Othello example, let us take up the 2-adic relatives <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math>
Returning to the Othello example, let us take up the 2-adic relatives <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math>
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math> to be given by the following equation:
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}</math> to be given by the following equation:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|}
|}
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math> The terms of this service are:
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math> The terms of this service are:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{*{15}{c}}
<math>\begin{array}{*{15}{c}}
1
& = & \mathrm{B}
& = & \mathrm{B}
& +\!\!, & \mathrm{C}
& +\!\!, & \mathrm{C}
& +\!\!, & \mathrm{D}
& +\!\!, & \mathrm{D}
& +\!\!, & \mathrm{E}
& +\!\!, & \mathrm{E}
\end{array}\!</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\mathit{l}\mathbf{1}
\mathit{l}1
& = &
& = &
\text{lover of anything}
\text{lover of anything}
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\mathit{s}\mathbf{1}
\mathit{s}1
& = &
& = &
\text{servant of anything}
\text{servant of anything}
\mathit{l}\mathit{s}
\mathit{l}\mathit{s}
& = &
& = &
\text{lover of a servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
\text{lover of a servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = &
& = &
\mathit{s}\mathit{l}
\mathit{s}\mathit{l}
& = &
& = &
\text{servant of a lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
\text{servant of a lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = &
& = &
|}
|}
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>
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===
===Commentary Note 8.5===
|}
|}
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.
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 t|o signify.
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
\begin{bmatrix}
\begin{bmatrix}
0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1
0 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 1
\end{bmatrix}\!
\end{bmatrix}
</math>
</math>
|}
|}
<math>\begin{matrix}
<math>\begin{matrix}
\mathit{s}\mathrm{w} & = & \text{servant of a woman} & =
\mathit{s}\mathrm{w} & = & \text{servant of a woman} & =
\end{matrix}\!</math>
\end{matrix}</math>
|-
|-
|
|
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathit{l}\mathit{s} & = & \text{lover of a servant of ---} & =
\mathit{l}\mathit{s} & = & \operatorname{lover~of~a~servant~of}~(~) =
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathit{s}\mathit{l} & = & \text{servant of a lover of ---} & =
\mathit{s}\mathit{l} & = & \operatorname{servant~of~a~lover~of}~(~) =
\end{matrix}</math>
\end{matrix}</math>
|-
|-
<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>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>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>
<p>Then "man that is black" will be written:</p>
|-
|-
| align="center" | <math>\mathrm{m},\!\mathrm{b}</math>
| align="center" | <math>\mathrm{m},\!\mathrm{b}</math>
<math>\begin{array}{*{11}{c}}
<math>\begin{array}{*{11}{c}}
\mathrm{m,}
\mathrm{m,}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
& = & \mathrm{C}:\mathrm{C}
& = & \mathrm{C}:\mathrm{C}
& +\!\!, & \mathrm{I}:\mathrm{I}
& +\!\!, & \mathrm{I}:\mathrm{I}
\\[6pt]
\\[6pt]
\mathrm{n,}
\mathrm{n,}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
& = & \mathrm{C}:\mathrm{C}
& = & \mathrm{C}:\mathrm{C}
& +\!\!, & \mathrm{D}:\mathrm{D}
& +\!\!, & \mathrm{D}:\mathrm{D}
\\[6pt]
\\[6pt]
\mathrm{w,}
\mathrm{w,}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
& = & \mathrm{B}:\mathrm{B}
& = & \mathrm{B}:\mathrm{B}
& +\!\!, & \mathrm{D}:\mathrm{D}
& +\!\!, & \mathrm{D}:\mathrm{D}
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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>
<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>
<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>
<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>
<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>
<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>
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\mathbf{1,}
\mathbf{1,}
& = & \text{anything that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{anything that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[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}
& = & \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]
\\[9pt]
\mathrm{m,}
\mathrm{m,}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{n,}
\mathrm{n,}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{w,}
\mathrm{w,}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
===Commentary Note 9.5===
===Commentary Note 9.5===
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''.
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. This is why I call it the ''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.
An idempotent element <math>x\!</math> is given by the abstract condition that <math>xx = x,\!</math> but we commonly encounter such elements 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.
Let's see how this looks in the matrix and graph pictures of absolute and relative terms:
Let's see how all of this looks from the graphical and matrical perspectives.
Absolute terms:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{*{17}{l}}
<math>\begin{array}{*{17}{l}}
\mathbf{1} & = & \text{anything} & = &
\mathbf{1}
\mathrm{B} & +\!\!, &
& = & \text{anything}
\mathrm{C} & +\!\!, &
& = & \mathrm{B}
\mathrm{D} & +\!\!, &
& +\!\!, & \mathrm{C}
\mathrm{E} & +\!\!, &
& +\!\!, & \mathrm{D}
\mathrm{I} & +\!\!, &
& +\!\!, & \mathrm{E}
\mathrm{J} & +\!\!, &
& +\!\!, & \mathrm{I}
\mathrm{O}
& +\!\!, & \mathrm{J}
& +\!\!, & \mathrm{O}
\\[6pt]
\\[6pt]
\mathrm{m} & = & \text{man} & = &
\mathrm{m}
\mathrm{C} & +\!\!, &
& = & \text{man}
\mathrm{I} & +\!\!, &
& = & \mathrm{C}
\mathrm{J} & +\!\!, &
& +\!\!, & \mathrm{I}
\mathrm{O}
& +\!\!, & \mathrm{J}
& +\!\!, & \mathrm{O}
\\[6pt]
\\[6pt]
\mathrm{n} & = & \text{noble} & = &
\mathrm{n}
\mathrm{C} & +\!\!, &
& = & \text{noble}
\mathrm{D} & +\!\!, &
& = & \mathrm{C}
\mathrm{O}
& +\!\!, & \mathrm{D}
& +\!\!, & \mathrm{O}
\\[6pt]
\\[6pt]
\mathrm{w} & = & \text{woman} & = &
\mathrm{w}
\mathrm{B} & +\!\!, &
& = & \text{woman}
\mathrm{D} & +\!\!, &
& = & \mathrm{B}
\mathrm{E}
& +\!\!, & \mathrm{D}
\end{array}</math>
& +\!\!, & \mathrm{E}
|}
\end{array}</math>
|}
Previously, we represented absolute terms as column arrays. The above four terms are given by the columns of the following table:
Previously, we represented absolute terms as column vectors. The above four terms are given by the columns of the following table:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{c|cccc}
<math>\begin{array}{c|cccc}
\text{ } & \mathbf{1} & \mathrm{m} & \mathrm{n} & \mathrm{w} \\
\text{ } & \mathbf{1} & \mathrm{m} & \mathrm{n} & \mathrm{w}
\text{---} & \text{---} & \text{---} & \text{---} & \text{---} \\
\\
\mathrm{B} & 1 & 0 & 0 & 1 \\
\text{---} & \text{---} & \text{---} & \text{---} & \text{---}
\mathrm{C} & 1 & 1 & 1 & 0 \\
\\
\mathrm{D} & 1 & 0 & 1 & 1 \\
\mathrm{B} & 1 & 0 & 0 & 1
\mathrm{E} & 1 & 0 & 0 & 1 \\
\\
\mathrm{I} & 1 & 1 & 0 & 0 \\
\mathrm{C} & 1 & 1 & 1 & 0
\mathrm{J} & 1 & 1 & 0 & 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
\mathrm{O} & 1 & 1 & 1 & 0
\end{array}</math>
\end{array}</math>
|}
|}
One way to represent sets in the bigraph picture is simply to mark the nodes in some way, like so:
{| align="center" cellpadding="10" width="90%"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 4.1.jpg]] || (4.1)
|
<pre>
B C D E I J O
1 + + + + + + +
B C D E I J O
m o + o o + + +
B C D E I J O
n o + + o o o +
B C D E I J O
w + o + + o o o
</pre>
|}
|}
Diagonal extensions of the absolute terms:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\mathbf{1,}
\mathbf{1,}
& = & \text{anything that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{anything that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[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}
& = & \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]
\\[9pt]
\mathrm{m,}
\mathrm{m,}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{n,}
\mathrm{n,}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{w,}
\mathrm{w,}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}
\\[6pt]
\\[6pt]
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
|}
|}
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:
Cast into the bigraph picture of 2-adic relations, the diagonal extension of an absolute term takes on a very distinctive sort of ''straight-laced'' character:
{| align="center" cellspacing="6" width="90%"
|
<pre>
B C D E I J O
u o o o o o o o
| | | | | | |
1, | | | | | | |
| | | | | | |
u o o o o o o o
B C D E I J O
</pre>
|}
{| align="center" cellpadding="10" width="90%"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 5.1.jpg]] || (5.1)
|
|-
<pre>
| [[Image:LOR 1870 Figure 5.2.jpg]] || (5.2)
B C D E I J O
|-
u o o o o o o o
| [[Image:LOR 1870 Figure 5.3.jpg]] || (5.3)
| | | |
|-
m, | | | |
| [[Image:LOR 1870 Figure 5.4.jpg]] || (5.4)
| | | |
u o o o o o o o
B C D E I J O
</pre>
|}
{| align="center" cellspacing="6" width="90%"
|
<pre>
B C D E I J O
u o o o o o o o
| | |
n, | | |
| | |
u o o o o o o o
B C D E I J O
</pre>
|}
{| align="center" cellspacing="6" width="90%"
|
<pre>
B C D E I J O
u o o o o o o o
| | |
w, | | |
| | |
u o o o o o o o
B C D E I J O
</pre>
|}
|}
===Commentary Note 9.6===
===Commentary Note 9.6===
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.
Just to be doggedly persistent about it all, 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 graphical and matrical representations.
====Example 1====
====Example 1====
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
| <math>\mathbf{1,}\mathbf{1} ~=~ \mathbf{1}\!</math>
| <math>\mathbf{1,}\mathbf{1} ~=~ \mathbf{1}</math>
|-
|-
| <math>\text{anything that is anything} ~=~ \text{anything}</math>
| <math>\text{anything that is anything} ~=~ \text{anything}</math>
|}
|}
{| align="center" cellpadding="10" width="100%"
{| align="center" cellspacing="6" width="90%"
| width="2%" |
|
| width="48%" | [[Image:LOR 1870 Figure 6.1.jpg]]
<pre>
| width="50%" | (6.1)
B C D E I J O
|}
+ + + + + + + 1
| | | | | | |
| | | | | | | 1,
| | | | | | |
o o o o o o o =
+ + + + + + + 1
B C D E I J O
</pre>
|}
====Example 2====
====Example 2====
|}
|}
{| align="center" cellpadding="10" width="100%"
{| align="center" cellspacing="6" width="90%"
| width="2%" |
|
| width="48%" | [[Image:LOR 1870 Figure 6.2.jpg]]
<pre>
| width="50%" | (6.2)
B C D E I J O
o + o o + + + m
| | | | | | |
| | | | | | | 1,
| | | | | | |
o o o o o o o =
o + o o + + + m
B C D E I J O
</pre>
|}
|}
|}
|}
{| align="center" cellpadding="10" width="100%"
{| align="center" cellspacing="6" width="90%"
| width="2%" |
|
| width="48%" | [[Image:LOR 1870 Figure 6.3.jpg]]
<pre>
| width="50%" | (6.3)
B C D E I J O
+ + + + + + + 1
| | | |
| | | | m,
| | | |
o o o o o o o =
o + o o + + + m
B C D E I J O
</pre>
|}
|}
|}
|}
{| align="center" cellpadding="10" width="100%"
{| align="center" cellspacing="6" width="90%"
| width="2%" |
|
| width="48%" | [[Image:LOR 1870 Figure 6.4.jpg]]
<pre>
| width="50%" | (6.4)
B C D E I J O
o + + o o o + n
| | | |
| | | | m,
| | | |
o o o o o o o =
o + o o o o + m,n
B C D E I J O
</pre>
|}
|}
|}
|}
{| align="center" cellpadding="10" width="100%"
{| align="center" cellspacing="6" width="90%"
| width="2%" |
|
| width="48%" | [[Image:LOR 1870 Figure 6.5.jpg]]
<pre>
| width="50%" | (6.5)
B C D E I J O
o + o o + + + m
| | |
| | | n,
| | |
o o o o o o o =
o + o o o o + n,m
B C D E I J O
</pre>
|}
|}
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.
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.
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.
We have covered the way that Peirce uses his invention of the 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 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.
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.
==Selection 10==
==Selection 10==
|
|
|-
|-
| and in general;
| and in general,
| align="right" | <math>x,\!\mathfrak{2}.y</math>
| align="right" | <math>x,\!\mathfrak{2}.y</math>
| align="center" | <math>=\!</math>
| align="center" | <math>=\!</math>
===Commentary Note 10.1===
===Commentary Note 10.1===
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.
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.
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.
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.
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.
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 we when relate relations, possibly complex, to other relations, possibly simple.
===Commentary Note 10.2===
===Commentary Note 10.2===
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.
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.
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>
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 1 and 2, where I have drawn in the corresponding lines of identity between the subjacent marks of reference: <code>!, @, #</code>.
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 7.0.jpg]] || (7)
o-------------------------------------------------o
| |
| |
| 'l'__! !'s'__@ @w |
| o o o o |
| \ / \ / |
| \ / o |
| \ / @ |
| o |
| ! |
| |
| |
o-------------------------------------------------o
Figure 1. Lover of a Servant of a Woman
</pre>
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 8.0.jpg]] || (8)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| `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
</pre>
|}
|}
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.
One way to approach the problem of "information fusion" in Peirce's syntax is to soften the distinction between 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 relational composition that emphasizes the set-theoretic constraints involved in the construction of a composite.
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>
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 3 and Figure 4 present two ways of picturing the constraints that are involved in constructing the relational composition <math>L \circ M \subseteq X \times Z.</math>
<br>
<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%"
{| 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>
|+ <math>\text{Table 3. Relational Composition}\!</math>
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| <math>Z\!</math>
| <math>Z\!</math>
|-
|-
| style="border-right:1px solid black" | <math>L \circ M\!</math>
| style="border-right:1px solid black" | <math>L \circ M</math>
| <math>X\!</math>
| <math>X\!</math>
|
|
<br>
<br>
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 way to read Table 3 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>
Figure 10 shows a different way of viewing the same situation.
Figure 4 shows a different way of viewing the same situation.
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 10.jpg]] || (10)
o-------------------------------------------------o
| |
| 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
</pre>
|}
|}
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:
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:
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 7.0.jpg]] || (11)
o-------------------------------------------------o
| |
| |
| 'l'__! !'s'__@ @w |
| o o o o |
| \ / \ / |
| \ / o |
| \ / @ |
| o |
| ! |
| |
| |
o-------------------------------------------------o
Figure 1. Lover of a Servant of a Woman
</pre>
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 8.0.jpg]] || (12)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| `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
</pre>
|}
|}
<br>
<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%"
{| 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>
|+ '''Table 3. Relational Composition'''
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
|
|
|-
|-
| style="border-right:1px solid black" | <math>S\!</math>
| style="border-right:1px solid black" | <math>M\!</math>
|
|
| <math>Y\!</math>
| <math>Y\!</math>
| <math>Z\!</math>
| <math>Z\!</math>
|-
|-
| style="border-right:1px solid black" | <math>L \circ S\!</math>
| style="border-right:1px solid black" | <math>L \circ M</math>
| <math>X\!</math>
| <math>X\!</math>
|
|
<br>
<br>
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 14.jpg]] || (14)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| 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
</pre>
|}
|}
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.
Figures 1 and 2 exhibit examples of relative multiplication 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 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 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.
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.
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.
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.
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.
Figure 5 demonstrates this possibility, posing a transitional case between the style of syntax in Figure 1 and the picture of composition in Figure 4.
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 15.jpg]] || (15)
o-----------------------------------------------------------o
| |
| L o S |
| ____________O____________ |
| / \ |
| / L S \ |
| / @ @ \ |
| / / \ / \ \ |
| / / \ / \ \ |
| o o o o o o |
| X X Y Y Z Z |
| 1,__! !'l'__@ @'s'__# #1 |
| o o o o o o |
| \ / \ / \ / |
| \ / \ / \ / |
| \ / \ / \ / |
| O O O |
| !1! !1! !1! |
| |
o-----------------------------------------------------------o
Figure 5. Anything that is a Lover of a Servant of Anything
</pre>
|}
|}
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.
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===
===Commentary Note 10.4===
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.
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 3 as illustrating the ''spreadsheet'' model of relational composition, while Figure 4 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 5 may be called the ''universal bracketing'' of relatives as relations.
===Commentary Note 10.5===
===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.
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.
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|}
|}
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>
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 Jeste (J) is secretly in love with Desdemona (D).
Then we begin with the modified data set:
Then we begin with the modified data set:
\mathit{l}, ~=
\mathit{l}, ~=
\\[6pt]
\\[6pt]
\text{lover that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)} ~=
\text{lover that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)} ~=
\\[6pt]
\\[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})
(\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})
Now what are we to make of that?
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:
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%"
{| align="center" cellspacing="6" width="90%"
|}
|}
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.
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:
It follows that:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
===Commentary Note 10.6===
===Commentary Note 10.6===
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.
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 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, and there are places where Peirce verges on systems that are very similar, but I am making a deliberate effort to remain here as close as possible within the syntactoplastic chronism of his 1870 Logic of Relatives. So let us make use of the smoother transitions that are afforded by the paradigmatic Figures and Tables that I drew up earlier.
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.
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 instance, the three that I have redrawn below.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 8.0.jpg]] || (16)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| `g`__!__@ !'l'__# #w @h |
| o o o o o o |
| \ \ / \ / / |
| \ \/ O / |
| \ /\______ ______/ |
| O O |
| |
| |
o-------------------------------------------------o
Figure 6. Giver of a Horse to a Lover of a Woman
</pre>
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 17.0.jpg]] || (17)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| `g`__!__@ !'o'__# #@h |
| o o o o oo |
| \ \ / \ // |
| \ \/ O/ |
| \ /\____ ____/ |
| O O |
| |
| |
o-------------------------------------------------o
Figure 7. Giver of a Horse to an Owner of It
</pre>
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 18.jpg]] || (18)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| 'l',__!__@ !'s'__# #@w |
| o o o o oo |
| \ \ / \ // |
| \ \/ O/ |
| \ /\____ ____/ |
| O O |
| |
| |
o-------------------------------------------------o
Figure 8. Lover that is a Servant of a Woman
</pre>
|}
|}
===Commentary Note 10.7===
===Commentary Note 10.7===
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.
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 2-adic compositions.
We may begin with the mark-up shown in Figure 19.
We may begin with the mark-up shown in Figure 6.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 8.0.jpg]] || (19)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| `g`__!__@ !'l'__# #w @h |
| o o o o o o |
| \ \ / \ / / |
| \ \/ O / |
| \ /\______ ______/ |
| O O |
| |
| |
o-------------------------------------------------o
Figure 6. Giver of a Horse to a Lover of a Woman
</pre>
|}
|}
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.
If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a 3-adic ''giving'' relation <math>G \subseteq T \times U \times V</math> with a 2-adic ''loving'' relation <math>L \subseteq U \times W</math> so as to obtain a specialized sort of 3-adic relation <math>(G \circ L) \subseteq T \times W \times V.</math> The applicable constraints on tuples are shown in Table 9.
<br>
<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:75%"
{| 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>
|+ <math>\text{Table 9. Composite of Triadic and Dyadic Relations}\!</math>
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:20%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:20%" |
| <math>T\!</math>
| <math>T\!</math>
| <math>U\!</math>
| <math>U\!</math>
|
| <math>V\!</math>
| <math>V\!</math>
|-
|-
| style="border-right:1px solid black" | <math>L\!</math>
| style="border-right:1px solid black" | <math>L\!</math>
|
|
| <math>U\!</math>
| <math>U\!</math>
| <math>W\!</math>
|
|
|-
|-
| style="border-right:1px solid black" | <math>G \circ L</math>
| style="border-right:1px solid black" | <math>G \circ L</math>
| <math>T\!</math>
| <math>T\!</math>
|
|
| <math>W\!</math>
| <math>V\!</math>
| <math>V\!</math>
|}
|}
<br>
<br>
The hypergraph picture of the abstract composition is given in Figure 21.
The hypergraph picture of the abstract composition is given in Figure 10.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 21.jpg]] || (21)
| align="center" |
<pre>
o---------------------------------------------------------------------o
| |
| G o L |
| ___________O___________ |
| / \ \ |
| / G L \ \ |
| / O O \ \ |
| / /|\ / \ \ \ |
| / / | \ / \ \ \ |
| / / | \ / \ \ \ |
| / / | \ / \ \ \ |
| 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 |
| \ / \ \ / \ / / |
| O \ \/ O / |
| !1! \ /\ !1! / |
| \ / \_______ _______/ |
| O O |
| !1! !1! |
| |
o---------------------------------------------------------------------o
Figure 10. Anything that is a Giver of Anything to a Lover of Anything
</pre>
|}
|}
===Commentary Note 10.8===
===Commentary Note 10.8===
In taking up the next example of relational composition, let's exchange the relation <math>\mathit{t} = \text{trainer of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}</math> for Peirce's relation <math>\mathit{o} = \text{owner of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)},</math> simply for the sake of avoiding conflicts in the symbols that we use. In this way, Figure 7 is transformed into Figure 11.
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| `g`__!__@ !'t'__# #@h |
| o o o o oo |
| \ \ / \ // |
| \ \/ O/ |
| \ /\____ ____/ |
| O O |
| |
| |
o-------------------------------------------------o
Figure 11. Giver of a Horse to a Trainer of It
</pre>
|}
Now here's an interesting point, in fact, a critical transition point, 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 23.
The hypergraph picture of the abstract composition is given in Figure 12.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 23.jpg]] || (23)
| align="center" |
<pre>
o---------------------------------------------------------------------o
| |
| G o T |
| _________________O_________________ |
| / \ |
| / G T \ |
| / O O \ |
| / /|\ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| 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 |
| \ / \ \ / | / |
| \ / \ \/ | / |
| \ / \ /\ | / |
| \ / \ / \__________|__________/ |
| O O O |
| !1! !1! !1! |
| |
o---------------------------------------------------------------------o
Figure 12. Anything that is a Giver of Anything to a Trainer of It
</pre>
|}
|}
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.
If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a 3-adic ''giving'' relation <math>G \subseteq X \times Y \times Z</math> with a 2-adic ''training'' relation <math>T \subseteq Y \times Z</math> in such a way as to determine a certain 2-adic relation <math>(G \circ T) \subseteq X \times Z.</math> Table 13 schematizes the associated constraints on tuples.
<br>
<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%"
{| 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>
|+ <math>\text{Table 13. Another Brand of Composition}\!</math>
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
===Commentary Note 10.9===
===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 use of the concepts of identity and teridentity is not to identify 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 functioned in the last example, where they had the opportunity to show their behavior in something like their natural habitat.
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.
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.
The use of the teridentity concept in the case 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.
===Commentary Note 10.10===
===Commentary Note 10.10===
Figure 8 depicts the last of the three examples involving the composition of 3-adic relatives with 2-adic relatives:
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 18.jpg]] || (25)
| align="center" |
<pre>
o-------------------------------------------------o
| |
| |
| 'l',__!__@ !'s'__# #@w |
| o o o o oo |
| \ \ / \ // |
| \ \/ @/ |
| \ /\____ ____/ |
| O O |
| |
| |
o-------------------------------------------------o
Figure 8. Lover that is a Servant of a Woman
</pre>
|}
|}
The hypergraph picture of the abstract composition is given in Figure 26.
The hypergraph picture of the abstract composition is given in Figure 14.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 26.jpg]] || (26)
| align="center" |
<pre>
o---------------------------------------------------------------------o
| |
| L , S |
| __________________O__________________ |
| / \ |
| / L_, S \ |
| / O O \ |
| / /|\ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| / / | \ / \ \ |
| 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 |
| \ / \ \ / | / |
| \ / \ \ | / |
| \ / \ / \ | / |
| \ / \ / \___________|___________/ |
| O O O |
| !1! !1! !1! |
| |
o---------------------------------------------------------------------o
Figure 14. Anything that's a Lover of Anything and that's a Servant of It
</pre>
|}
|}
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.
This example illustrates the way that Peirce analyzes the logical conjunction, we might even say the ''parallel conjunction'', of a pair of 2-adic 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.
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.
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 2-adic ''loving'' relation <math>L \subseteq X \times Y</math> to the corresponding 3-adic ''being and loving'' relation <math>L \subseteq X \times X \times Y,</math> which is then composed in a specific way with a 2-adic ''serving'' relation <math>S \subseteq X \times Y,</math> so as to determine the 2-adic relation <math>L,\!S \subseteq X \times Y.</math> Table 15 schematizes the associated constraints on tuples.
<br>
<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%"
{| 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>
|+ <math>\text{Table 15. Conjunction Via Composition}\!</math>
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
===Commentary Note 10.11===
===Commentary Note 10.11===
I 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:
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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.
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.
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>
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>(X\!:\!Y\!:\!Z)(Y\!:\!Z)(Z)\!</math> are the only ones that produce a non-null result, namely, <math>(X\!:\!Y\!:\!Z)(Y\!:\!Z)(Z) = X.\!</math>
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
\mathbf{1},\!, ~=~
\mathbf{1},\!, =
\\[6pt]
\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}
\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>
\end{array}</math>
|}
|}
==Selection 11==
==Selection 11==
===Commentary Note 11.1===
===Commentary Note 11.1===
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 reached a suitable place to pause in our reading of Peirce's text — actually, it is more like a place to run as fast as we can along a parallel track — where I can pay off a few IOU's that I've used to pave the way to this point.
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.
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.
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.
At this point we have two good pictures of how to compute the 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 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.
But we do not have a comparable picture of how to compute the richer variety of relational compositions that involve 3-adic 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, general enough to articulate the “generative potency” of Peirce's 1870 Logic of Relatives.
Therefore, let us inaugurate a systematic study of relational composition, general enough to explicate the "generative potency" of Peirce's 1870 LOR.
===Commentary Note 11.2===
===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.
Let's bring together the various things that Peirce has said about the "number of function" up to this point in the paper.
====NOF 1====
====NOF 1====
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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>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>
<p>(Peirce, CP 3.65).</p>
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<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>
<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>
<p>If we have an equation of the form:</p>
<p>If we have an equation of the form:</p>
===Commentary Note 11.3===
===Commentary Note 11.3===
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.
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 that will no longer abide its assigned place under the rug.
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.
Functions have long been understood, from well before Peirce's time to ours, as special cases of 2-adic relations, so the "number of" function itself is 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 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.
It is possible to work all this out in a very nice way within a very general context 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 types of compromise formations.
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.
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.
===Commentary Note 11.4===
===Commentary Note 11.4===
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.
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.
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:
I am optimistic that the some of the tethering material that I spun along the "Relations In General" thread will help us to track the equivalential and functional properties of special relations in a way that will not weigh too heavily on the embedding of syntax in 1-dimensional strings on 2-dimensional pages. But I cannot see far enough ahead to foresee all the consequences of trying this tack, and so it must remain a bit experimental.
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 immedeiate 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 2-adic relative term — say <math>~v~</math> — where:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.\!</math>
| <math>v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.</math>
|}
|}
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.
To set the 2-adic 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 "terms". Then the 2-adic relation <math>~V~</math> is evidently a function from <math>~S~</math> to <math>\mathbb{R}.</math> We might think to use the plain letter <math>{}^{\backprime\backprime} v {}^{\prime\prime}</math> to denote this function, as <math>v : S \to \mathbb{R},</math> but I worry this may be a chaos waiting to happen. Also, I think we should anticipate the very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain <math>~S~</math> that we choose, so it is probably better to account the 2-adic 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.
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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>q : X \to Y\!</math> means that <math>q\!</math> is functional at <math>X.\!</math>
| <math>q : X \to Y</math> means that <math>~q~</math> is functional at <math>~X.~</math>
|-
|-
| <math>q : X \leftarrow Y\!</math> means that <math>q\!</math> is functional at <math>Y.\!</math>
| <math>q : X \leftarrow Y</math> means that <math>~q~</math> is functional at <math>~X.~</math>
|-
|-
| <math>q : X \rightharpoonup Y\!</math> means that <math>q\!</math> is pre-functional at <math>X.\!</math>
| <math>q : X \rightharpoonup Y</math> means that <math>~q~</math> is pre-functional at <math>~X.~</math>
|-
|-
| <math>q : X \leftharpoonup Y\!</math> means that <math>q\!</math> is pre-functional at <math>Y.\!</math>
| <math>q : X \leftharpoonup Y</math> means that <math>~q~</math> is pre-functional at <math>~Y.~</math>
|}
|}
For now, I will pretend 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 2-adic relation <math>V \subseteq \mathbb{R} \times S,</math> and associated with the 2-adic 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.
===Commentary Note 11.5===
===Commentary Note 11.5===
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 right form of diagram can be a great aid in rendering complex matters comprehensible, so let's extract the overly compressed bits of the "Relations In General" thread that we need to illuminate Peirce's 1870 "Logic Of Relatives" and draw what icons we can within the current frame.
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.
For the immediate present, we may start with 2-adic relations and describe the customary species of relations and functions in terms of their local and numerical incidence properties.
Let <math>P \subseteq X \times Y\!</math> be an arbitrary dyadic relation. The following properties of <math>P\!</math> can be defined:
Let <math>P \subseteq X \times Y</math> be an arbitrary 2-adic relation. The following properties of <math>~P~</math> can be defined:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|}
|}
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>
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 alternative lower case name, say <math>{}^{\backprime\backprime} p {}^{\prime\prime},</math> and writing <math>p : X \rightharpoonup Y.</math>
Just by way of formalizing the definition:
Just by way of formalizing the definition:
& \iff &
& \iff &
P ~\text{is tubular at}~ Y.
P ~\text{is tubular at}~ Y.
\end{array}\!</math>
\end{array}</math>
|}
|}
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:
To illustrate these properties, let us fashion a generic enough example of a 2-adic 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"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 30.jpg]] || (30)
|
<pre>
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
</pre>
|}
|}
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.
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.
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.
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.
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>
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 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>
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 1.
Clearly, then, the relation <math>E\!</math> cannot qualify as a pre-function, much less as a function on either of its relational domains.
Clearly, then, the relation <math>~E~</math> cannot qualify as a pre-function, much less as a function on either of its relational domains.
===Commentary Note 11.6===
===Commentary Note 11.6===
<math>E_1\!</math> exemplifies the quality of ''totality at <math>X.\!</math>''
<math>E_1\!</math> exemplifies the quality of ''totality at <math>X.\!</math>''
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 31.jpg]] || (31)
|
<pre>
0 1 2 3 4 5 6 7 8 9
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
</pre>
|}
|}
<math>E_2\!</math> exemplifies the quality of ''totality at <math>Y.\!</math>''
<math>E_2\!</math> exemplifies the quality of ''totality at <math>Y.\!</math>''
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 32.jpg]] || (32)
|
<pre>
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
|\ \ |\ /|\ \ \ | |\ \
| \ \ | / | \ \ \ | | \ \ E_2
| \ \|/ \| \ \ \| | \ \
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
</pre>
|}
|}
<math>E_3\!</math> exemplifies the quality of ''tubularity at <math>X.\!</math>''
<math>E_3\!</math> exemplifies the quality of ''tubularity at <math>X.\!</math>''
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 33.jpg]] || (33)
|
<pre>
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
\ | / \ \ | |
\ | / \ \ | | E_3
\|/ \ \| |
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
</pre>
|}
|}
<math>E_4\!</math> exemplifies the quality of ''tubularity at <math>Y.\!</math>''
<math>E_4\!</math> exemplifies the quality of ''tubularity at <math>Y.\!</math>''
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 34.jpg]] || (34)
|
<pre>
0 1 2 3 4 5 6 7 8 9
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
</pre>
|}
|}
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>
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>
===Commentary Note 11.7===
===Commentary Note 11.7===
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.
We come now to the very special cases of 2-adic 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 RIG material.
To begin, let's recall the definition of a ''local flag'':
To begin, let's recall the definition of a ''local flag'':
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>L_{x \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.\!</math>
| <math>L_{x \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.</math>
|}
|}
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>
In the case of a 2-adic 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 2-adic 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>
In light of these considerations, the local flags of a dyadic relation <math>L \subseteq X \times Y\!</math> may be formulated as follows:
In light of these considerations, the local flags of a 2-adic relation <math>L \subseteq X \times Y</math> may be formulated as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
& = &
& = &
\text{the ordered pairs in}~ L ~\text{that are incident with}~ v \in Y.
\text{the ordered pairs in}~ L ~\text{that are incident with}~ v \in Y.
\end{array}\!</math>
\end{array}</math>
|}
|}
u \cdot L
u \cdot L
& = &
& = &
\mathrm{proj}_2 (u \star L)
\operatorname{proj}_2 (u \star L)
\\[6pt]
\\[6pt]
& = &
& = &
L \cdot v
L \cdot v
& = &
& = &
\mathrm{proj}_1 (L \star v)
\operatorname{proj}_1 (L \star v)
\\[6pt]
\\[6pt]
& = &
& = &
& = &
& = &
\text{the elements of}~ X ~\text{that are}~ L\text{-related to}~ v.
\text{the elements of}~ X ~\text{that are}~ L\text{-related to}~ v.
\end{array}\!</math>
\end{array}</math>
|}
|}
A sufficient illustration is supplied by the earlier example <math>E.\!</math>
A sufficient illustration is supplied by the earlier example <math>E.\!</math>
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 30.jpg]] || (35)
|
<pre>
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
</pre>
|}
|}
The local flag <math>E_{3 \,\text{at}\, X}\!</math> is displayed here:
The local flag <math>E_{3 \,\text{at}\, X}</math> is displayed here:
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 36 ISW.jpg]] || (36)
|
<pre>
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
</pre>
|}
|}
The local flag <math>E_{2 \,\text{at}\, Y}\!</math> is displayed here:
The local flag <math>E_{2 \,\text{at}\, Y}</math> is displayed here:
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 37 ISW.jpg]] || (37)
|
<pre>
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
</pre>
|}
|}
Next let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.
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>
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.
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.
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
& \iff &
& \iff &
|L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j.
|L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j.
\end{array}\!</math>
\end{array}</math>
|}
|}
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>
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>
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:
For example, let <math>G = \{ r, s, t \}\!</math> and <math>H = \{ 1, \ldots, 9 \},\!</math> and consider the 2-adic relation <math>F \subseteq G \times H</math> that is bigraphed here:
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 38.jpg]] || (38)
|
<pre>
r s t
o o o G
/|\ /|\ /|\
/ | \ / | \ / | \ F
/ | \ / | \ / | \
o o o o o o o o o H
1 2 3 4 5 6 7 8 9
</pre>
|}
|}
===Commentary Note 11.9===
===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.
Among the variety of conceivable regularities affecting 2-adic 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:
Let <math>P \subseteq X \times Y</math> be an arbitrary 2-adic relation. The following properties of <math>~P~</math> can be defined:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
& \iff &
& \iff &
P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y.
P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y.
\end{array}\!</math>
\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:
We have already looked at 2-adic 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%"
{| align="center" cellspacing="6" width="90%"
& \iff &
& \iff &
P ~\text{is tubular at}~ Y.
P ~\text{is tubular at}~ Y.
\end{array}\!</math>
\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.
We arrive by way of this winding stair at the special stamps of 2-adic 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>
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:
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%"
{| align="center" cellspacing="6" width="90%"
& \iff &
& \iff &
P ~\text{is}~ 1\text{-regular at}~ Y.
P ~\text{is}~ 1\text{-regular at}~ Y.
\end{array}\!</math>
\end{array}</math>
|}
|}
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:
For example, let <math>X = Y = \{ 0, \ldots, 9 \}\!</math> and let <math>F \subseteq X \times Y</math> be the 2-adic relation that is depicted in the bigraph below:
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 39.jpg]] || (39)
|
<pre>
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
\ / /|\ \ | |\ \
\ / | \ \ | | \ \ F
/ \ / | \ \ | | \ \
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
</pre>
|}
|}
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>
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>
===Commentary Note 11.10===
===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:
In the case of a 2-adic 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%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
f ~\text{is surjective} & \iff & f ~\text{is total at}~ Y.
f ~\text{is surjective}
& \iff &
f ~\text{is total at}~ Y.
\\[6pt]
\\[6pt]
f ~\text{is injective} & \iff & f ~\text{is tubular at}~ Y.
f ~\text{is injective}
& \iff &
f ~\text{is tubular at}~ Y.
\\[6pt]
\\[6pt]
f ~\text{is bijective} & \iff & f ~\text{is}~ 1\text{-regular at}~ Y.
f ~\text{is bijective}
\end{array}\!</math>
& \iff &
f ~\text{is}~ 1\text{-regular at}~ y.
\end{array}</math>
|}
|}
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.
For example, the function <math>f : X \to Y</math> that is 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.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 40.jpg]] || (40)
|
<pre>
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
| \ | / \ \ | | \ /
| \ | / \ \ | | \ f
| \|/ \ \| | / \
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
</pre>
|}
|}
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>
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>
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 41.jpg]] || (41)
|
<pre>
0 1 2 3 4 5 6 7 8 9
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
</pre>
|}
|}
The function <math>h : Y^\prime \to Y\!</math> is injective.
The function <math>h : Y^\prime \to Y</math> is injective.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 42.jpg]] || (42)
|
<pre>
0 2 5 6 7 8 9
o o o o o o o Y'
| | \ / | \ /
| | \ | \ h
| | / \ | / \
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
</pre>
|}
|}
The function <math>m : X \to Y\!</math> is bijective.
The function <math>m : X \to Y</math> is bijective.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 43.jpg]] || (43)
|
<pre>
0 1 2 3 4 5 6 7 8 9
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
</pre>
|}
|}
===Commentary Note 11.11===
===Commentary Note 11.11===
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.
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.
By way of extending a few very tentative planks, let us experiment with the following definitions:
By way of extending a few very tentative planks, let us experiment with the following definitions:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<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>^{\backprime\backprime} p ^{\prime\prime}</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>
<p>A relative term <math>^{\backprime\backprime} p ^{\prime\prime}</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>
|}
|}
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.
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.
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.
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.
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 44.jpg]] || (44)
o-----------------------------------------------------------o
| |
| P o Q |
| ____________O____________ |
| / \ |
| / P Q \ |
| / O O \ |
| / / \ / \ \ |
| / / \ / \ \ |
| o o o o o o |
| X X Y Y Z Z |
| 1,__! !'p'__@ @'q'__# #1 |
| o o o o o o |
| \ / \ / \ / |
| \ / \ / \ / |
| \ / \ / \ / |
| O O O |
| !1! !1! !1! |
| |
o-----------------------------------------------------------o
Figure 16. Anything that is a 'p' of a 'q' of Anything
</pre>
|}
|}
From this we extract the ''hypergraph picture'' of relational composition:
From this we extract the ''hypergraph picture'' of relational composition:
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 45.jpg]] || (45)
o-----------------------------------------------------------o
| |
| P P o Q Q |
| O O O |
| / \ / \ / \ |
| / \ / \ / \ |
| o o o o o o |
| X Y X Z Y Z |
| o o o o o o |
| \ \ / \ / / |
| \ \ / \ / / |
| \ / \ / |
| \ / \ / \ / |
| \ / \___ ___/ \ / |
| O O O |
| !1! !1! !1! |
| |
o-----------------------------------------------------------o
Figure 17. Relational Composition P o Q
</pre>
|}
|}
{| 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%"
{| 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>
|+ <math>\text{Table 18. Relational Composition}~ P \circ Q</math>
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
===Commentary Note 11.12===
===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.
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 pair of 2-adic relations is again a 2-adic relation — we know that the relational composition of two 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 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.
Just for novelty's sake, let's try to prove this for relations that are functional on correlates.
The task is this — We are given a pair of dyadic relations:
The task is this — We are given a pair of 2-adic relations:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>P \subseteq X \times Y \quad \text{and} \quad Q \subseteq Y \times Z\!</math>
| <math>P \subseteq X \times Y \quad \text{and} \quad Q \subseteq Y \times Z</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>P : X \gets Y \quad \text{and} \quad Q : Y \gets Z\!</math>
| <math>P : X \leftarrow Y \quad \text{and} \quad Q : Y \leftarrow Z</math>
|}
|}
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>
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 \leftarrow Z.</math>
It always helps to begin by recalling the pertinent definitions.
It always helps to begin by recalling the pertinent definitions.
For a dyadic relation <math>L \subseteq X \times Y,\!</math> we have:
For a 2-adic relation <math>L \subseteq X \times Y,</math> we have:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
L ~\text{is a function}~ L : X \gets Y
L ~\text{is a function}~ L : X \leftarrow Y
& \iff &
& \iff &
L ~\text{is}~ 1\text{-regular at}~ Y.
L ~\text{is}~ 1\text{-regular at}~ Y.
|}
|}
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:
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>
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}\!</math>
| <math>(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}</math>
|}
|}
So let's begin.
So let us begin.
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<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>
<p><math>P : X \leftarrow Y,</math> or the fact <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>
|-
|-
|
|
<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><math>Q : Y \leftarrow 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>
<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 \leftarrow Z.</math>
|}
|}
===Commentary Note 11.13===
===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.
As we make our way toward the foothills of Peirce's 1870 LOR, there are several pieces of equipment that we must not leave the plains without, namely, the utilities that are variously referred to as ''arrows'', ''morphisms'', ''homomorphisms'', ''structure-preserving maps'', and several other names, depending on the altitude of abstraction that one happens to be traversing at the moment in question. As a moderate to middling but not too beaten track, let us take up a few ways of defining a morphism that will serve us in the present discussion.
Suppose we are given three functions <math>J, K, L~\!</math> that satisfy the following conditions:
Suppose we have three functions <math>J, K, L\!</math> given by the type descriptors and the equation that follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{array}{lcccl}
<math>\begin{array}{lcccl}
J & : & X & \gets & Y
J & : & X & \leftarrow & Y
\\[6pt]
\\[6pt]
K & : & X & \gets & X \times X
K & : & X & \leftarrow & X \times X
\\[6pt]
\\[6pt]
L & : & Y & \gets & Y \times Y
L & : & Y & \leftarrow & Y \times Y
\end{array}</math>
\end{array}</math>
|-
|-
Our sagittarian leitmotif can be rubricized in the following slogan:
Our sagittarian leitmotif can be rubricized in the following slogan:
{| align="center" cellspacing="12" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\textit{The~image~of~the~ligature~is~the~compound~of~the~images.}</math>
|
<p>The image of the ligature is the compound of the images.</p>
|-
|-
| (Where <math>J\!</math> is the ''image'', <math>K\!</math> is the ''compound'', and <math>L\!</math> is the ''ligature''.)
|
<p>Where <math>J\!</math> is the ''image'', <math>K\!</math> is the ''compound'', and <math>L\!</math> is the ''ligature''.</p>
|}
|}
Figure 47 presents us with a picture of the situation in question.
Figure 19 presents us with a picture of the situation in question.
<br>
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
o-----------------------------------------------------------o
| |
| K L |
| O O |
| /|\ /|\ |
| / | \ / | \ |
| v | \ v | \ |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| ^ ^ ^ / / / |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| O O O |
| J J J |
| |
o-----------------------------------------------------------o
Figure 19. Structure Preserving Transformation J : K <- L
</pre>
|}
Here, I have used arrowheads to indicate the relational domains at which each of the relations <math>J, K, L\!</math> happens to be functional.
Table 48 gives the constraint matrix version of the same thing.
Table 20 gives the constraint matrix version of the same thing.
<br>
<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%"
{| 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 48.} ~~ \text{Arrow Equation:} ~~ J(L(u, v)) = K(Ju, Jv)\!</math>
|+ <math>\text{Table 20. Arrow Equation:}~~ J(L(u, v)) = K(Ju, Jv)</math>
|-
|-
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
| style="border-right:1px solid black; border-bottom:1px solid black; width:25%" |
===Commentary Note 11.14===
===Commentary Note 11.14===
First, a correction. Ignore for now the gloss that I gave in regard to Figure 19:
<br>
{| align="center" cellspacing="6" width="90%"
|
<p>Here, I have used arrowheads to indicate the relational domains at which each of the relations <math>J, K, L\!</math> happens to be functional.</p>
|}
It is more like the feathers of the arrows that serve to mark the relational domains at which the relations <math>J, K, L\!</math> 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 <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 2-ary operations of arithmetical multiplication and arithmetical summation, respectively.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 49.jpg]] || (49)
| align="center" |
<pre>
o-----------------------------------------------------------o
| |
| {+} {.} |
| O O |
| /|\ /|\ |
| / | \ / | \ |
| v | \ v | \ |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| ^ ^ ^ / / / |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| O O O |
| J J J |
| |
o-----------------------------------------------------------o
Figure 21. Logarithm Arrow J : {+} <- {.}
</pre>
|}
|}
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:
Thus, where the ''image'' <math>J\!</math> is the logarithm map, the ''compound'' <math>K\!</math> is the numerical sum, and the the ''ligature'' <math>L\!</math> is the numerical product, one obtains the immemorial mnemonic motto:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<p><math>\textit{The~image~of~the~product~is~the~sum~of~the~images.}</math></p>
<p>The image of the product is the sum of the images.</p>
|-
|-
|
|
J(u \cdot v) & = & J(u) + J(v)
J(u \cdot v) & = & J(u) + J(v)
\\[12pt]
\\[12pt]
J(L(u, v)) & = & K(Ju, Jv)
J(L(u, v)) & = & K(Ju, Jv)
\end{array}</math>
\end{array}</math>
|}
|}
===Commentary Note 11.15===
===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.
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 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.
The ''structure'' that is 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 <math>J,\!</math> we have the data:
For example, in the previous case of the logarithm map <math>J,\!</math> we have the data:
|
|
<math>\begin{array}{lcccll}
<math>\begin{array}{lcccll}
J & : & \mathbb{R} & \gets & \mathbb{R}
J & : & \mathbb{R} & \leftarrow & \mathbb{R}
& \text{(properly restricted)}
& \text{(properly restricted)}
\\[6pt]
\\[6pt]
K & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R}
K & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R}
& \text{where}~ K(r, s) = r + s
& \text{where}~ K(r, s) = r + s
\\[6pt]
\\[6pt]
L & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R}
L & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R}
& \text{where}~ L(u, v) = u \cdot v
& \text{where}~ L(u, v) = u \cdot v
\end{array}</math>
\end{array}</math>
|}
|}
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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
J
J
& : &
& : &
[+] \gets [\,\cdot\,]
[+] \leftarrow [\,\cdot\,]
\\[6pt]
\\[6pt]
[+]
[+]
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <p><math>\textit{The~image~of~the~sum~is~the~sum~of~the~images.}</math></p>
| <p>The image of the sum is the sum of the images.</p>
|-
|-
| <p><math>\textit{The~image~of~the~product~is~the~sum~of~the~images.}</math></p>
| <p>The image of the product is the sum of the images.</p>
|}
|}
Figure 50 presents a generic picture for groups <math>G\!</math> and <math>H.\!</math>
Figure 22 presents a generic picture for groups <math>G\!</math> and <math>H.\!</math>
{| align="center" cellspacing="6" width="90%"
| align="center" |
{| align="center" cellpadding="10"
<pre>
| [[Image:LOR 1870 Figure 50.jpg]] || (50)
o-----------------------------------------------------------o
| |
| G H |
| O O |
| /|\ /|\ |
| / | \ / | \ |
| v | \ v | \ |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| ^ ^ ^ / / / |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| O O O |
| J J J |
| |
o-----------------------------------------------------------o
Figure 22. Group Homomorphism J : G <- H
</pre>
|}
|}
===Commentary Note 11.16===
===Commentary Note 11.16===
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>
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 <math>t\!</math> by writing it in square brackets, as <math>[t].\!</math> It is convenient to have a prefix notation for this function, and since Peirce reserves <math>\mathit{n}\!</math> for <math>\operatorname{not},\!</math> 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]].
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 collected in [[Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives#Commentary_Note_11.2|Section 11.2]].
'''NOF 1'''
'''NOF 1'''
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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>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>
<p>(Peirce, CP 3.65).</p>
|}
|}
We may formalize the role of the "number of" function 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 so-called ''syntactic domain'', that is ample enough to hold all of the terms whose numbers we might wish to evaluate in a given discussion, and where <math>\mathbb{R}</math> is the real number domain.
Transcribing Peirce's example:
Transcribing Peirce's example:
|-
|-
| and
| and
| <math>\mathit{t} = \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.</math>
| <math>\mathit{t} = \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</math>
|
|
|-
|-
|}
|}
That is, in a universe of perfect human dentition, the number of the relative term <math>\text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math>
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.
The 2-adic relative term <math>\mathit{t}\!</math> determines a 2-adic relation <math>T \subseteq U \times V,</math> where <math>U\!</math> and <math>V\!</math> are two universes of discourse, possibly the same one, that contain among other things all the teeth and all the people that happen to be under discussion, respectively.
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>
A rough indication of the bigraph for <math>T\!</math> might be drawn as follows, where I have tried to sketch in just the toothy part of <math>U\!</math> and the peoply part of <math>V.\!</math>
{| align="center" cellpadding="10"
{| align="center" cellspacing ="6" width="90%"
| [[Image:LOR 1870 Figure 51.jpg]] || (51)
|
<pre>
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 ...
</pre>
|}
|}
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>
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>[\mathit{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.
Finally, one observes that this component of <math>T\!</math> is a function in the direction <math>T : U \to V,</math> since we are counting only those teeth that ideally occupy exactly one mouth of a tooth-bearing creature.
===Commentary Note 11.17===
===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.
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.
With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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>
<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>
<p>(Peirce, CP 3.66).</p>
<p>(Peirce, CP 3.66).</p>
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.
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.
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:
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> specifically, this:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|}
|}
The “number of” map <math>v : (S, <_1) \to (\mathbb{R}, <_2)</math> has just this character, as exemplified in the case at hand:
The "number of" map <math>v : (S, <_1) \to (\mathbb{R}, <_2)</math> has just this character, as exemplified in the case at hand:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{f} & < & \mathrm{m} & \Rightarrow & [\mathrm{f}] & < & [\mathrm{m}]
\mathrm{f} & < & \mathrm{m} & \Rightarrow & [\mathrm{f}] & < & [\mathrm{m}]
\\[6pt]
\\[6pt]
\mathrm{f} & < & \mathrm{m} & \Rightarrow & v(\mathrm{f}) & < & v(\mathrm{m})
\mathrm{f} & < & \mathrm{m} & \Rightarrow & v\mathrm{f} & < & v\mathrm{m}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
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.
Here, to be more exacting, the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the left is read as ''proper subsumption'', that is, the relation of being a subset but not being equal, while the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the right is read as the usual ''less than'' relation.
===Commentary Note 11.18===
===Commentary Note 11.18===
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?”
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?".
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>
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>
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
===Commentary Note 11.19===
===Commentary Note 11.19===
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:
Up to this point in the 1870 LOR, Peirce has introduced the "number of" measure on logical terms and discussed the extent to which this measure, <math>v : S \to \mathbb{R}</math> such that <math>v : s \mapsto [s],</math> exhibits a couple of important measure-theoretic principles:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| valign="top" | 1.
| valign="top" | 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.
| 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.
| 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 "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.
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 3.1'''
'''NOF 3.1'''
|}
|}
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 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.
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.
'''NOF 3.2'''
'''NOF 3.2'''
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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>
<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 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five.</p>
<p>(Peirce, CP 3.67).</p>
<p>(Peirce, CP 3.67).</p>
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.
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 <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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
v(x ~+\!\!,~ y) & = & vx ~+~ vy & ?
\end{matrix}</math>
\end{matrix}</math>
|}
|}
|
|
<math>\begin{matrix}
<math>\begin{matrix}
[x ~+\!\!,~ y] & = & [x] ~+~ [y] & ?
\end{matrix}</math>
\end{matrix}</math>
|}
|}
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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
v(x ~+\!\!,~ y) & \le & v(x) ~+~ v(y)
v(x ~+\!\!,~ y) & \le & vx ~+~ vy
\end{matrix}</math>
\end{matrix}</math>
|}
|}
|}
|}
In sum, therefore, it can be said: ''It also serves that only preserves a due respect for the function of a vacuum in nature.''
In sum, therefore, it also serves that only preserves a due respect for the function of a vacuum in nature.
===Commentary Note 11.20===
===Commentary Note 11.20===
We arrive at the last of Peirce's statements about the “number of” map that we singled out above:
We arrive at the last of Peirce's statements about the "number of" map that we singled out above:
'''NOF 4.1'''
'''NOF 4.1'''
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<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>
<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>
<p>If we have an equation of the form:</p>
<p>If we have an equation of the form:</p>
|}
|}
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.
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.
The proviso for the equation <math>[xy] = [x][y]\!</math> to hold is this:
The proviso for the equation <math>[xy] = [x][y]\!</math> to hold is this:
|}
|}
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.
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:
Transcribing Peirce's example:
|-
|-
| and
| and
| <math>\mathit{t} = \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.\!</math>
| <math>\mathit{t} = \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</math>
|
|
|-
|-
|}
|}
That is to say, the number of the relative term <math>\text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\!</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>
That is, in a universe of perfect human dentition, the number of the relative term <math>\text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math>
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.
The 2-adic relative term <math>\mathit{t}\!</math> determines a 2-adic relation <math>T \subseteq U \times V,</math> where <math>U\!</math> and <math>V\!</math> are two universes of discourse, possibly the same one, that contain among other things all the teeth and all the people that happen to be under discussion, respectively.
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.
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 bigraphical composition below shows all pertinent facts of the case.
{| align="center" cellpadding="10"
{| align="center" cellspacing ="6" width="90%"
| [[Image:LOR 1870 Figure 52.jpg]] || (52)
|
<pre>
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
</pre>
|}
|}
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)},\!</math> which in turn determines the idempotent representation of Frenchmen as a subset of mankind, <math>F \subseteq Y \times Y.\!</math>
In this picture the order of relational composition flows up the page. For convenience, the absolute term <math>\mathrm{f} = \text{Frenchman}\!</math> has been converted by means of the comma functor to give the idempotent representation <math>\mathrm{f,} = \text{Frenchman that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</math> In this way 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:
By way of a legend for the figure, we have the following data:
|}
|}
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,
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%"
{| align="center" cellspacing="6" width="90%"
| <math>[\mathit{t}\mathrm{f}] = [\mathit{t}][\mathrm{f}],\!</math>
| <math>[\mathit{t}\mathrm{f}] = [\mathit{t}][\mathrm{f}]\!</math>
|}
|}
whose transpose gives the equation,
whose transpose gives:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>[\mathit{t}] = \frac{[\mathit{t}\mathrm{f}]}{[\mathrm{f}]},\!</math>
| <math>[\mathit{t}] = \frac{[\mathit{t}\mathrm{f}]}{[\mathrm{f}]}</math>
|}
|}
is every bit as true as the defining equation in this circumstance, namely,
is every bit as true as the defining equation in this circumstance, namely:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>[\mathit{t}] = \frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.\!</math>
| <math>[\mathit{t}] = \frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.</math>
|}
|}
===Commentary Note 11.21===
===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.
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'''
'''NOF 4.4'''
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
|
<p>So if men are just as apt to be black as things in general,</p>
<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>
| align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]</math>
|-
|-
|
|
|}
|}
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:
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%"
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}).\!</math>
| <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b})</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ {\mathrm{P}(\mathrm{b}\mathrm{m}) \over \mathrm{P}(\mathrm{m})}.\!</math>
| <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) ~=~ \frac{\operatorname{P}(\mathrm{b}\mathrm{m})}{\operatorname{P}(\mathrm{m})}</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}).\!</math>
| <math>\operatorname{P}(\mathrm{b}\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b}|\mathrm{m})\operatorname{P}(\mathrm{m})</math>
|}
|}
Thus we may derive the equivalent statement:
{| align="center" cellspacing="6" width="90%"
{| 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>
| <math>\operatorname{P}(\mathrm{b}\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b}|\mathrm{m})\operatorname{P}(\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b})\operatorname{P}(\mathrm{m})</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}].\!</math>
| <math>[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}]</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{P}(\mathrm{m}\mathrm{b}) ~=~ \mathrm{P}(\mathrm{m})\mathrm{P}(\mathrm{b}).\!</math>
| <math>\operatorname{P}(\mathrm{m}\mathrm{b}) ~=~ \operatorname{P}(\mathrm{m})\operatorname{P}(\mathrm{b}).</math>
|}
|}
Let's see if this checks out.
Let's see if this checks out.
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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
[\mathrm{m,}\mathrm{b}]
[\mathrm{m,}\mathrm{b}]
& = &
& = &
\mathrm{P}(\mathrm{m}\mathrm{b}) N
\operatorname{P}(\mathrm{m}\mathrm{b}) N
\\[6pt]
\\[6pt]
& = &
& = &
\mathrm{P}(\mathrm{m})\mathrm{P}(\mathrm{b}) N
\operatorname{P}(\mathrm{m})\operatorname{P}(\mathrm{b}) N
\\[6pt]
\\[6pt]
& = &
& = &
\mathrm{P}(\mathrm{m})[\mathrm{b}]
\operatorname{P}(\mathrm{m})[\mathrm{b}]
\\[6pt]
\\[6pt]
& = &
& = &
[\mathrm{m,}][\mathrm{b}].
[\mathrm{m,}][\mathrm{b}]
\end{array}</math>
\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.
As a result, we have to interpret <math>[\mathrm{m,}]\!</math> = "the average number of men per things in general" as <math>\operatorname{P}(\mathrm{m})</math> = "the probability of a thing in general being a man". This seems to make sense.
===Commentary Note 11.22===
===Commentary Note 11.22===
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
|
<p>So if men are just as apt to be black as things in general,</p>
<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>
| align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]</math>
|-
|-
|
|
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.
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.
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”.
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".
Here are the relevant data:
Here are the relevant data:
|
|
<math>\begin{array}{*{15}{l}}
<math>\begin{array}{*{15}{l}}
\mathrm{b} & = & \mathrm{O}
\mathbf{1}
& = & \mathrm{B}
& +\!\!, & \mathrm{C}
& +\!\!, & \mathrm{D}
& +\!\!, & \mathrm{E}
& +\!\!, & \mathrm{I}
& +\!\!, & \mathrm{J}
& +\!\!, & \mathrm{O}
\\[6pt]
\\[6pt]
\mathrm{m} & = &
\mathbf{1,}
\mathrm{C} & +\!\!, &
& = & \mathrm{B}\!:\!\mathrm{B}
\mathrm{I} & +\!\!, &
& +\!\!, & \mathrm{C}\!:\!\mathrm{C}
\mathrm{J} & +\!\!, &
& +\!\!, & \mathrm{D}\!:\!\mathrm{D}
\mathrm{O}
& +\!\!, & \mathrm{E}\!:\!\mathrm{E}
& +\!\!, & \mathrm{I}\!:\!\mathrm{I}
& +\!\!, & \mathrm{J}\!:\!\mathrm{J}
& +\!\!, & \mathrm{O}\!:\!\mathrm{O}
\\[6pt]
\mathrm{b}
& = & \mathrm{O}
\\[6pt]
\\[6pt]
\mathrm{b,}
\mathrm{B} & +\!\!, &
& = & \mathrm{O}\!:\!\mathrm{O}
\mathrm{E} & +\!\!, &
\mathrm{I} & +\!\!, &
\\[6pt]
\\[6pt]
\mathrm{m,} & = &
\mathrm{m}
\mathrm{C\!:\!C} & +\!\!, &
& = & \mathrm{C}
\mathrm{I\!:\!I} & +\!\!, &
& +\!\!, & \mathrm{I}
\mathrm{J\!:\!J} & +\!\!, &
& +\!\!, & \mathrm{J}
\mathrm{O\!:\!O}
& +\!\!, & \mathrm{O}
\\[6pt]
\\[6pt]
\mathbf{1,} & = &
\mathrm{m,}
\mathrm{B\!:\!B} & +\!\!, &
& = & \mathrm{C}\!:\!\mathrm{C}
\mathrm{C\!:\!C} & +\!\!, &
& +\!\!, & \mathrm{I}\!:\!\mathrm{I}
\mathrm{D\!:\!D} & +\!\!, &
& +\!\!, & \mathrm{J}\!:\!\mathrm{J}
\mathrm{E\!:\!E} & +\!\!, &
& +\!\!, & \mathrm{O}\!:\!\mathrm{O}
\mathrm{I\!:\!I} & +\!\!, &
\mathrm{J\!:\!J} & +\!\!, &
\mathrm{O\!:\!O}
\end{array}</math>
\end{array}</math>
|}
|}
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.
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.
Should this hold, the consequence would be:
Should this hold, the consequence would be:
|}
|}
Once again, it is convenient to represent the absolute term <math>\mathrm{b} = \text{black}\!</math> by means the corresponding idempotent rheme, <math>\mathrm{b,} = \text{black that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</math> In this way it can be taken as a selective from the universe of discourse.
Consider the bigraph for the composition:
Consider the bigraph for the composition:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{m,}\mathrm{b,} ~=~ \text{man that is black that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.</math>
| <math>\mathrm{m,}\mathrm{b,} ~=~ \text{man that is black that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</math>
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 53.jpg]] || (53)
|
<pre>
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
</pre>
|}
|}
|}
|}
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>
This is equivalent to the implication <math>\mathrm{b} \Rightarrow \mathrm{m},</math> the content of which statement Peirce would have written in the following form:
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:
{| align="center" cellspacing="6" width="90%"
| <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 of the general formulas and particular facts together, we arrive at following summation of situation in the ''Othello'' case:
If the fair sampling condition were true, it would have the following consequence:
If the fair sampling condition were true, it would have the following consequence:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>[\mathrm{m,}] ~=~ \frac{[\mathrm{m,}\mathbf{1}]}{[\mathbf{1}]} ~=~ \frac{[\mathrm{m}]}{[\mathbf{1}]} ~=~ \frac{4}{7}.\!</math>
| <math>[\mathrm{m,}] ~=~ \frac{[\mathrm{m,}\mathbf{1}]}{[\mathbf{1}]} ~=~ \frac{[\mathrm{m}]}{[\mathbf{1}]} ~=~ \frac{4}{7}.</math>
|}
|}
In sum, it is not the case in the ''Othello'' example that “men are just as apt to be black as things in general”.
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: <math>\mathrm{P}(\mathrm{m}) = \frac{4}{7}</math> and <math>\mathrm{P}(\mathrm{b}) = \frac{1}{7}.</math>
Expressed in terms of probabilities: <math>\operatorname{P}(\mathrm{m}) = \frac{4}{7}</math> and <math>\operatorname{P}(\mathrm{b}) = \frac{1}{7}.</math>
If these were independent terms we would have: <math>\mathrm{P}(\mathrm{m}\mathrm{b}) = \frac{4}{49}.</math>
If these were independent terms we would have: <math>\operatorname{P}(\mathrm{m}\mathrm{b}) = \frac{4}{49}.</math>
In point of fact, however, we have: <math>\mathrm{P}(\mathrm{m}\mathrm{b}) = \mathrm{P}(\mathrm{b}) = \frac{1}{7}.</math>
In point of fact, however, we have: <math>\operatorname{P}(\mathrm{m}\mathrm{b}) = \operatorname{P}(\mathrm{b}) = \frac{1}{7}.</math>
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>
Another way to see it is to observe that: <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) = \frac{1}{4}</math> while <math>\operatorname{P}(\mathrm{b}) = \frac{1}{7}.</math>
===Commentary Note 11.23===
===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.
'''NOF 4.4'''
'''NOF 4.4'''
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
|
<p>So if men are just as apt to be black as things in general,</p>
<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>
| align="center" | <math>[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]</math>
|-
|-
|
|
|}
|}
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.
In different lights the formula <math>[\mathrm{m,}\mathrm{b}] = [\mathrm{m,}][\mathrm{b}]\!</math> presents itself as an ''aimed arrow'', ''fair sample'', or ''statistical independence'' condition. The concept of independence was illustrated above by means of a case where that property fails. For ease of reference, the details of that counterexample are summarized below.
{| align="center" cellpadding="10"
{| align="center" cellspacing="6" width="90%"
| [[Image:LOR 1870 Figure 53.jpg]] || (54)
|
<pre>
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
</pre>
|}
|}
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>
The condition that "men are just as apt to be black as things in general" can be expressed in terms of conditional probabilities as <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) = \operatorname{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>
In the Othello example, it is enough to observe that <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) = \tfrac{1}{4}</math> while <math>\operatorname{P}(\mathrm{b}) = \tfrac{1}{7}</math> in order to recognize the bias or dependency of the sampling map.
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>
This reduction of a conditional probability to an absolute probability in the form <math>\operatorname{P}(A|Z) = \operatorname{P}(A)</math> is a familiar disguise, and yet in practice one of the ways that we most commonly come to recognize the condition of independence <math>\operatorname{P}(AZ) = \operatorname{P}(A)P(Z),</math> via the definition of a conditional probability according to the rule <math>\operatorname{P}(A|Z) = \frac{\operatorname{P}(AZ)}{\operatorname{P}(Z)}.</math> To recall the familiar consequences, the definition of conditional probability plus the independence condition yields <math>\operatorname{P}(A|Z) = \frac{\operatorname{P}(AZ)}{P(Z)} = \frac{\operatorname{P}(A)\operatorname{P}(Z)}{\operatorname{P}(Z)},</math> to wit, <math>\operatorname{P}(A|Z) = \operatorname{P}(A).</math>
As Hamlet discovered, there's a lot to be learned from turning a crank.
As Hamlet discovered, there's a lot to be learned from turning a crank.
===Commentary Note 11.24===
===Commentary Note 11.24===
We come to the end of the “number of” examples that we found on our agenda at this point in the text:
We come to the end of the "number of" examples that we found on our agenda at this point in the text:
'''NOF 4.5'''
'''NOF 4.5'''
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
|
<p>It is to be observed that</p>
<p>It is to be observed that:</p>
|-
|-
| align="center" | <math>[\mathit{1}] ~=~ 1.</math>
| align="center" | <math>[\mathit{1}] ~=~ 1.</math>
|}
|}
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>
There are problems with the printing of the text at this point. Let us first recall the conventions that I am using in this transcription, specifically, <math>\mathit{1}\!</math> for the italic 1 that signifies the 2-adic identity relation and <math>\mathfrak{1}</math> for the "antique figure one" that Peirce defines as <math>\mathit{1}_\infty = \text{something}.</math>
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>
CP 3 gives <math>[\mathit{1}] = \mathfrak{1},</math> which I cannot make any 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>
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:
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:
|}
|}
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>
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>
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.
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.
==Selection 12==
==Selection 12==
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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. 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>
<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></p>
<p>Thus</p>
|-
| align="center" | <math>\mathit{l}^\mathrm{w}\!</math>
|-
|
<p>will be a lover of every woman.</p>
<p>Then</p>
|-
| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math>
|-
|
<p>will denote whatever stands to every woman in the relation of servant of every lover of hers;</p>
<p>and</p>
|-
| align="center" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math>
|-
|
<p>will denote whatever is a servant of everything that is lover of a woman.</p>
<p>So that</p>
|-
|-
| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.\!</math>
| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.</math>
|-
|-
|
|
| height="40" | <math>X\!</math> is a set singled out in a particular discussion as the ''universe of discourse''.
| 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>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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.\!</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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.\!</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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:10, 17 November 2015 (UTC)}.</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| 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>
| height="40" | <math>\mathfrak{W} = (\mathfrak{W}_x) = \operatorname{Mat}(W) = \operatorname{Mat}(\mathrm{w})</math> is the 1-dimensional matrix representation of the set <math>W\!</math> and the term <math>\mathrm{w}.\!</math>
|-
|-
| 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>
| height="40" | <math>\mathfrak{L} = (\mathfrak{L}_{xy}) = \operatorname{Mat}(L) = \operatorname{Mat}(\mathit{l})</math> is the 2-dimensional matrix representation of the relation <math>L\!</math> and the relative term <math>\mathit{l}.\!</math>
|-
|-
| 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>
| height="40" | <math>\mathfrak{S} = (\mathfrak{S}_{xy}) = \operatorname{Mat}(S) = \operatorname{Mat}(\mathit{s})</math> is the 2-dimensional matrix representaion of the relation <math>S\!</math> and the relative term <math>\mathit{s}.\!</math>
|}
|}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
u \star L
u \star L
& = & L_{u \,\text{at}\, 1}
& = &
L_{u \,\text{at}\, 1}
\\[6pt]
\\[6pt]
& = & \{ (u, x) \in L \}
& = & \{ (u, x) \in L \}
\\[6pt]
\\[6pt]
& = & \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.
& = &
\text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.
\\[9pt]
\\[9pt]
L \star v
L \star v
& = & L_{v \,\text{at}\, 2}
& = &
L_{v \,\text{at}\, 2}
\\[6pt]
\\[6pt]
& = & \{ (x, v) \in L \}
& = &
\{ (x, v) \in L \}
\\[6pt]
\\[6pt]
& = & \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}.
& = &
\end{array}\!</math>
\text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}.
\end{array}</math>
|}
|}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
u \cdot L
u \cdot L
& = & \mathrm{proj}_2 (u \star L)
& = &
\operatorname{proj}_2 (u \star L)
\\[6pt]
\\[6pt]
& = & \{ x \in X : (u, x) \in L \}
& = &
\{ x \in X : (u, x) \in L \}
\\[6pt]
\\[6pt]
& = & \text{loved by}~ u.
& = &
\text{loved by}~ u.
\\[9pt]
\\[9pt]
L \cdot v
L \cdot v
& = & \mathrm{proj}_1 (L \star v)
& = &
\operatorname{proj}_1 (L \star v)
\\[6pt]
\\[6pt]
& = & \{ x \in X : (x, v) \in L \}
& = &
\{ x \in X : (x, v) \in L \}
\\[6pt]
\\[6pt]
& = & \text{lover of}~ v.
& = &
\end{array}\!</math>
\text{lover of}~ v.
\end{array}</math>
|}
|}
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| 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>
<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></p>
<p>Thus</p>
|-
| align="center" | <math>\mathit{l}^\mathrm{w}\!</math>
|-
|
<p>will be a lover of every woman.</p>
<p>(Peirce, CP 3.77).</p>
<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 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”.
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".
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:
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:
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>
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>
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 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>\operatorname{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:
{| align="center" cellspacing="6" width="90%"
{| 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>
| height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \operatorname{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x</math>
|}
|}
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:
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>\mathrm{J}\!</math> is a lover of every woman is to say that <math>\mathrm{J}\!</math> loves <math>\mathrm{K}\!</math> if <math>\mathrm{K}\!</math> is a woman. This can be rendered in symbols as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>j ~\text{loves}~ k ~\Leftarrow~ k ~\text{is a woman}</math>
| height="60" | <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math>
|}
|}
Interpreting the formula <math>\mathit{l}^\mathrm{w}\!</math> as <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math> highlights the form of the 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.
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:
The operations of the forms <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%"
{| align="center" cellspacing="6" width="90%"
|
|
<math>
<math>
\begin{array}{ccc}
\begin{matrix}
0^0 & = & 1
\\
0^0 & = & 1 \\
0^1 & = & 0
0^1 & = & 0 \\
\\
1^0 & = & 1 \\
1^0 & = & 1
\\
1^1 & = & 1
1^1 & = & 1
\end{array}
\end{matrix}
\qquad\qquad\qquad
\qquad\qquad\qquad
\begin{array}{ccc}
\begin{matrix}
0\!\Leftarrow\!0 & = & 1
\\
0\!\Leftarrow\!0 & = & 1 \\
0\!\Leftarrow\!1 & = & 0
0\!\Leftarrow\!1 & = & 0 \\
\\
1\!\Leftarrow\!0 & = & 1 \\
1\!\Leftarrow\!0 & = & 1
\\
1\!\Leftarrow\!1 & = & 1
1\!\Leftarrow\!1 & = & 1
\end{array}
\end{matrix}
</math>
</math>
|}
|}
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.
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:
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>\mathfrak{L} = \operatorname{Mat}(\mathit{l})</math> and <math>\mathfrak{W} = \operatorname{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>\mathfrak{L}^\mathfrak{W} = \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})},</math> that produces the matrix <math>\operatorname{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%"
{| 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>
| height="60" | <math>\mathfrak{L}^\mathfrak{W} ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \operatorname{Mat}(\mathit{l}^\mathrm{w})</math>
|}
|}
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:
The fact that <math>\mathit{l}^\mathrm{w}\!</math> denotes the elements of a subset of <math>X\!</math> means that the matrix <math>\mathfrak{L}^\mathfrak{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>\mathfrak{L}^\mathfrak{W}</math> at the index <math>x \in X</math> is written <math>(\mathfrak{L}^\mathfrak{W})_x</math> and computed as follows:
{| align="center" cellspacing="6" width="90%"
{| 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>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_x ~=~ \prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math>
|}
|}
===Commentary Note 12.3===
===Commentary Note 12.3===
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”.
We now have two ways of computing a logical involution that raises a 2-adic relative term to the power of a 1-adic absolute term, for example, <math>\mathit{l}^\mathrm{w}\!</math> for "lover of every woman".
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:
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:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x\!</math>
| 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:
The second method operates in the matrix representation, expressing the value of the matrix <math>\mathfrak{L}^\mathfrak{W}</math> with respect to an argument <math>u\!</math> as a product of coefficient powers:
{| align="center" cellspacing="6" width="90%"
{| 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>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_x ~=~ \prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math>
|}
|}
====Example 6====
====Example 6====
The Figure below represents a universe of discourse <math>X\!</math> that is subject to the following data:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|}
|}
{| align="center" cellspacing="6" width="90%"
|
{| align="center" cellpadding="10" width="100%"
<pre>
| width="3%" |
a b c d e f g h i
| width="47%" | [[Image:LOR 1870 Figure 55.jpg]]
o o o o o o o o o X
| width="50%" | (55)
| |
| | W,
| |
o o o o o o o o o X
\ \ / / \ | / \ \ / /
\ / / \ | / \ \ / L
\ / \ / \|/ \ / \ /
o o o o o o o o o X
a b c d e f g h i
</pre>
|}
|}
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.
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.
Computing the denotation of <math>\mathit{l}^\mathrm{w}\!</math> by way of the set-theoretic formula, we can show our work as follows:
Computing the denotation of <math>\mathit{l}^\mathrm{w}\!</math> by way of the set-theoretic formula, we can show our work as follows:
|}
|}
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:
With the above picture in mind, we can visualize the computation of <math>(\mathfrak{L}^\mathfrak{W})_x = \textstyle\prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math> as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| valign="top" | 1.
| valign="top" | 1.
| Pick a specific <math>u\!</math> in the bottom row of the Figure.
| Pick a specific <math>x\!</math> in the bottom row of the Figure.
|-
|-
| valign="top" | 2.
| valign="top" | 2.
| Pan across the elements <math>v\!</math> in the middle row of the Figure.
| Pan across the elements <math>p\!</math> in the middle row of the Figure.
|-
|-
| valign="top" | 3.
| 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>
| If <math>x\!</math> links to <math>p\!</math> then <math>\mathfrak{L}_{xp} = 1,</math> otherwise <math>\mathfrak{L}_{xp} = 0.</math>
|-
|-
| valign="top" | 4.
| 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>
| If <math>p\!</math> in the middle row links to <math>p\!</math> in the top row then <math>\mathfrak{W}_p = 1,</math> otherwise <math>\mathfrak{W}_p = 0.</math>
|-
|-
| valign="top" | 5.
| 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.
| Compute the value <math>\mathfrak{L}_{xp}^{\mathfrak{W}_p} = (\mathfrak{L}_{xp} >\!\!\!-~ \mathfrak{W}_p)</math> for each <math>p\!</math> in the middle row.
|-
|-
| valign="top" | 6.
| 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>
| If any of the values <math>\mathfrak{L}_{xp}^{\mathfrak{W}_p}</math> is <math>0\!</math> then the product <math>\textstyle\prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math> is <math>0,\!</math> otherwise it is <math>1.\!</math>
|}
|}
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>
As a general observation, we know that the value of <math>(\mathfrak{L}^\mathfrak{W})_x</math> goes to <math>0\!</math> just as soon as we find an <math>p \in X</math> such that <math>\mathfrak{L}_{xp} = 0</math> and <math>\mathfrak{W}_p = 1,</math> in other words, such that <math>(x, p) \notin L</math> but <math>p \in W.</math> If there is no such <math>p,\!</math> then <math>(\mathfrak{L}^\mathfrak{W})_x = 1.</math>
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:
Running through the program for each <math>x \in X,</math> the only case that produces a non-zero result is <math>(\mathfrak{L}^\mathfrak{W})_e = 1.</math> That portion of the work can be sketched as follows:
{| align="center" cellspacing="6" width="90%"
{| 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>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_e ~=~ \prod_{p \in X} \mathfrak{L}_{ep}^{\mathfrak{W}_p} ~=~ 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>
|}
|}
===Commentary Note 12.4===
===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>
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, the law that states <math>(a^b)^c = a^{bc}.\!</math>
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
|
<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>
<p>Then</p>
|-
| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w}\!</math>
|-
|
<p>will denote whatever stands to every woman in the relation of servant of every lover of hers;</p>
<p>and</p>
|-
| align="center" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})}\!</math>
|-
|
<p>will denote whatever is a servant of everything that is lover of a woman.</p>
<p>So that</p>
|-
|-
| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.\!</math>
| align="center" | <math>(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.</math>
|-
|-
|
|
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| 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>
| height="60" | <math>\mathit{s}^{(\mathit{l}\mathrm{w})} ~=~ \bigcap_{x \in LW} \operatorname{proj}_1 (S \star x) ~=~ \bigcap_{x \in LW} S \cdot x</math>
|}
|}
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.
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 2-adic relative term to the power of a 2-adic relative term. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.
====Example 7====
====Example 7====
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{| align="center" cellpadding="10" width="100%"
{| align="center" cellspacing="6" width="90%"
| width="3%" |
|
| width="47%" | [[Image:LOR 1870 Figure 56.jpg]]
<pre>
| width="50%" | (56)
a b c d e f g h i
o o o o o o o o o X
\ \ / / \ | / \ \ / /
\ / / \ | / \ \ / L
\ / \ / \|/ \ / \ /
o o o o o o o o o X
\ / \ | / \ | / \ /
\ / \ | / \ | / \ / S
\ / \|/ \|/ \ /
o o o o o o o o o X
a b c d e f g h i
</pre>
|}
|}
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.
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.
The computational requirements are evidently met by the following formula:
The computational requirements are evidently met by the following formula:
{| align="center" cellspacing="6" width="90%"
{| 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>
| height="60" | <math>(\mathfrak{S}^\mathfrak{L})_{xy} ~=~ \prod_{p \in X} \mathfrak{S}_{xp}^{\mathfrak{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>
In other words, <math>(\mathfrak{S}^\mathfrak{L})_{xy} = 0</math> if and only if there exists a <math>p \in X</math> such that <math>\mathfrak{S}_{xp} = 0</math> and <math>\mathfrak{L}_{py} = 1.</math>
===Commentary Note 12.5===
===Commentary Note 12.5===
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>(\mathsf{S}^\mathsf{L})^\mathsf{W} ~=~ \mathsf{S}^{\mathsf{L}\mathsf{W}}</math>
| height="60" | <math>(\mathfrak{S}^\mathfrak{L})^\mathfrak{W} ~=~ \mathfrak{S}^{\mathfrak{L}\mathfrak{W}}</math>
|}
|}
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>
If <math>\mathfrak{A}</math> and <math>\mathfrak{B}</math> are two 1-dimensional matrices over the same index set <math>X,\!</math> then <math>\mathfrak{A} = \mathfrak{B}</math> if and only if <math>\mathfrak{A}_x = \mathfrak{B}_x</math> for every <math>x \in X.</math> Thus, a routine way to check the validity of <math>(\mathfrak{S}^\mathfrak{L})^\mathfrak{W} = \mathfrak{S}^{\mathfrak{L}\mathfrak{W}}</math> is to check whether the following equation holds for arbitrary <math>x \in X.</math>
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>((\mathsf{S}^\mathsf{L})^\mathsf{W})_x ~=~ (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x</math>
| height="60" | <math>((\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_x ~=~ (\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_x</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="200" |
| height="80" |
<math>
((\mathfrak{S}^\mathfrak{L})^\mathfrak{W})_x ~=~
\prod_{p \in X} (\mathfrak{S}^\mathfrak{L})_{xp}^{\mathfrak{W}_p} ~=~
\prod_{p \in X} (\prod_{q \in X} \mathfrak{S}_{xq}^{\mathfrak{L}_{qp}})^{\mathfrak{W}_p} ~=~
\prod_{p \in X} \prod_{q \in X} \mathfrak{S}_{xq}^{\mathfrak{L}_{qp}\mathfrak{W}_p}
</math>
|}
{| align="center" cellspacing="6" width="90%"
| height="80" |
<math>
<math>
(\mathfrak{S}^{\mathfrak{L}\mathfrak{W}})_x ~=~
\prod_{q \in X} \mathfrak{S}_{xq}^{(\mathfrak{L}\mathfrak{W})_q} ~=~
\prod_{q \in X} \mathfrak{S}_{xq}^{\sum_{p \in X} \mathfrak{L}_{qp} \mathfrak{W}_p} ~=~
\prod_{q \in X} \prod_{p \in X} \mathfrak{S}_{xq}^{\mathfrak{L}_{qp} \mathfrak{W}_p}
(\mathsf{S}^{\mathsf{L}\mathsf{W}})_x
\prod_{q \in X} \mathsf{S}_{xq}^{(\mathsf{L}\mathsf{W})_q}
\prod_{q \in X} \mathsf{S}_{xq}^{\sum_{p \in X} \mathsf{L}_{qp} \mathsf{W}_p}
\prod_{q \in X} \prod_{p \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp} \mathsf{W}_p}
</math>
</math>
|}
|}
==References==
==References==
* 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.
* [[George Boole|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.
* 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)].
* [[Charles Sanders Peirce|Peirce, C.S.]] (1870), [[Logic of Relatives (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). [http://books.google.com/books?id=fFnWmf5oLaoC Online].
* 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).
* 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).
* 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).
* 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).
* Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY.
* Mili, A., Desharnais, J., Mili, F., with Frappier, M. (1994), ''Computer Program Construction'', Oxford University Press, New York, NY.
==See Also==
==See Also==
