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'''Author's Note.'''  The text that follows is a collection of notes that will eventually be developed into a paper on [[Charles Sanders Peirce]]'s [[Logic of Relatives (1870)|1870 memoir on the logic of relative terms]].  [[User:Jon Awbrey|Jon Awbrey]] 06:06, 8 October 2007 (PDT)
 
'''Author's Note.'''  The text that follows is a collection of notes that will eventually be developed into a paper on [[Charles Sanders Peirce]]'s [[Logic of Relatives (1870)|1870 memoir on the logic of relative terms]].  [[User:Jon Awbrey|Jon Awbrey]] 06:06, 8 October 2007 (PDT)
 +
 +
----
 +
 +
<pre>
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
IDS -- LOR
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Logic Of Relatives
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The letters of the alphabet will denote logical signs.
 +
| Now logical terms are of three grand classes.
 +
|
 +
| 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'.
 +
|
 +
| 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'.
 +
|
 +
| The third class embraces terms whose logical form involves the
 +
| conception of bringing things into relation, and which require
 +
| the addition of more than one term to complete the denotation.
 +
| They discriminate not only with consciousness of discrimination,
 +
| but with consciousness of its origin.  They regard  an object
 +
| as medium or third between two others, that is as conjugative;
 +
| as giver of --- to ---, or buyer of --- for --- from ---.
 +
| These may be termed 'conjugative terms'.
 +
|
 +
| The conjugative term involves the conception of 'third', the relative that of
 +
| second or 'other', the absolute term simply considers 'an' object.  No fourth
 +
| class of terms exists involving the conception of 'fourth', because when that
 +
| of 'third' is introduced, since it involves the conception of bringing objects
 +
| into relation, all higher numbers are given at once, inasmuch as the conception
 +
| of bringing objects into relation is independent of the number of members of the
 +
| relationship.  Whether this 'reason' for the fact that there is no fourth class
 +
| of terms fundamentally different from the third is satisfactory of not, the fact
 +
| itself is made perfectly evident by the study of the logic of relatives.
 +
|
 +
| C.S. Peirce, CP 3.63
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 relations",
 +
his definition of sign relations, and in the pragmatic maxim itself.
 +
 +
One might also call attention to the following two statements:
 +
 +
| Now logical terms are of three grand classes.
 +
 +
| No fourth class of terms exists involving the conception of 'fourth',
 +
| because when that of 'third' is introduced, since it involves the
 +
| conception of bringing objects into relation, all higher numbers are
 +
| given at once, inasmuch as the conception of bringing objects into
 +
| relation is independent of the number of members of the relationship.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Numbers Corresponding to Letters
 +
|
 +
| I propose to use the term "universe" to denote that class of individuals
 +
| 'about' which alone the whole discourse is understood to run.  The universe,
 +
| therefore, in this sense, as in Mr. De Morgan's, is different on different
 +
| occasions.  In this sense, moreover, discourse may run upon something which
 +
| is not a subjective part of the universe;  for instance, upon the qualities
 +
| or collections of the individuals it contains.
 +
|
 +
| I propose to assign to all logical terms, numbers;  to an absolute term,
 +
| the number of individuals it denotes;  to a relative term, the average
 +
| number of things so related to one individual.  Thus in a universe of
 +
| perfect men ('men'), the number of "tooth of" would be 32.  The number
 +
| of a relative with two correlates would be the average number of things
 +
| so related to a pair of individuals;  and so on for relatives of higher
 +
| numbers of correlates.  I propose to denote the number of a logical term
 +
| by enclosing the term in square brackets, thus ['t'].
 +
|
 +
| C.S. Peirce, CP 3.65
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Peirce's remarks at CP 3.65 are so replete with remarkable ideas,
 +
some of them so taken for granted in mathematical discourse that
 +
they usually escape explicit mention, and others so suggestive
 +
of things to come in a future remote from his time of writing,
 +
and yet so smoothly introduced in passing that it's all too
 +
easy to overlook their consequential significance, that I
 +
can do no better here than to highlight these ideas in
 +
other words, whose main advantage is to be a little
 +
more jarring to the mind's sensibilities.
 +
 +
| Numbers Corresponding to Letters
 +
|
 +
| I propose to use the term "universe" to denote that class of individuals
 +
| 'about' which alone the whole discourse is understood to run.  The universe,
 +
| therefore, in this sense, as in Mr. De Morgan's, is different on different
 +
| occasions.  In this sense, moreover, discourse may run upon something which
 +
| is not a subjective part of the universe;  for instance, upon the qualities
 +
| or collections of the individuals it contains.
 +
|
 +
| I propose to assign to all logical terms, numbers;  to an absolute term,
 +
| the number of individuals it denotes;  to a relative term, the average
 +
| number of things so related to one individual.  Thus in a universe of
 +
| perfect men ('men'), the number of "tooth of" would be 32.  The number
 +
| of a relative with two correlates would be the average number of things
 +
| so related to a pair of individuals;  and so on for relatives of higher
 +
| numbers of correlates.  I propose to denote the number of a logical term
 +
| by enclosing the term in square brackets, thus ['t'].
 +
|
 +
| C.S. Peirce, 'Collected Papers', CP 3.65
 +
 +
1.  This mapping of letters to numbers, or logical terms to mathematical quantities,
 +
    is the very core of what "quantification theory" is all about, and definitely
 +
    more to the point than the mere "innovation" of using distinctive symbols
 +
    for the so-called "quantifiers".  We will speak of this more later on.
 +
 +
2.  The mapping of logical terms to numerical measures,
 +
    to express it in current language, would probably be
 +
    recognizable as some kind of "morphism" or "functor"
 +
    from a logical domain to a quantitative co-domain.
 +
 +
3.  Notice that Peirce follows the mathematician's usual practice,
 +
    then and now, of making the status of being an "individual" or
 +
    a "universal" relative to a discourse in progress.  I have come
 +
    to appreciate more and more of late how radically different this
 +
    "patchwork" or "piecewise" approach to things is from the way of
 +
    some philosophers who seem to be content with nothing less than
 +
    many worlds domination, which means that they are never content
 +
    and rarely get started toward the solution of any real problem.
 +
    Just my observation, I hope you understand.
 +
 +
4.  It is worth noting that Peirce takes the "plural denotation"
 +
    of terms for granted, or what's the number of a term for,
 +
    if it could not vary apart from being one or nil?
 +
 +
5.  I also observe that Peirce takes the individual objects of a particular
 +
    universe of discourse in a "generative" way, not a "totalizing" way,
 +
    and thus they afford us with the basis for talking freely about
 +
    collections, constructions, properties, qualities, subsets,
 +
    and "higher types", as the phrase is mint.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs of Inclusion, Equality, Etc.
 +
|
 +
| I shall follow Boole in taking the sign of equality to signify identity.
 +
| Thus, if v denotes the Vice-President of the United States, and p the
 +
| President of the Senate of the United States,
 +
|
 +
| v = 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.
 +
| The sign "less than" is to be so taken that
 +
|
 +
| f < m
 +
|
 +
| 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 '=' and '<' that the sign '-<' (or '=<', "as small as")
 +
| will mean "is".  Thus,
 +
|
 +
| f -< m
 +
|
 +
| means "every Frenchman is a man", without saying whether there are any
 +
| other men or not.  So,
 +
|
 +
| 'm' -< 'l'
 +
|
 +
| will mean that every mother of anything is a lover of the same thing;
 +
| although this interpretation in some degree anticipates a convention to
 +
| be made further on.  These significations of '=' and '<' plainly conform
 +
| to the indispensable conditions.  Upon the transitive character of these
 +
| relations the syllogism depends, for by virtue of it, from
 +
|
 +
| f -< m
 +
|
 +
| and
 +
|
 +
| m -< a,
 +
|
 +
| we can infer that
 +
|
 +
| f -< a;
 +
|
 +
| that is, from every Frenchman being a man and every
 +
| man being an animal, that every Frenchman is an animal.
 +
|
 +
| But not only do the significations of '=' and '<' here adopted fulfill all
 +
| absolute requirements, but they have the supererogatory virtue of being very
 +
| nearly the same as the common significations.  Equality is, in fact, nothing
 +
| but the identity of two numbers;  numbers that are equal are those which are
 +
| predicable of the same collections, just as terms that are identical are those
 +
| which are predicable of the same classes.  So, to write 5 < 7 is to say that 5
 +
| is part of 7, just as to write f < m is to say that Frenchmen are part of men.
 +
| Indeed, if f < m, then the number of Frenchmen is less than the number of men,
 +
| and if v = p, then the number of Vice-Presidents is equal to the number of
 +
| Presidents of the Senate;  so that the numbers may always be substituted
 +
| for the terms themselves, in case no signs of operation occur in the
 +
| equations or inequalities.
 +
|
 +
| C.S. Peirce, CP 3.66
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The quantifier mapping from terms to their numbers that Peirce signifies
 +
by means of the square bracket notation has one of its principal uses in
 +
providing a basis for the computation of frequencies, probabilities, and
 +
all of the other statistical measures that can be constructed from these,
 +
and thus in affording what may be called a "principle of correspondence"
 +
between probability theory and its limiting case in the forms of logic.
 +
 +
This brings us once again to the relativity of contingency and necessity,
 +
as one way of approaching necessity is through the avenue of probability,
 +
describing necessity as a probability of 1, but the whole apparatus of
 +
probability theory only figures in if it is cast against the backdrop
 +
of probability space axioms, the reference class of distributions,
 +
and the sample space that we cannot help but to abdeuce upon the
 +
scene of observations.  Aye, there's the snake eyes.  And with
 +
them we can see that there is always an irreducible quantum
 +
of facticity to all our necessities.  More plainly spoken,
 +
it takes a fairly complex conceptual infrastructure just
 +
to begin speaking of probabilities, and this setting
 +
can only be set up by means of abductive, fallible,
 +
hypothetical, and inherently risky mental acts.
 +
 +
Pragmatic thinking is the logic of abduction, which is just another
 +
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.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Addition
 +
|
 +
| The sign of addition is taken by Boole so that
 +
|
 +
| x + y
 +
|
 +
| denotes everything denoted by x, and, 'besides',
 +
| everything denoted by y.
 +
|
 +
| Thus
 +
|
 +
| m + w
 +
|
 +
| denotes all men, and, besides, all women.
 +
|
 +
| This signification for this sign is needed for
 +
| connecting the notation of logic with that of the
 +
| theory of probabilities.  But if there is anything
 +
| which is denoted by both terms of the sum, the latter
 +
| no longer stands for any logical term on account of
 +
| its implying that the objects denoted by one term
 +
| are to be taken 'besides' the objects denoted by
 +
| the other.
 +
|
 +
| For example,
 +
|
 +
| f + u
 +
|
 +
| means all Frenchmen besides all violinists, and,
 +
| therefore, considered as a logical term, implies
 +
| that all French violinists are 'besides themselves'.
 +
|
 +
| For this reason alone, in a paper which is published
 +
| in the Proceedings of the Academy for March 17, 1867,
 +
| I preferred to take as the regular addition of logic
 +
| a non-invertible process, such that
 +
|
 +
| m +, b
 +
|
 +
| stands for all men and black things, without any implication that
 +
| the black things are to be taken besides the men;  and the study of
 +
| the logic of relatives has supplied me with other weighty reasons for
 +
| the same determination.
 +
|
 +
| Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in
 +
| a tract called 'Pure Logic, or the Logic of Quality' [1864], had anticipated me in
 +
| substituting the same operation for Boole's addition, although he rejects Boole's
 +
| operation entirely and writes the new one with a '+' sign while withholding from
 +
| it the name of addition.
 +
|
 +
| It is plain that both the regular non-invertible addition
 +
| and the invertible addition satisfy the absolute conditions.
 +
| But the notation has other recommendations.  The conception
 +
| of 'taking together' involved in these processes is strongly
 +
| analogous to that of summation, the sum of 2 and 5, for example,
 +
| being the number of a collection which consists of a collection of
 +
| two and a collection of five.  Any logical equation or inequality
 +
| in which no operation but addition is involved may be converted
 +
| into a numerical equation or inequality by substituting the
 +
| numbers of the several terms for the terms themselves --
 +
| provided all the terms summed are mutually exclusive.
 +
|
 +
| Addition being taken in this sense,
 +
| 'nothing' is to be denoted by 'zero',
 +
| for then
 +
|
 +
| x +, 0 = x,
 +
|
 +
| whatever is denoted by x;  and this is the definition
 +
| of 'zero'.  This interpretation is given by Boole, and
 +
| is very neat, on account of the resemblance between the
 +
| ordinary conception of 'zero' and that of nothing, and
 +
| because we shall thus have
 +
|
 +
| [0] = 0.
 +
|
 +
| C.S. Peirce, CP 3.67
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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:
 +
 +
http://suo.ieee.org/ontology/msg04350.html
 +
http://suo.ieee.org/ontology/msg04351.html
 +
 +
In setting up a correspondence between "letters" and "numbers",
 +
my sense is that Peirce is "nocking an arrow", or constructing
 +
some kind of structure-preserving map from a logical domain to
 +
a numerical domain, and this interpretation is here reinforced
 +
by the careful attention that he gives to the conditions under
 +
which precisely which aspects of structure are preserved, plus
 +
his telling recognition of the criterial fact that zeroes are
 +
preserved by the mapping.  But here's the catch, the arrow is
 +
from the qualitative domain to the quantitative domain, which
 +
is just the opposite of what I tend to expect, since I think
 +
of quantitative measures as preserving more information than
 +
qualitative measures.  To curtail the story, it is possible
 +
to sort this all out, but that is a story for another day.
 +
 +
Other than that, I just want to red flag the beginnings
 +
of another one of those "failures to communicate" that
 +
so dogged the disciplines in the 20th Century, namely,
 +
the fact that Peirce seemed to have an inkling about
 +
the problems that would be caused by using the plus
 +
sign for inclusive disjunction, but, as it happens,
 +
his advice was overridden by the usages in various
 +
different communities, rendering the exchange of
 +
information among engineering, mathematical, and
 +
philosophical specialties a minefield in place
 +
of mindfield to this very day.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication
 +
|
 +
| I shall adopt for the conception of multiplication
 +
| 'the application of a relation', in such a way that,
 +
| for example, 'l'w 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.
 +
|
 +
| 's'(m +, w) will, then, denote whatever is
 +
| servant of anything of the class composed
 +
| of men and women taken together.  So that:
 +
|
 +
| 's'(m +, w)  =  's'm +, 's'w.
 +
|
 +
| ('l' +, 's')w will denote whatever is
 +
| lover or servant to a woman, and:
 +
|
 +
| ('l' +, 's')w  =  'l'w +, 's'w.
 +
|
 +
| ('sl')w will denote whatever stands to
 +
| a woman in the relation of servant of
 +
| a lover, and:
 +
|
 +
| ('sl')w  =  's'('l'w).
 +
|
 +
| Thus all the absolute conditions
 +
| of multiplication are satisfied.
 +
|
 +
| The term "identical with ---" is a unity
 +
| for this multiplication.  That is to say,
 +
| if we denote "identical with ---" by !1!
 +
| we have:
 +
|
 +
| 'x'!1!  =  'x',
 +
|
 +
| whatever relative term 'x' may be.
 +
| For what is a lover of something
 +
| identical with anything, is the
 +
| same as a lover of that thing.
 +
|
 +
| C.S. Peirce, CP 3.68
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Peirce in 1870 is five years down the road from the Peirce of 1865-1866
 +
who lectured extensively on the role of sign relations in the logic of
 +
scientific inquiry, articulating their involvement in the three types
 +
of inference, and inventing the concept of "information" to explain
 +
what it is that signs convey in the process.  By this time, then,
 +
the semiotic or sign relational approach to logic is so implicit
 +
in his way of working that he does not always take the trouble
 +
to point out its distinctive features at each and every turn.
 +
So let's take a moment to draw out a few of these characters.
 +
 +
Sign relations, like any non-trivial brand of 3-adic relations,
 +
can become overwhelming to think about once the cardinality of
 +
the object, sign, and interpretant domains or the complexity
 +
of the relation itself ascends beyond the simplest examples.
 +
Furthermore, most of the strategies that we would normally
 +
use to control the complexity, like neglecting one of the
 +
domains, in effect, projecting the 3-adic sign relation
 +
onto one of its 2-adic faces, or focusing on a single
 +
ordered triple of the form <o, s, i> at a time, can
 +
result in our receiving a distorted impression of
 +
the sign relation's true nature and structure.
 +
 +
I find that it helps me to draw, or at least to imagine drawing,
 +
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:
 +
v = p, where "v" denotes the Vice-President of the United States,
 +
and "p" denotes the President of the Senate of the United States.
 +
 +
o-----------------------------o-----------------------------o
 +
|  Objective Framework (OF)  | Interpretive Framework (IF) |
 +
o-----------------------------o-----------------------------o
 +
|          Objects          |            Signs            |
 +
o-----------------------------o-----------------------------o
 +
|                                                          |
 +
|                                o "v"                    |
 +
|                                /                          |
 +
|                              /                          |
 +
|                              /                            |
 +
|          o ... o-----------@                            |
 +
|                              \                            |
 +
|                              \                          |
 +
|                                \                          |
 +
|                                o "p"                    |
 +
|                                                          |
 +
o-----------------------------o-----------------------------o
 +
 +
Depending on whether we interpret the terms "v" and "p" as applying to
 +
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.
 +
 +
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.
 +
 +
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
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication (cont.)
 +
|
 +
| A conjugative term like 'giver' naturally requires two correlates,
 +
| one denoting the thing given, the other the recipient of the gift.
 +
|
 +
| We must be able to distinguish, in our notation, the
 +
| giver of A to B from the giver to A of B, 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.
 +
|
 +
| Let `g` 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:
 +
|
 +
| `g`xy
 +
|
 +
| will denote a giver to x of y.
 +
|
 +
| But according to the notation,
 +
| x here multiplies y, so that
 +
| if we put for x owner ('o'),
 +
| and for y horse (h),
 +
|
 +
| `g`'o'h
 +
|
 +
| appears to denote the giver of a horse
 +
| to an owner of a horse.  But let the
 +
| individual horses be H, H', H", etc.
 +
|
 +
| Then:
 +
|
 +
| h  =  H +, H' +, H" +, etc.
 +
|
 +
| `g`'o'h  =  `g`'o'(H +, H' +, H" +, etc.)
 +
|
 +
|          =  `g`'o'H +, `g`'o'H' +, `g`'o'H" +, etc.
 +
|
 +
| Now this last member must be interpreted as a giver
 +
| of a horse to the owner of 'that' horse, and this,
 +
| therefore must be the interpretation of `g`'o'h.
 +
|
 +
| This is always very important.
 +
|
 +
| 'A term multiplied by two relatives shows that
 +
|  the same individual is in the two relations.'
 +
|
 +
| If we attempt to express the giver of a horse to
 +
| a lover of a woman, and for that purpose write:
 +
|
 +
| `g`'l'wh,
 +
|
 +
| we have written giver of a woman to a lover of her,
 +
| and if we add brackets, thus,
 +
|
 +
| `g`('l'w)h,
 +
|
 +
| we abandon the associative principle of multiplication.
 +
|
 +
| 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.
 +
|
 +
| Then, the giver of a horse to a lover of a woman may be written:
 +
|
 +
| `g`_12 'l'_1 w h  =  `g`_11 'l'_2 h w  =  `g`_2(-1) h 'l'_1 w.
 +
|
 +
| Of course a negative number indicates that
 +
| the former correlate follows the latter
 +
| by the corresponding positive number.
 +
|
 +
| A subjacent 'zero' makes the term itself the correlate.
 +
|
 +
| Thus,
 +
|
 +
| 'l'_0
 +
|
 +
| denotes the lover of 'that' lover or the lover of himself, just as
 +
| `g`'o'h denotes that the horse is given to the owner of itself, for
 +
| to make a term doubly a correlate is, by the distributive principle,
 +
| to make each individual doubly a correlate, so that:
 +
|
 +
| 'l'_0  =  L_0 +, L_0' +, L_0" +, etc.
 +
|
 +
| A subjacent sign of infinity may
 +
| indicate that the correlate is
 +
| indeterminate, so that:
 +
|
 +
| 'l'_oo
 +
|
 +
| will denote a lover of something.
 +
| We shall have some confirmation
 +
| of this presently.
 +
|
 +
| If the last subjacent number is a 'one'
 +
| it may be omitted.  Thus we shall have:
 +
|
 +
| 'l'_1  =  'l',
 +
|
 +
| `g`_11  =  `g`_1  =  `g`.
 +
|
 +
| This enables us to retain our former expressions 'l'w, `g`'o'h, etc.
 +
|
 +
| C.S. Peirce, CP 3.69-70
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Peirce's way of representing sets as sums may seem archaic, but it is
 +
quite often used, and is actually the tool of choice in many branches
 +
of algebra, combinatorics, computing, and statistics to this very day.
 +
 +
Peirce's application to logic is fairly novel, and the degree of his
 +
elaboration of the logic of relative terms is certainly original with
 +
him, but this particular genre of representation, commonly going under
 +
the handle of "generating functions", goes way back, well before anyone
 +
thought to stick a flag in set theory as a separate territory or to try
 +
to fence off our native possessions of it with expressly decreed axioms.
 +
And back in the days when computers were people, before we had the sorts
 +
of "electronic register machines" that we take so much for granted today,
 +
mathematicians were constantly using generating functions as a rough and
 +
ready type of addressable memory to sort, store, and keep track of their
 +
accounts of a wide variety of formal objects of thought.
 +
 +
Let us look at a few simple examples of generating functions,
 +
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, {a, b, c}, 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:
 +
 +
(1 + a)(1 + b)(1 + c).
 +
 +
The factor (1 + a) represents the option that we have, in choosing
 +
a subset of {a, b, c}, to leave the 'a' out (signified by the "1"),
 +
or else to include it (signified by the "a"), and likewise for the
 +
other elements 'b' and 'c' in their turns.
 +
 +
Probably on account of all those years I flippered away
 +
playing the oldtime pinball machines, I tend to imagine
 +
a product like this being displayed in a vertical array:
 +
 +
(1 + a)
 +
(1 + b)
 +
(1 + c)
 +
 +
I picture this as a playboard with six "bumpers",
 +
the ball chuting down the board in such a career
 +
that it strikes exactly one of the two bumpers
 +
on each and every one of the three levels.
 +
 +
So a trajectory of the ball where it
 +
hits the "a" bumper on the 1st level,
 +
hits the "1" bumper on the 2nd level,
 +
hits the "c" bumper on the 3rd level,
 +
and then exits the board, represents
 +
a single term in the desired product
 +
and corresponds to the subset {a, c}.
 +
 +
Multiplying out (1 + a)(1 + b)(1 + c), one obtains:
 +
 +
1 + a + b + c + ab + ac + bc + abc.
 +
 +
And this informs us that the subsets of choice are:
 +
 +
{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication (cont.)
 +
|
 +
| The associative principle does not hold in this counting
 +
| of factors.  Because it does not hold, these subjacent
 +
| numbers are frequently inconvenient in practice, and
 +
| I therefore use also another mode of showing where
 +
| the correlate of a term is to be found.  This is
 +
| by means of the marks of reference, † ‡ || § ¶,
 +
| which are placed subjacent to the relative
 +
| term and before and above the correlate.
 +
| Thus, giver of a horse to a lover of
 +
| a woman may be written:
 +
|
 +
| `g`_†‡ †'l'_|| ||w ‡h.
 +
|
 +
| The asterisk I use exclusively to refer to the last
 +
| correlate of the last relative of the algebraic term.
 +
|
 +
| Now, considering the order of multiplication to be: --
 +
| a term, a correlate of it, a correlate of that correlate,
 +
| etc. -- there is no violation of the associative principle.
 +
| The only violations of it in this mode of notation are that
 +
| in thus passing from relative to correlate, we skip about
 +
| among the factors in an irregular manner, and that we
 +
| cannot substitute in such an expression as `g`'o'h
 +
| a single letter for 'o'h.
 +
|
 +
| I would suggest that such a notation may be found useful in treating other
 +
| cases of non-associative multiplication.  By comparing this with what was
 +
| said above [in CP 3.55] concerning functional multiplication, it appears
 +
| that multiplication by a conjugative term is functional, and that the
 +
| letter denoting such a term is a symbol of operation.  I am therefore
 +
| using two alphabets, the Greek and Kennerly, where only one was
 +
| necessary.  But it is convenient to use both.
 +
|
 +
| C.S. Peirce, CP 3.71-72
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
NB.  On account of the invideous circumstance that various
 +
listservers balk at Peirce's "marks of reference" -- or is
 +
it only the Microsoft Cryptkeeper's kryptonizing of them? --
 +
I will make the following substitutions in Peirce's text:
 +
 +
@  =  dagger symbol
 +
#  =  double dagger
 +
|| =  parallel sign
 +
$  =  section symbol
 +
%  =  paragraph mark
 +
 +
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:
 +
 +
o---------------------------------------o
 +
|                                      |
 +
|            @        ||                |
 +
|          / \      /  \              |
 +
|          o  o    o    o              |
 +
|      `g`_@#  @'l'_||  ||w  #h        |
 +
|          o                o          |
 +
|            \______________/          |
 +
|                  #                  |
 +
|                                      |
 +
o---------------------------------------o
 +
Giver of a Horse to a Lover of a Woman
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 8
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication (cont.)
 +
|
 +
| 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.
 +
|
 +
| 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.
 +
|
 +
| Then:
 +
|
 +
| "man that is black"
 +
|
 +
| will be written
 +
|
 +
| m,b.
 +
|
 +
| But not only may any absolute term be thus regarded as a relative term,
 +
| but any relative term may in the same way be regarded as a relative with
 +
| one correlate more.  It is convenient to take this additional correlate
 +
| as the first one.
 +
|
 +
| Then:
 +
|
 +
| 'l','s'w
 +
|
 +
| will denote a lover of a woman that is a servant of that woman.
 +
|
 +
| The comma here after 'l' should not be considered as altering at
 +
| all the meaning of 'l', but as only a subjacent sign, serving to
 +
| alter the arrangement of the correlates.
 +
|
 +
| In point of fact, since a comma may be added in this way to any
 +
| relative term, it may be added to one of these very relatives
 +
| formed by a comma, and thus by the addition of two commas
 +
| an absolute term becomes a relative of two correlates.
 +
|
 +
| So:
 +
|
 +
| m,,b,r
 +
|
 +
| interpreted like
 +
|
 +
| `g`'o'h
 +
|
 +
| means a man that is a rich individual and
 +
| is a black that is that rich individual.
 +
|
 +
| But this has no other meaning than:
 +
|
 +
| m,b,r
 +
|
 +
| or a man that is a black that is rich.
 +
|
 +
| Thus we see that, after one comma is added, the
 +
| addition of another does not change the meaning
 +
| at all, so that whatever has one comma after it
 +
| must be regarded as having an infinite number.
 +
|
 +
| If, therefore, 'l',,'s'w is not the same as 'l','s'w (as it plainly is not,
 +
| because the latter means a lover and servant of a woman, and the former a
 +
| lover of and servant of and same as a woman), this is simply because the
 +
| writing of the comma alters the arrangement of the correlates.
 +
|
 +
| And if we are to suppose that absolute terms are multipliers
 +
| at all (as mathematical generality demands that we should},
 +
| we must regard every term as being a relative requiring
 +
| an infinite number of correlates to its virtual infinite
 +
| series "that is --- and is --- and is --- etc."
 +
|
 +
| Now a relative formed by a comma of course receives its
 +
| subjacent numbers like any relative, but the question is,
 +
| What are to be the implied subjacent numbers for these
 +
| implied correlates?
 +
|
 +
| Any term may be regarded as having an
 +
| infinite number of factors, those
 +
| at the end being 'ones', thus:
 +
|
 +
| 'l','s'w  =  'l','s'w,!1!,!1!,!1!,!1!,!1!,!1!,!1!, etc.
 +
|
 +
| A subjacent number may therefore be as great as we please.
 +
|
 +
| But all these 'ones' denote the same identical individual denoted
 +
| by w;  what then can be the subjacent numbers to be applied to 's',
 +
| for instance, on account of its infinite "that is"'s?  What numbers
 +
| can separate it from being identical with w?  There are only two.
 +
| The first is 'zero', which plainly neutralizes a comma completely,
 +
| since
 +
|
 +
| 's',_0 w  =  's'w
 +
|
 +
| and the other is infinity;  for as 1^oo is indeterminate
 +
| in ordinary algbra, so it will be shown hereafter to be
 +
| here, so that to remove the correlate by the product of
 +
| an infinite series of 'ones' is to leave it indeterminate.
 +
|
 +
| Accordingly,
 +
|
 +
| m,_oo
 +
|
 +
| should be regarded as expressing 'some' man.
 +
|
 +
| Any term, then, is properly to be regarded as having an infinite
 +
| number of commas, all or some of which are neutralized by zeros.
 +
|
 +
| "Something" may then be expressed by:
 +
|
 +
| !1!_oo.
 +
|
 +
| I shall for brevity frequently express this by an antique figure one (`1`).
 +
|
 +
| "Anything" by:
 +
|
 +
| !1!_0.
 +
|
 +
| I shall often also write a straight 1 for 'anything'.
 +
|
 +
| C.S. Peirce, CP 3.73
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 8.1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
To my way of thinking, CP 3.73 is one of the most remarkable passages
 +
in the history of logic.  In this first pass over its deeper contents
 +
I won't be able to accord it much more than a superficial dusting off.
 +
 +
As always, it is probably best to begin with a concrete example.
 +
So let us initiate a discourse, whose universe X may remind us
 +
a little of the cast of characters in Shakespeare's 'Othello'.
 +
 +
X  =  {Bianca, Cassio, Clown, Desdemona, Emilia, Iago, Othello}.
 +
 +
The universe X 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 staying a bit closer to Peirce's conventions about capitalization,
 +
let us rename the universe "u", the Clown "Jeste", and then rewrite the
 +
above description of the universe of discourse in the following fashion:
 +
 +
u  =  {B, C, D, E, I, J, O}.
 +
 +
This specification of the universe of discourse could be
 +
summed up in Peirce's notation by the following equation:
 +
 +
1  =  B +, C +, D +, E +, I +, J +, O.
 +
 +
Within this discussion, then, the "individual terms" are
 +
"B", "C", "D", "E", "I", "J", "O", each of which denotes
 +
in a singular fashion the corresponding individual in X.
 +
 +
As "general terms" of this discussion,
 +
we might begin with the following set:
 +
 +
"b"  =  "black"
 +
 +
"m"  =  "man"
 +
 +
"w"  =  "woman"
 +
 +
In Peirce's notation, the denotation of a general term
 +
can be expressed by means of an equation between terms:
 +
 +
b  =  O
 +
 +
m  =  C +, I +, J +, O
 +
 +
w  =  B +, D +, E
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 8.2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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:
 +
 +
1  =  B +, C +, D +, E +, I +, J +, O
 +
 +
b  =  O
 +
 +
m  =  C +, I +, J +, O
 +
 +
w  =  B +, D +, E
 +
 +
This much provides a basis for collection of absolute terms that
 +
I plan to use in this example.  Let us now consider how we might
 +
represent a sufficiently exemplary collection of relative terms.
 +
 +
If we consider the genesis of relative terms, for example, "lover of ---",
 +
"betrayer to --- of ---", or "winner over of --- to --- from ---", we may
 +
regard these fill-in-the-blank forms as being derived by way of a kind of
 +
"rhematic abstraction" from the corresponding instances of absolute terms.
 +
 +
In other words:
 +
 +
1.  The relative term "lover of ---" can be constructed by abstracting
 +
    the absolute term "Emilia" from the absolute term "lover of Emilia".
 +
    Since Iago is a lover of Emilia, the relate-correlate pair denoted
 +
    by "Iago:Emilia" is a summand of the relative term "lover of ---".
 +
 +
2.  The relative term "betrayer to --- of ---" can be constructed
 +
    by abstracting the absolute terms "Othello" and "Desdemona"
 +
    from the absolute term "betrayer to Othello of Desdemona".
 +
    In as much as Iago is a betrayer to Othello of Desdemona,
 +
    the relate-correlate-correlate triple denoted by "I:O:D"
 +
    belongs to the relative term "betrayer to --- of ---".
 +
 +
3.  The relative term "winner over of --- to --- from ---" can be constructed
 +
    by abstracting the absolute terms "Othello", "Iago", and "Cassio" from the
 +
    absolute term "winner over of Othello to Iago from Cassio".  Since Iago is
 +
    a winner over of Othello to Iago from Cassio, the elementary relative term
 +
    "I:O:I:C" belongs to the relative term "winner over of --- to --- from ---".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 8.3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Speaking very strictly, we need to be careful to
 +
distinguish a "relation" from a "relative term".
 +
 +
1.  The relation is an 'object' of thought
 +
    that may be regarded "in extension" as
 +
    a set of ordered tuples that are known
 +
    as its "elementary relations".
 +
 +
2.  The relative term is a 'sign' that denotes certain objects,
 +
    called its "relates", as these are determined in relation
 +
    to certain other objects, called its "correlates".  Under
 +
    most circumstances, one may also regard the relative term
 +
    as denoting the corresponding relation.
 +
 +
Returning to the Othello example, let us take up the
 +
2-adic relatives "lover of ---" and "servant of ---".
 +
 +
Ignoring the many splendored nuances appurtenant to the idea of love,
 +
we may regard the relative term 'l' for "lover of ---" to be given by
 +
the following equation:
 +
 +
'l'  =  B:C +, C:B +, D:O +, E:I +, I:E +, O:D.
 +
 +
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 's' for "servant of ---".  The terms of this service are:
 +
 +
's'  =  C:O +, E:D +, I:O +, J:D +, J:O.
 +
 +
The term I:C may also be implied, but, since it is
 +
so hotly arguable, I will leave it out of the toll.
 +
 +
One more thing that we need to be duly wary about:
 +
There are many different conventions in the field
 +
as to the ordering of terms in their applications,
 +
and it happens that different conventions will be
 +
more convenient under different circumstances, so
 +
there does not appear to be much of a chance that
 +
any one of them can be canonized once and for all.
 +
 +
In the current reading, we are applying relative terms
 +
from right to left, and so our conception of relative
 +
multiplication, or relational composition, will need
 +
to be adjusted accordingly.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 8.4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
To familiarize ourselves with the forms of calculation
 +
that are available in Peirce's notation, let us compute
 +
a few of the simplest products that we find at hand in
 +
the Othello case.
 +
 +
Here are the absolute terms:
 +
 +
1  =  B +, C +, D +, E +, I +, J +, O
 +
 +
b  =  O
 +
 +
m  =  C +, I +, J +, O
 +
 +
w  =  B +, D +, E
 +
 +
Here are the 2-adic relative terms:
 +
 +
'l'  =  B:C +, C:B +, D:O +, E:I +, I:E +, O:D
 +
 +
's'  =  C:O +, E:D +, I:O +, J:D +, J:O
 +
 +
Here are a few of the simplest products among these terms:
 +
 +
'l'1 = "lover of anybody"
 +
 +
    = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(B +, C +, D +, E +, I +, J +, O)
 +
 +
    = B +, C +, D +, E +, I +, O
 +
 +
    = "anybody except J"
 +
 +
'l'b = "lover of a black"
 +
 +
    = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)O
 +
 +
    = D
 +
 +
'l'm = "lover of a man"
 +
 +
    = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(C +, I +, J +, O)
 +
 +
    = B +, D +, E
 +
 +
'l'w = "lover of a woman"
 +
 +
    = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(B +, D +, E)
 +
 +
    = C +, I +, O
 +
 +
's'1 = "servant of anybody"
 +
 +
    = (C:O +, E:D +, I:O +, J:D +, J:O)(B +, C +, D +, E +, I +, J +, O)
 +
 +
    = C +, E +, I +, J
 +
 +
's'b = "servant of a black"
 +
 +
    = (C:O +, E:D +, I:O +, J:D +, J:O)O
 +
 +
    = C +, I +, J
 +
 +
's'm = "servant of a man"
 +
 +
    = (C:O +, E:D +, I:O +, J:D +, J:O)(C +, I +, J +, O)
 +
 +
    = C +, I +, J
 +
 +
's'w = "servant of a woman"
 +
 +
    = (C:O +, E:D +, I:O +, J:D +, J:O)(B +, D +, E)
 +
 +
    = E +, J
 +
 +
'ls' = "lover of a servant of ---"
 +
 +
    = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)(C:O +, E:D +, I:O +, J:D +, J:O)
 +
 +
    = B:O +, E:O +, I:D
 +
 +
'sl' = "servant of a lover of ---"
 +
 +
    = (C:O +, E:D +, I:O +, J:D +, J:O)(B:C +, C:B +, D:O +, E:I +, I:E +, O:D)
 +
 +
    = C:D +, E:O +, I:D +, J:D +, J:O
 +
 +
Among other things, one observes that the
 +
relative terms 'l' and 's' do not commute,
 +
that is to say, 'ls' is not equal to 'sl'.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 8.5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Since multiplication by a 2-adic relative term
 +
is a logical analogue of matrix multiplication
 +
in linear algebra, all of the products that we
 +
computed above can be represented in terms of
 +
logical matrices and logical vectors.
 +
 +
Here are the absolute terms again, followed by
 +
their representation as "coefficient tuples",
 +
otherwise thought of as "coordinate vectors".
 +
 +
1  =  B +, C +, D +, E +, I +, J +, O
 +
 +
  =  <1, 1, 1, 1, 1, 1, 1>
 +
 +
b  =  O
 +
 +
  =  <0, 0, 0, 0, 0, 0, 1>
 +
 +
m  =  C +, I +, J +, O
 +
 +
  =  <0, 1, 0, 0, 1, 1, 1>
 +
 +
w  =  B +, D +, E
 +
 +
  =  <1, 0, 1, 1, 0, 0, 0>
 +
 +
Since we are going to be regarding these tuples as "column vectors",
 +
it is convenient to arrange them into a table of the following form:
 +
 +
  | 1 b m w
 +
---o---------
 +
B | 1 0 0 1
 +
C | 1 0 1 0
 +
D | 1 0 0 1
 +
E | 1 0 0 1
 +
I | 1 0 1 0
 +
J | 1 0 1 0
 +
O | 1 1 1 0
 +
 +
Here are the 2-adic relative terms again, followed by
 +
their representation as coefficient matrices, in this
 +
case bordered by row and column labels to remind us
 +
what the coefficient values are meant to signify.
 +
 +
'l' = B:C +, C:B +, D:O +, E:I +, I:E +, O:D =
 +
 +
'l'| B C D E I J O
 +
---o---------------
 +
B | 0 1 0 0 0 0 0
 +
C | 1 0 0 0 0 0 0
 +
D | 0 0 0 0 0 0 1
 +
E | 0 0 0 0 1 0 0
 +
I | 0 0 0 1 0 0 0
 +
J | 0 0 0 0 0 0 0
 +
O | 0 0 1 0 0 0 0
 +
 +
's' = C:O +, E:D +, I:O +, J:D +, J:O =
 +
 +
's'| B C D E I J O
 +
---o---------------
 +
B | 0 0 0 0 0 0 0
 +
C | 0 0 0 0 0 0 1
 +
D | 0 0 0 0 0 0 0
 +
E | 0 0 1 0 0 0 0
 +
I | 0 0 0 0 0 0 1
 +
J | 0 0 1 0 0 0 1
 +
O | 0 0 0 0 0 0 0
 +
 +
Here are the matrix representations of
 +
the products that we calculated before:
 +
 +
'l'1 = "lover of anybody" =
 +
 +
| 0 1 0 0 0 0 0 | | 1 |  | 1 |
 +
| 1 0 0 0 0 0 0 | | 1 |  | 1 |
 +
| 0 0 0 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 0 0 1 0 0 | | 1 | = | 1 |
 +
| 0 0 0 1 0 0 0 | | 1 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 1 |  | 1 |
 +
 +
'l'b = "lover of a black" =
 +
 +
| 0 1 0 0 0 0 0 | | 0 |  | 0 |
 +
| 1 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 0 0 0 0 1 | | 0 |  | 1 |
 +
| 0 0 0 0 1 0 0 | | 0 | = | 0 |
 +
| 0 0 0 1 0 0 0 | | 0 |  | 0 |
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 1 |  | 0 |
 +
 +
'l'm = "lover of a man" =
 +
 +
| 0 1 0 0 0 0 0 | | 0 |  | 1 |
 +
| 1 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 0 0 0 0 1 | | 0 |  | 1 |
 +
| 0 0 0 0 1 0 0 | | 0 | = | 1 |
 +
| 0 0 0 1 0 0 0 | | 1 |  | 0 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 1 |  | 0 |
 +
 +
'l'w = "lover of a woman" =
 +
 +
| 0 1 0 0 0 0 0 | | 1 |  | 0 |
 +
| 1 0 0 0 0 0 0 | | 0 |  | 1 |
 +
| 0 0 0 0 0 0 1 | | 1 |  | 0 |
 +
| 0 0 0 0 1 0 0 | | 1 | = | 0 |
 +
| 0 0 0 1 0 0 0 | | 0 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 0 |  | 1 |
 +
 +
's'1 = "servant of anybody" =
 +
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 0 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 1 | = | 1 |
 +
| 0 0 0 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 1 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
 +
's'b = "servant of a black" =
 +
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 0 0 0 0 1 | | 0 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 0 | = | 0 |
 +
| 0 0 0 0 0 0 1 | | 0 |  | 1 |
 +
| 0 0 1 0 0 0 1 | | 0 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
 +
's'm = "servant of a man" =
 +
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 0 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 0 | = | 0 |
 +
| 0 0 0 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 1 0 0 0 1 | | 1 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
 +
's'w = "servant of a woman" =
 +
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 0 0 0 0 1 | | 0 |  | 0 |
 +
| 0 0 0 0 0 0 0 | | 1 |  | 0 |
 +
| 0 0 1 0 0 0 0 | | 1 | = | 1 |
 +
| 0 0 0 0 0 0 1 | | 0 |  | 0 |
 +
| 0 0 1 0 0 0 1 | | 0 |  | 1 |
 +
| 0 0 0 0 0 0 0 | | 0 |  | 0 |
 +
 +
'ls' = "lover of a servant of ---" =
 +
 +
| 0 1 0 0 0 0 0 | | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 1 |
 +
| 1 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |
 +
| 0 0 0 0 0 0 1 | | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
 +
| 0 0 0 0 1 0 0 | | 0 0 1 0 0 0 0 | = | 0 0 0 0 0 0 1 |
 +
| 0 0 0 1 0 0 0 | | 0 0 0 0 0 0 1 |  | 0 0 1 0 0 0 0 |
 +
| 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 1 |  | 0 0 0 0 0 0 0 |
 +
| 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
 +
 +
'sl' = "servant of a lover of ---" =
 +
 +
| 0 0 0 0 0 0 0 | | 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |
 +
| 0 0 0 0 0 0 1 | | 1 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |
 +
| 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |
 +
| 0 0 1 0 0 0 0 | | 0 0 0 0 1 0 0 | = | 0 0 0 0 0 0 1 |
 +
| 0 0 0 0 0 0 1 | | 0 0 0 1 0 0 0 |  | 0 0 1 0 0 0 0 |
 +
| 0 0 1 0 0 0 1 | | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 1 |
 +
| 0 0 0 0 0 0 0 | | 0 0 1 0 0 0 0 |  | 0 0 0 0 0 0 0 |
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 8.6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The foregoing has hopefully filled in enough background that we
 +
can begin to make sense of the more mysterious parts of CP 3.73.
 +
 +
| 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.
 +
|
 +
| 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.
 +
|
 +
| Then:
 +
|
 +
| "man that is black"
 +
|
 +
| will be written
 +
|
 +
| m,b.
 +
|
 +
| C.S. Peirce, CP 3.73
 +
 +
In any system where elements are organized according to types,
 +
there tend to be any number of ways in which elements of one
 +
type are naturally associated with elements of another type.
 +
If the association is anything like a logical equivalence,
 +
but with the first type being "lower" and the second type
 +
being "higher" in some sense, then one frequently speaks
 +
of a "semantic ascent" from the lower to the higher type.
 +
 +
For instance, it is very common in mathematics to associate an element m
 +
of a set M with the constant function f_m : X -> M such that f_m (x) = m
 +
for all x in X, where X is an arbitrary set.  Indeed, the correspondence
 +
is so close that one often uses the same name "m" for the element m in M
 +
and the function m = f_m : X -> M, relying on the context or an explicit
 +
type indication to tell them apart.
 +
 +
For another instance, we have the "tacit extension" of a k-place relation
 +
L c X_1 x ... x X_k to a (k+1)-place relation L' c X_1 x ... x X_k+1 that
 +
we get by letting L' = L x X_k+1, that is, by maintaining the constraints
 +
of L on the first k variables and letting the last variable wander freely.
 +
 +
What we have here, if I understand Peirce correctly, is another such
 +
type of natural extension, sometimes called the "diagonal extension".
 +
This associates a k-adic relative or a k-adic relation, counting the
 +
absolute term and the set whose elements it denotes as the cases for
 +
k = 0, with a series of relatives and relations of higher adicities.
 +
 +
A few examples will suffice to anchor these ideas.
 +
 +
Absolute terms:
 +
 +
m  =  "man"                =  C +, I +, J +, O
 +
 +
n  =  "noble"              =  C +, D +, O
 +
 +
w  =  "woman"              =  B +, D +, E
 +
 +
Diagonal extensions:
 +
 +
m,  =  "man that is ---"    =  C:C +, I:I +, J:J +, O:O
 +
 +
n,  =  "noble that is ---"  =  C:C +, D:D +, O:O
 +
 +
w,  =  "woman that is ---"  =  B:B +, D:D +, E:E
 +
 +
Sample products:
 +
 +
m,n  =  "man that is noble" 
 +
 +
    =  (C:C +, I:I +, J:J +, O:O)(C +, D +, O)
 +
 +
    =  C +, O
 +
 +
n,m  =  "noble that is man"
 +
 +
    =  (C:C +, D:D +, O:O)(C +, I +, J +, O)
 +
 +
    =  C +, O
 +
 +
n,w  =  "noble that is woman"
 +
 +
    =  (C:C +, D:D +, O:O)(B +, D +, E)
 +
 +
    =  D
 +
 +
w,n  =  "woman that is noble"
 +
 +
    =  (B:B +, D:D +, E:E)(C +, D +, O)
 +
 +
    =  D
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication (cont.)
 +
|
 +
| It is obvious that multiplication into
 +
| a multiplicand indicated by a comma is
 +
| commutative <1>, that is,
 +
|
 +
| 's','l'  =  'l','s'.
 +
|
 +
| This multiplication is effectively the same as
 +
| that of Boole in his logical calculus.  Boole's
 +
| unity is my 1, that is, it denotes whatever is.
 +
|
 +
| <1>.  It will often be convenient to speak of the whole operation of
 +
| affixing a comma and then multiplying as a commutative multiplication,
 +
| the sign for which is the comma.  But though this is allowable, we shall
 +
| fall into confusion at once if we ever forget that in point of fact it is
 +
| not a different multiplication, only it is multiplication by a relative
 +
| whose meaning -- or rather whose syntax -- has been slightly altered;
 +
| and that the comma is really the sign of this modification of the
 +
| foregoing term.
 +
|
 +
| C.S. Peirce, CP 3.74
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let us backtrack a few years, and consider how Boole explained his
 +
twin conceptions of "selective operations" and "selective symbols".
 +
 +
| 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".
 +
|
 +
| Now it is perfectly apparent that if the operations above described
 +
| had been performed in a converse order, the result would have been the
 +
| same.  Whether we begin by forming the conception of "'men'", and then by
 +
| a second intellectual act limit that conception to "white men", or whether
 +
| we begin by forming the conception of "white objects", and then limit it to
 +
| such of that class as are "men", is perfectly indifferent so far as the result
 +
| is concerned.  It is obvious that the order of the mental processes would be
 +
| equally indifferent if for the words "white" and "men" we substituted any
 +
| other descriptive or appellative terms whatever, provided only that their
 +
| meaning was fixed and absolute.  And thus the indifference of the order
 +
| of two successive acts of the faculty of Conception, the one of which
 +
| furnishes the subject upon which the other is supposed to operate,
 +
| is a general condition of the exercise of that faculty.  It is
 +
| a law of the mind, and it is the real origin of that law of
 +
| the literal symbols of Logic which constitutes its formal
 +
| expression (1) Chap. II, [namely, xy = yx].
 +
|
 +
| It is equally clear that the mental operation above described is of such
 +
| a nature that its effect is not altered by repetition.  Suppose that by
 +
| a definite act of conception the attention has been fixed upon men, and
 +
| that by another exercise of the same faculty we limit it to those of the
 +
| race who are white.  Then any further repetition of the latter mental act,
 +
| by which the attention is limited to white objects, does not in any way
 +
| modify the conception arrived at, viz., that of white men.  This is also
 +
| an example of a general law of the mind, and it has its formal expression
 +
| in the law ((2) Chap. II) of the literal symbols [namely, x^2 = x].
 +
|
 +
| Boole, 'Laws of Thought', pp. 44-45.
 +
|
 +
| George Boole,
 +
|'An Investigation of the Laws of Thought,
 +
| On Which are Founded the Mathematical
 +
| Theories of Logic and Probabilities',
 +
| Reprinted, Dover, New York, NY, 1958.
 +
| Originally published, Macmillan, 1854.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In setting up his discussion of selective operations and
 +
their corresponding selective symbols, Boole writes this:
 +
 +
| The operation which we really perform is one of 'selection according to
 +
| a prescribed principle or idea'.  To what faculties of the mind such an
 +
| operation would be referred, according to the received classification of
 +
| its powers, it is not important to inquire, but I suppose that it would be
 +
| considered as dependent upon the two faculties of Conception or Imagination,
 +
| and Attention.  To the one of these faculties might be referred the formation
 +
| of the general conception;  to the other the fixing of the mental regard upon
 +
| those individuals within the prescribed universe of discourse which answer to
 +
| the conception.  If, however, as seems not improbable, the power of Attention
 +
| is nothing more than the power of continuing the exercise of any other faculty
 +
| of the mind, we might properly regard the whole of the mental process above
 +
| described as referrible to the mental faculty of Imagination or Conception,
 +
| the first step of the process being the conception of the Universe itself,
 +
| and each succeeding step limiting in a definite manner the conception
 +
| thus formed.  Adopting this view, I shall describe each such step,
 +
| or any definite combination of such steps, as a 'definite act
 +
| of conception'.
 +
|
 +
| Boole, 'Laws of Thought', p. 43.
 +
|
 +
| George Boole,
 +
|'An Investigation of the Laws of Thought,
 +
| On Which are Founded the Mathematical
 +
| Theories of Logic and Probabilities',
 +
| Reprinted, Dover, New York, NY, 1958.
 +
| Originally published, Macmillan, 1854.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In algebra, an "idempotent element" x is one that obeys the
 +
"idempotent law", that is, it satisfies the equation xx = x.
 +
Under most circumstances, it is usual to write this x^2 = x.
 +
 +
If the algebraic system in question falls under the additional laws
 +
that are necessary to carry out the requisite transformations, then
 +
x^2 = x is convertible into x - x^2 = 0, and this into x(1 - x) = 0.
 +
 +
If the algebraic system in question happens to be a boolean algebra,
 +
then the equation x(1 - x) = 0 says that x & ~x is identically false,
 +
in effect, a statement of the classical principle of non-contradiction.
 +
 +
We have already seen how Boole found rationales for the commutative law and
 +
the idempotent law by contemplating the properties of "selective operations".
 +
 +
It is time to bring these threads together, which we can do by considering the
 +
so-called "idempotent representation" of sets.  This will give us one of the
 +
best ways to understand the significance that Boole attached to selective
 +
operations.  It will also link up with the statements that Peirce makes
 +
about his adicity-augmenting comma operation.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Boole rationalized the properties of what we now dub "boolean multiplication",
 +
roughly equivalent to logical conjunction, in terms of the laws that apply to
 +
selective operations.  Peirce, in his turn, taking a very significant step of
 +
analysis that has seldom been recognized for what it would lead to, much less
 +
followed, does not consider this multiplication to be a fundamental operation,
 +
but derives it as a by-product of relative multiplication by a comma relative.
 +
Thus, Peirce makes logical conjunction a special case of relative composition.
 +
 +
This opens up a very wide field of investigation,
 +
"the operational significance of logical terms",
 +
one might say, but it will be best to advance
 +
bit by bit, and to lean on simple examples.
 +
 +
Back to Venice, and the close-knit party
 +
of absolutes and relatives that we were
 +
entertaining when last we were there.
 +
 +
Here is the list of absolute terms that we were considering before,
 +
to which I have thrown in 1, the universe of "anybody or anything",
 +
just for good measure:
 +
 +
1  =  "anybody"              =  B +, C +, D +, E +, I +, J +, O
 +
 +
m  =  "man"                  =  C +, I +, J +, O
 +
 +
n  =  "noble"                =  C +, D +, O
 +
 +
w  =  "woman"                =  B +, D +, E
 +
 +
Here is the list of "comma inflexions" or "diagonal extensions" of these terms:
 +
 +
1,  =  "anybody that is ---"  =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
 +
 +
m,  =  "man that is ---"      =  C:C +, I:I +, J:J +, O:O
 +
 +
n,  =  "noble that is ---"    =  C:C +, D:D +, O:O
 +
 +
w,  =  "woman that is ---"    =  B:B +, D:D +, E:E
 +
 +
One observes that the diagonal extension of 1
 +
is the same thing as the identity relation !1!.
 +
 +
Inspired by this identification of "1," with "!1!", and because
 +
the affixed commas of the diagonal extensions tend to get lost
 +
in the ordinary commas of punctuation, I will experiment with
 +
using the alternative notations:
 +
 +
m,  =  !m!
 +
n,  =  !n!
 +
w,  =  !w!
 +
 +
Working within our smaller sample of absolute terms,
 +
we have already computed the sorts of products that
 +
apply the diagonal extension of an absolute term to
 +
another absolute term, for instance, these products:
 +
 +
m,n  =  !m!n  =  "man that is noble"    =  C +, O
 +
n,m  =  !n!m  =  "noble that is man"    =  C +, O
 +
n,w  =  !n!w  =  "noble that is woman"  =  D
 +
w,n  =  !w!n  =  "woman that is noble"  =  D
 +
 +
This exercise gave us a bit of practical insight into
 +
why the commutative law holds for logical conjunction.
 +
 +
Further insight into the laws that govern this realm of logic,
 +
and the underlying reasons why they apply, might be gained by
 +
systematically working through the whole variety of different
 +
products that are generated by the operational means in sight,
 +
namely, the products indicated by {1, m, n, w}<,>{1, m, n, w}.
 +
 +
But before we try to explore this territory more systematically,
 +
let us equip ourselves with the sorts of graphical and matrical
 +
representations that we discovered to provide us with such able
 +
assists to the intuition in so many of our previous adventures.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 x is given by the abstract condition that xx = x,
 +
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 all of this looks from the graphical and matrical perspectives.
 +
 +
Absolute terms:
 +
 +
1  =  "anybody"  =  B +, C +, D +, E +, I +, J +, O
 +
 +
m  =  "man"      =  C +, I +, J +, O
 +
 +
n  =  "noble"    =  C +, D +, O
 +
 +
w  =  "woman"    =  B +, D +, E
 +
 +
Previously, we represented absolute terms as column vectors.
 +
The above four terms are given by the columns of this table:
 +
 +
  | 1 m n w
 +
---o---------
 +
B | 1 0 0 1
 +
C | 1 1 1 0
 +
D | 1 0 1 1
 +
E | 1 0 0 1
 +
I | 1 1 0 0
 +
J | 1 1 0 0
 +
O | 1 1 1 0
 +
 +
One way to represent sets in the bigraph picture
 +
is simply to mark the nodes in some way, like so:
 +
 +
    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
 +
 +
Diagonal extensions of the absolute terms:
 +
 +
1,  =  "anybody that is ---"  =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
 +
 +
m,  =  "man that is ---"      =  C:C +, I:I +, J:J +, O:O
 +
 +
n,  =  "noble that is ---"    =  C:C +, D:D +, O:O
 +
 +
w,  =  "woman that is ---"    =  B:B +, D:D +, E:E
 +
 +
Naturally enough, the diagonal extensions are represented by diagonal matrices:
 +
 +
!1!| B C D E I J O
 +
---o---------------
 +
B | 1 0 0 0 0 0 0
 +
C | 0 1 0 0 0 0 0
 +
D | 0 0 1 0 0 0 0
 +
E | 0 0 0 1 0 0 0
 +
I | 0 0 0 0 1 0 0
 +
J | 0 0 0 0 0 1 0
 +
O | 0 0 0 0 0 0 1
 +
 +
!m!| B C D E I J O
 +
---o---------------
 +
B | 0 0 0 0 0 0 0
 +
C | 0 1 0 0 0 0 0
 +
D | 0 0 0 0 0 0 0
 +
E | 0 0 0 0 0 0 0
 +
I | 0 0 0 0 1 0 0
 +
J | 0 0 0 0 0 1 0
 +
O | 0 0 0 0 0 0 1
 +
 +
!n!| B C D E I J O
 +
---o---------------
 +
B | 0 0 0 0 0 0 0
 +
C | 0 1 0 0 0 0 0
 +
D | 0 0 1 0 0 0 0
 +
E | 0 0 0 0 0 0 0
 +
I | 0 0 0 0 0 0 0
 +
J | 0 0 0 0 0 0 0
 +
O | 0 0 0 0 0 0 1
 +
 +
!w!| B C D E I J O
 +
---o---------------
 +
B | 1 0 0 0 0 0 0
 +
C | 0 0 0 0 0 0 0
 +
D | 0 0 1 0 0 0 0
 +
E | 0 0 0 1 0 0 0
 +
I | 0 0 0 0 0 0 0
 +
J | 0 0 0 0 0 0 0
 +
O | 0 0 0 0 0 0 0
 +
 +
Cast into the bigraph picture of 2-adic relations,
 +
the diagonal extension of an absolute term takes on
 +
a very distinctive sort of "straight-laced" character:
 +
 +
    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
 +
 +
    B  C  D  E  I  J  O
 +
u  o  o  o  o  o  o  o
 +
        |          |  |  |
 +
m,      |          |  |  |
 +
        |          |  |  |
 +
u  o  o  o  o  o  o  o
 +
    B  C  D  E  I  J  O
 +
 +
    B  C  D  E  I  J  O
 +
u  o  o  o  o  o  o  o
 +
        |  |              |
 +
n,      |  |              |
 +
        |  |              |
 +
u  o  o  o  o  o  o  o
 +
    B  C  D  E  I  J  O
 +
 +
    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
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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.  Anything That Is Anything
 +
 +
1,1  =  1
 +
 +
"anything that is anything"  =  "anything"
 +
 +
B  C  D  E  I  J  O
 +
+  +  +  +  +  +  +  1
 +
|  |  |  |  |  |  |
 +
|  |  |  |  |  |  |  1,
 +
|  |  |  |  |  |  |
 +
o  o  o  o  o  o  o  =
 +
 +
+  +  +  +  +  +  +  1
 +
B  C  D  E  I  J  O
 +
 +
| 1 0 0 0 0 0 0 | | 1 |    | 1 |
 +
| 0 1 0 0 0 0 0 | | 1 |    | 1 |
 +
| 0 0 1 0 0 0 0 | | 1 |    | 1 |
 +
| 0 0 0 1 0 0 0 | | 1 |  =  | 1 |
 +
| 0 0 0 0 1 0 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 1 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 0 1 | | 1 |    | 1 |
 +
 +
Example 2.  Anything That Is Man
 +
 +
1,m  =  m
 +
 +
"anything that is man"  =  "man"
 +
 +
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
 +
 +
| 1 0 0 0 0 0 0 | | 0 |    | 0 |
 +
| 0 1 0 0 0 0 0 | | 1 |    | 1 |
 +
| 0 0 1 0 0 0 0 | | 0 |    | 0 |
 +
| 0 0 0 1 0 0 0 | | 0 |  =  | 0 |
 +
| 0 0 0 0 1 0 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 1 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 0 1 | | 1 |    | 1 |
 +
 +
Example 3.  Man That Is Anything
 +
 +
m,1  =  m
 +
 +
"man that is anything"  =  "man"
 +
 +
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
 +
 +
| 0 0 0 0 0 0 0 | | 1 |    | 0 |
 +
| 0 1 0 0 0 0 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |    | 0 |
 +
| 0 0 0 0 0 0 0 | | 1 |  =  | 0 |
 +
| 0 0 0 0 1 0 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 1 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 0 1 | | 1 |    | 1 |
 +
 +
Example 4.  Man That Is Noble
 +
 +
m,n  =  "man that is noble"
 +
 +
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
 +
 +
| 0 0 0 0 0 0 0 | | 0 |    | 0 |
 +
| 0 1 0 0 0 0 0 | | 1 |    | 1 |
 +
| 0 0 0 0 0 0 0 | | 1 |    | 0 |
 +
| 0 0 0 0 0 0 0 | | 0 |  =  | 0 |
 +
| 0 0 0 0 1 0 0 | | 0 |    | 0 |
 +
| 0 0 0 0 0 1 0 | | 0 |    | 0 |
 +
| 0 0 0 0 0 0 1 | | 1 |    | 1 |
 +
 +
Example 5.  Noble That Is Man
 +
 +
n,m  =  "noble that is man"
 +
 +
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
 +
 +
| 0 0 0 0 0 0 0 | | 0 |    | 0 |
 +
| 0 1 0 0 0 0 0 | | 1 |    | 1 |
 +
| 0 0 1 0 0 0 0 | | 0 |    | 0 |
 +
| 0 0 0 0 0 0 0 | | 0 |  =  | 0 |
 +
| 0 0 0 0 0 0 0 | | 1 |    | 0 |
 +
| 0 0 0 0 0 0 0 | | 1 |    | 0 |
 +
| 0 0 0 0 0 0 1 | | 1 |    | 1 |
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 9.7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 when we propose these connections.
 +
 +
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.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 10
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication (cont.)
 +
|
 +
| The sum 'x' + 'x' generally denotes no logical term.
 +
| But 'x',_oo + 'x',_oo may be considered as denoting
 +
| some two 'x's.
 +
|
 +
| It is natural to write:
 +
|
 +
| 'x' + 'x'  =  !2!.'x'
 +
|
 +
| and
 +
|
 +
| 'x',_oo + 'x',_oo  =  !2!.'x',_oo
 +
|
 +
| where the dot shows that this multiplication is invertible.
 +
|
 +
| We may also use the antique figures so that:
 +
|
 +
| !2!.'x',_oo  =  `2`'x'
 +
|
 +
| just as
 +
|
 +
| !1!_oo  =  `1`.
 +
|
 +
| Then `2` alone will denote some two things.
 +
|
 +
| But this multiplication is not in general commutative,
 +
| and only becomes so when it affects a relative which
 +
| imparts a relation such that a thing only bears it
 +
| to 'one' thing, and one thing 'alone' bears it to
 +
| a thing.
 +
|
 +
| For instance, the lovers of two women are not
 +
| the same as two lovers of women, that is:
 +
|
 +
| 'l'`2`.w
 +
|
 +
| and
 +
|
 +
| `2`.'l'w
 +
|
 +
| are unequal;
 +
|
 +
| but the husbands of two women are the
 +
| same as two husbands of women, that is:
 +
|
 +
| 'h'`2`.w  =  `2`.'h'w
 +
|
 +
| and in general:
 +
|
 +
| 'x',`2`.'y'  =  `2`.'x','y'.
 +
|
 +
| C.S. Peirce, CP 3.75
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 careully 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 we when relate relations,
 +
possibly complex, to other relations, possibly simple.
 +
 +
But that will take another cup of java ('c'j) ---
 +
or maybe two, `2`'c'j = (!2!.'c',_oo)j ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 jacent 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 #, $, %.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        'l'__#      #'s'__$  $w                |
 +
|            o      o    o  o                |
 +
|              \    /      \ /                  |
 +
|              \  /        o                  |
 +
|                \ /          $                  |
 +
|                o                              |
 +
|                #                              |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 1.  Lover of a Servant of a Woman
 +
 +
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
 +
 +
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 L c X x Y, M c Y x Z.
 +
Table 3 and Figure 4 present a couple of ways of picturing the constraints
 +
that are involved in constructing the relational composition L o M c X x Z.
 +
 +
Table 3.  Relational Composition
 +
o---------o---------o---------o---------o
 +
|        #  !1!  |  !1!  |  !1!  |
 +
o=========o=========o=========o=========o
 +
|    L    #    X    |    Y    |        |
 +
o---------o---------o---------o---------o
 +
|    M    #        |    Y    |    Z    |
 +
o---------o---------o---------o---------o
 +
|  L o M  #    X    |        |    Z    |
 +
o---------o---------o---------o---------o
 +
 +
The way to read Table 3 is to imagine that you are
 +
playing a game that involves placing tokens on the
 +
squares of a board that is marked in just this way.
 +
The rules are that you have to place a single token
 +
on each marked square in the middle of the board in
 +
such a way that all of the indicated constraints are
 +
satisfied.  That is to say, you have to place a token
 +
whose denomination is a value in the set X on each of
 +
the squares marked "X", and similarly for the squares
 +
marked "Y" and "Z", meanwhile leaving all of the blank
 +
squares empty.  Furthermore, the tokens placed in each
 +
row and column have to obey the relational constraints
 +
that are indicated at the heads of the corresponding
 +
row and column.  Thus, the two tokens from X have to
 +
denominate the very same value from X, and likewise
 +
for Y and Z, while the pairs of tokens on the rows
 +
marked "L" and "M" are required to denote elements
 +
that are in the relations L and M, respectively.
 +
The upshot is that when just this much is done,
 +
that is, when the L, M, and !1! relations are
 +
satisfied, then the row marked "L o M" will
 +
automatically bear the tokens of a pair of
 +
elements in the composite relation L o M.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                L    L o M    M                |
 +
|                @      @      @                |
 +
|              / \    / \    / \              |
 +
|              o  o  o  o  o  o              |
 +
|              X  Y  X  Z  Y  Z              |
 +
|              o  o  o  o  o  o              |
 +
|              \  \ /    \ /  /              |
 +
|                \  /      \  /                |
 +
|                \ / \__ __/ \ /                |
 +
|                  @    @    @                  |
 +
|                !1!  !1!  !1!                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 4.  Relational Composition
 +
 +
Figure 4 merely shows a different way of viewing the same situation.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        'l'__$      $'s'__%  %w                |
 +
|            o      o    o  o                |
 +
|              \    /      \ /                  |
 +
|              \  /        o                  |
 +
|                \ /          %                  |
 +
|                o                              |
 +
|                $                              |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 1.  Lover of a Servant of a Woman
 +
 +
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
 +
 +
Table 3.  Relational Composition
 +
o---------o---------o---------o---------o
 +
|        #  !1!  |  !1!  |  !1!  |
 +
o=========o=========o=========o=========o
 +
|    L    #    X    |    Y    |        |
 +
o---------o---------o---------o---------o
 +
|    S    #        |    Y    |    Z    |
 +
o---------o---------o---------o---------o
 +
|  L o S  #    X    |        |    Z    |
 +
o---------o---------o---------o---------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                L    L o S    S                |
 +
|                @      @      @                |
 +
|              / \    / \    / \              |
 +
|              o  o  o  o  o  o              |
 +
|              X  Y  X  Z  Y  Z              |
 +
|              o  o  o  o  o  o              |
 +
|              \  \ /    \ /  /              |
 +
|                \  /      \  /                |
 +
|                \ / \__ __/ \ /                |
 +
|                  @    @    @                  |
 +
|                !1!  !1!  !1!                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 4.  Relational Composition
 +
 +
Figures 1 and 2 exhibit examples of relative multiplication
 +
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 the 1870 Logic of Relatives, implicit lines of identity are invoked by
 +
the subjacent numbers and marks of reference only when a correlate of some
 +
relation is the relate of some relation.  Thus, the principal relate, which
 +
is not a correlate of any explicit relation, is not singled out in this way.
 +
 +
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 5 demonstrates this possibility, posing a transitional case between
 +
the style of syntax in Figure 1 and the picture of composition in Figure 4.
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                          L o S                          |
 +
|                ____________@____________                |
 +
|                /                        \                |
 +
|              /      L            S      \              |
 +
|              /      @            @      \              |
 +
|            /      / \          / \      \            |
 +
|            /      /  \        /  \      \            |
 +
|          o      o    o      o    o      o          |
 +
|          X      X    Y      Y    Z      Z          |
 +
|      1,__#      #'l'__$      $'s'__%      %1          |
 +
|          o      o    o      o    o      o          |
 +
|            \    /      \    /      \    /            |
 +
|            \  /        \  /        \  /            |
 +
|              \ /          \ /          \ /              |
 +
|              @            @            @              |
 +
|              !1!          !1!          !1!              |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 5.  Anything that is a Lover of a Servant of Anything
 +
 +
In this composite sketch, the diagonal extension of the universe 1
 +
is invoked up front to anchor an explicit line of identity for the
 +
leading relate of the composition, while the terminal argument "w"
 +
has been generalized to the whole universe 1, in effect, executing
 +
an act of abstraction.  This type of universal bracketing isolates
 +
the composing of the relations L and S to form the composite L o S.
 +
The three relational domains X, Y, Z may be distinguished from one
 +
another, or else rolled up into a single universe of discourse, as
 +
one prefers.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 "solitaire" or "spreadsheet" model
 +
of relational composition, while Figure 4 may be thought of as making
 +
a start toward the "hyper(di)graph" model of generalized compositions.
 +
I will explain the hypergraph model in some detail at a later point.
 +
The transitional form of analysis represented by Figure 5 may be
 +
called the "universal bracketing" of relatives as relations.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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.
 +
 +
| But not only may any absolute term be thus regarded as
 +
| a relative term, but any relative term may in the same
 +
| way be regarded as a relative with one correlate more.
 +
| It is convenient to take this additional correlate as
 +
| the first one.
 +
|
 +
| Then:
 +
|
 +
| 'l','s'w
 +
|
 +
| will denote a lover of a woman
 +
| that is a servant of that woman.
 +
|
 +
| The comma here after 'l' should not be considered
 +
| as altering at all the meaning of 'l', but as only
 +
| a subjacent sign, serving to alter the arrangement
 +
| of the correlates.
 +
|
 +
| C.S. Peirce, CP 3.73
 +
 +
Just to plant our feet on a more solid stage,
 +
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:
 +
 +
w  =  "woman"          =  B +, D +, E
 +
 +
'l'  =  "lover of ---"    =  B:C +, C:B +, D:O +, E:I +, I:E +, J:D +, O:D
 +
 +
's'  =  "servant of ---"  =  C:O +, E:D +, I:O +, J:D +, J:O
 +
 +
And next we derive the following results:
 +
 +
'l',  =  "lover that is --- of ---"
 +
 +
      =  B:B:C +, C:C:B +, D:D:O +, E:E:I +, I:I:E +, J:J:D +, O:O:D
 +
 +
'l','s'w  =  (B:B:C +, C:C:B +, D:D:O +, E:E:I +, I:I:E +, J:J:D +, O:O:D)
 +
 +
          x  (C:O +, E:D +, I:O +, J:D +, J:O)
 +
 +
          x  (B +, D +, E)
 +
 +
Now what are we to make of that?
 +
 +
If we operate in accordance with Peirce's example of `g`'o'h
 +
as the "giver of a horse to an owner of that horse", then we
 +
may assume that the associative law and the distributive law
 +
are by default in force, allowing us to derive this equation:
 +
 +
'l','s'w  =  'l','s'(B +, D +, E)
 +
 +
          =  'l','s'B +, 'l','s'D +, 'l','s'E
 +
 +
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 (R:S:T)(S:T)(T)
 +
is equal to R but that no other form of product yields
 +
a non-null result.  Scanning the implied terms of the
 +
triple product tells us that only the following case
 +
is non-null:  J = (J:J:D)(J:D)(D).  It follows that:
 +
 +
'l','s'w  =  "lover and servant of a woman"
 +
 +
          =  "lover that is a servant of a woman"
 +
 +
          =  "lover of a woman that is a servant of that woman"
 +
 +
          =  J
 +
 +
And so what Peirce says makes sense in this case.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 instance, the three
 +
that I have redrawn below.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        `g`__$__%    $'l'__*  *w  %h          |
 +
|              o  o    o    o  o    o          |
 +
|              \  \  /      \ /    /            |
 +
|                \  \/        @    /            |
 +
|                \ /\______ ______/              |
 +
|                  @        @                    |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 6.  Giver of a Horse to a Lover of a Woman
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        `g`__$__%    $'o'__*  *%h              |
 +
|              o  o    o    o  oo              |
 +
|              \  \  /      \ //                |
 +
|                \  \/        @/                |
 +
|                \ /\____ ____/                  |
 +
|                  @      @                      |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 7.  Giver of a Horse to an Owner of It
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        'l',__$__%    $'s'__*  *%w              |
 +
|              o  o    o    o  oo              |
 +
|              \  \  /      \ //                |
 +
|                \  \/        @/                |
 +
|                \ /\____ ____/                  |
 +
|                  @      @                      |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 8.  Lover that is a Servant of a Woman
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Here is what I get when I try to analyze Peirce's
 +
"giver of a horse to a lover of a woman" example
 +
along the same lines as the 2-adic compositions.
 +
 +
We may begin with the mark-up shown in Figure 6.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        `g`__$__%    $'l'__*  *w  %h          |
 +
|              o  o    o    o  o    o          |
 +
|              \  \  /      \ /    /            |
 +
|                \  \/        @    /            |
 +
|                \ /\______ ______/              |
 +
|                  @        @                    |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 6.  Giver of a Horse to a Lover of a Woman
 +
 +
If we analyze this in accord with the "spreadsheet" model
 +
of relational composition, the core of it is a particular
 +
way of composing a 3-adic "giving" relation G c T x U x V
 +
with a 2-adic "loving" relation L c U x W so as to obtain
 +
a specialized sort of 3-adic relation (G o L) c T x W x V.
 +
The applicable constraints on tuples are shown in Table 9.
 +
 +
Table 9.  Composite of Triadic and Dyadic Relations
 +
o---------o---------o---------o---------o---------o
 +
|        #  !1!  |  !1!  |  !1!  |  !1!  |
 +
o=========o=========o=========o=========o=========o
 +
|    G    #    T    |    U    |        |    V    |
 +
o---------o---------o---------o---------o---------o
 +
|    L    #        |    U    |    W    |        |
 +
o---------o---------o---------o---------o---------o
 +
|  G o L  #    T    |        |    W    |    V    |
 +
o---------o---------o---------o---------o---------o
 +
 +
The hypergraph picture of the abstract composition is given in Figure 10.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|                                G o L                                |
 +
|                      ___________@___________                      |
 +
|                      /                  \    \                      |
 +
|                    /  G              L  \    \                    |
 +
|                    /  @              @  \    \                    |
 +
|                  /  /|\            / \  \    \                  |
 +
|                  /  / | \          /  \  \    \                  |
 +
|                /  /  |  \        /    \  \    \                |
 +
|                /  /  |  \      /      \  \    \                |
 +
|              o  o    o    o    o        o  o    o              |
 +
|              T  T    U    V    U        W  W    V              |
 +
|            1,_#  #`g`_$____%    $'l'______*  *1  %1              |
 +
|              o  o    o    o    o        o  o    o              |
 +
|                \ /      \    \  /          \ /    /                |
 +
|                @        \    \/            @    /                |
 +
|                !1!        \  /\            !1!  /                  |
 +
|                            \ /  \_______ _______/                  |
 +
|                            @          @                          |
 +
|                            !1!        !1!                          |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 10.  Anything that is a Giver of Anything to a Lover of Anything
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.8
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In taking up the next example of relational composition,
 +
let's exchange the relation 't' = "trainer of ---" for
 +
Peirce's relation 'o' = "owner of ---", simply for the
 +
sake of avoiding conflicts in the symbols that we use.
 +
In this way, Figure 7 is transformed into Figure 11.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        `g`__$__%    $'t'__*  *%h              |
 +
|              o  o    o    o  oo              |
 +
|              \  \  /      \ //                |
 +
|                \  \/        @/                |
 +
|                \ /\____ ____/                  |
 +
|                  @      @                      |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 11.  Giver of a Horse to a Trainer of It
 +
 +
Now here's an interesting point, in fact, a critical transition point,
 +
that we see resting in potential but a stone's throw removed from the
 +
chronism, the secular neigborhood, the temporal vicinity of Peirce's
 +
1870 LOR, and it's a vertex that turns on the teridentity relation.
 +
 +
The hypergraph picture of the abstract composition is given in Figure 12.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|                                G o T                                |
 +
|                _________________@_________________                |
 +
|                /                                  \                |
 +
|              /        G              T            \              |
 +
|              /        @              @              \              |
 +
|            /        /|\            / \              \            |
 +
|            /        / | \          /  \              \            |
 +
|          /        /  |  \        /    \              \          |
 +
|          /        /  |  \      /      \              \          |
 +
|        o        o    o    o    o        o              o        |
 +
|        X        X    Y    Z    Y        Z              Z        |
 +
|      1,_#        #`g`_$____%    $'t'______%              %1        |
 +
|        o        o    o    o    o        o              o        |
 +
|          \      /      \    \  /          |            /          |
 +
|          \    /        \    \/          |            /          |
 +
|            \  /          \  /\          |          /            |
 +
|            \ /            \ /  \__________|__________/            |
 +
|              @              @              @                        |
 +
|            !1!            !1!            !1!                      |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 12.  Anything that is a Giver of Anything to a Trainer of It
 +
 +
If we analyze this in accord with the "spreadsheet" model
 +
of relational composition, the core of it is a particular
 +
way of composing a 3-adic "giving" relation G c X x Y x Z
 +
with a 2-adic "training" relation T c Y x Z in such a way
 +
as to determine a certain 2-adic relation (G o T) c X x Z.
 +
Table 13 schematizes the associated constraints on tuples.
 +
 +
Table 13.  Another Brand of Composition
 +
o---------o---------o---------o---------o
 +
|        #  !1!  |  !1!  |  !1!  |
 +
o=========o=========o=========o=========o
 +
|    G    #    X    |    Y    |    Z    |
 +
o---------o---------o---------o---------o
 +
|    T    #        |    Y    |    Z    |
 +
o---------o---------o---------o---------o
 +
|  G o T  #    X    |        |    Z    |
 +
o---------o---------o---------o---------o
 +
 +
So we see that the notorious teridentity relation,
 +
which I have left equivocally denoted by the same
 +
symbol as the identity relation !1!, is already
 +
implicit in Peirce's discussion at this point.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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.
 +
 +
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.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.10
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Figure 8 depicts the last of the three examples involving
 +
the composition of 3-adic relatives with 2-adic relatives:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        'l',__$__%    $'s'__*  *%w              |
 +
|              o  o    o    o  oo              |
 +
|              \  \  /      \ //                |
 +
|                \  \/        @/                |
 +
|                \ /\____ ____/                  |
 +
|                  @      @                      |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 8.  Lover that is a Servant of a Woman
 +
 +
The hypergraph picture of the abstract composition is given in Figure 14.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|                                L , S                                |
 +
|                __________________^__________________                |
 +
|              /                                    \              |
 +
|              /      L_,              S              \              |
 +
|            /        @              @              \            |
 +
|            /        /|\            / \              \            |
 +
|          /        / | \          /  \              \          |
 +
|          /        /  |  \        /    \              \          |
 +
|        /        /  |  \      /      \              \        |
 +
|        /        /    |    \    /        \              \        |
 +
|      o        o    o    o  o          o              o      |
 +
|      X        X    X    Y  X          Y              Y      |
 +
|    1,_#        #'l',_$_____%  $'t'________%              %1      |
 +
|      o        o    o    o  o          o              o      |
 +
|        \      /      \    \ /            |              /        |
 +
|        \    /        \    \            |            /        |
 +
|          \  /          \  / \            |            /          |
 +
|          \ /            \ /  \___________|___________/          |
 +
|            @              @                @                      |
 +
|          !1!            !1!              !1!                      |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 14.  Anything that's a Lover of Anything and that's a Servant of It
 +
 +
This example illustrates the way that Peirce analyzes the logical conjunction,
 +
we might even say the "parallel conjunction", of a couple of 2-adic relatives
 +
in terms of the comma extension and the same style of composition that we saw
 +
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 2-adic "loving" relation L c X x Y to the corresponding 3-adic
 +
"loving and being" relation L_, c X x X x Y, which is then composed
 +
in a specific way with a 2-adic "serving" relation S c X x Y, so as
 +
to determine the 2-adic relation L,S c X x Y.  Table 15 schematizes
 +
the associated constraints on tuples.
 +
 +
Table 15.  Conjunction Via Composition
 +
o---------o---------o---------o---------o
 +
|        #  !1!  |  !1!  |  !1!  |
 +
o=========o=========o=========o=========o
 +
|    L,  #    X    |    X    |    Y    |
 +
o---------o---------o---------o---------o
 +
|    S    #        |    X    |    Y    |
 +
o---------o---------o---------o---------o
 +
|  L , S  #    X    |        |    Y    |
 +
o---------o---------o---------o---------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 10.11
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I return to where we were in unpacking the contents of CP 3.73.
 +
Peirce remarks that the comma operator can be iterated at will:
 +
 +
| In point of fact, since a comma may be added in this way to any
 +
| relative term, it may be added to one of these very relatives
 +
| formed by a comma, and thus by the addition of two commas
 +
| an absolute term becomes a relative of two correlates.
 +
|
 +
| So:
 +
|
 +
| m,,b,r
 +
|
 +
| interpreted like
 +
|
 +
| `g`'o'h
 +
|
 +
| means a man that is a rich individual and
 +
| is a black that is that rich individual.
 +
|
 +
| But this has no other meaning than:
 +
|
 +
| m,b,r
 +
|
 +
| or a man that is a black that is rich.
 +
|
 +
| Thus we see that, after one comma is added, the
 +
| addition of another does not change the meaning
 +
| at all, so that whatever has one comma after it
 +
| must be regarded as having an infinite number.
 +
|
 +
| C.S. Peirce, CP 3.73
 +
 +
Again, let us check whether this makes sense
 +
on the stage of our small but dramatic model.
 +
 +
Let's say that Desdemona and Othello are rich,
 +
and, among the persons of the play, only they.
 +
 +
With this premiss we obtain a sample of absolute terms
 +
that is sufficiently ample to work through our example:
 +
 +
1    =  B +, C +, D +, E +, I +, J +, O
 +
 +
b    =  O
 +
 +
m    =  C +, I +, J +, O
 +
 +
r    =  D +, O
 +
 +
One application of the comma operator
 +
yields the following 2-adic relatives:
 +
 +
1,  =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
 +
 +
b,  =  O:O
 +
 +
m,  =  C:C +, I:I +, J:J +, O:O
 +
 +
r,  =  D:D +, O:O
 +
 +
Another application of the comma operator
 +
generates the following 3-adic relatives:
 +
 +
1,,  =  B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O
 +
 +
b,,  =  O:O:O
 +
 +
m,,  =  C:C:C +, I:I:I +, J:J:J +, O:O:O
 +
 +
r,,  =  D:D:D +, O:O:O
 +
 +
Assuming the associativity of multiplication among 2-adic relatives,
 +
we may compute the product m,b,r by a brute force method as follows:
 +
 +
m,b,r  =  (C:C +, I:I +, J:J +, O:O)(O:O)(D +, O)
 +
 +
      =  (C:C +, I:I +, J:J +, O:O)(O)
 +
 +
      =  O
 +
 +
This avers that a man that is black that is rich is Othello,
 +
which is true on the premisses of our universe of discourse.
 +
 +
The stock associations of `g`'o'h lead us to multiply out the
 +
product m,,b,r along the following lines, where the trinomials
 +
of the form (X:Y:Z)(Y:Z)(Z) are the only ones that produce any
 +
non-null result, specifically, of the form (X:Y:Z)(Y:Z)(Z) = X.
 +
 +
m,,b,r  =  (C:C:C +, I:I:I +, J:J:J +, O:O:O)(O:O)(D +, O)
 +
 +
        =  (O:O:O)(O:O)(O)
 +
 +
        =  O
 +
 +
So we have that m,,b,r = m,b,r.
 +
 +
In closing, observe that the teridentity relation has turned up again
 +
in this context, as the second comma-ing of the universal term itself:
 +
 +
1,,  =  B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 11
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Signs for Multiplication (concl.)
 +
|
 +
| The conception of multiplication we have adopted is that of
 +
| the application of one relation to another.  So, a quaternion
 +
| being the relation of one vector to another, the multiplication
 +
| of quaternions is the application of one such relation to a second.
 +
|
 +
| Even ordinary numerical multiplication involves the same idea, for
 +
| 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs, where
 +
| "triplet of" and "pair of" are evidently relatives.
 +
|
 +
| If we have an equation of the form:
 +
|
 +
| xy  =  z
 +
|
 +
| and there are just as many x's per y as there are,
 +
| 'per' things, things of the universe, then we have
 +
| also the arithmetical equation:
 +
|
 +
| [x][y]  =  [z].
 +
|
 +
| For instance, if our universe is perfect men, and there
 +
| are as many teeth to a Frenchman (perfect understood)
 +
| as there are to any one of the universe, then:
 +
|
 +
| ['t'][f]  =  ['t'f]
 +
|
 +
| holds arithmetically.
 +
|
 +
| So if men are just as apt to be black as things in general:
 +
|
 +
| [m,][b]  =  [m,b]
 +
|
 +
| where the difference between [m] and [m,] must not be overlooked.
 +
|
 +
| It is to be observed that:
 +
|
 +
| [!1!]  =  `1`.
 +
|
 +
| Boole was the first to show this connection between logic and
 +
| probabilities.  He was restricted, however, to absolute terms.
 +
| I do not remember having seen any extension of probability to
 +
| relatives, except the ordinary theory of 'expectation'.
 +
|
 +
| Our logical multiplication, then, satisfies the essential conditions
 +
| of multiplication, has a unity, has a conception similar to that of
 +
| admitted multiplications, and contains numerical multiplication as
 +
| a case under it.
 +
|
 +
| C.S. Peirce, CP 3.76
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have reached in our reading of Peirce's text a suitable place to pause --
 +
actually, it is more like to run as fast as we can along a parallel track --
 +
where I can due quietus make of a few IOU's that I've used to pave my way.
 +
 +
The more pressing debts that come to mind are concerned with the matter
 +
of Peirce's "number of" function, that maps a term t into a number [t],
 +
and with my justification for calling a certain style of illustration
 +
by the name of the "hypergraph" picture of relational composition.
 +
As it happens, there is a thematic relation between these topics,
 +
and so I can make my way forward by addressing them together.
 +
 +
At this point we have two good pictures of how to compute the
 +
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 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 explicate the "generative potency" of Peirce's 1870 LOR.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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.
 +
 +
| I propose to assign to all logical terms, numbers;
 +
| to an absolute term, the number of individuals it denotes;
 +
| to a relative term, the average number of things so related
 +
| to one individual.
 +
|
 +
| Thus in a universe of perfect men ('men'),
 +
| the number of "tooth of" would be 32.
 +
|
 +
| The number of a relative with two correlates would be the
 +
| average number of things so related to a pair of individuals;
 +
| and so on for relatives of higher numbers of correlates.
 +
|
 +
| I propose to denote the number of a logical term by
 +
| enclosing the term in square brackets, thus ['t'].
 +
|
 +
| C.S. Peirce, CP 3.65
 +
 +
NOF 2.
 +
 +
| But not only do the significations of '=' and '<' here adopted fulfill all
 +
| absolute requirements, but they have the supererogatory virtue of being very
 +
| nearly the same as the common significations.  Equality is, in fact, nothing
 +
| but the identity of two numbers;  numbers that are equal are those which are
 +
| predicable of the same collections, just as terms that are identical are those
 +
| which are predicable of the same classes.  So, to write 5 < 7 is to say that 5
 +
| is part of 7, just as to write f < m is to say that Frenchmen are part of men.
 +
| Indeed, if f < m, then the number of Frenchmen is less than the number of men,
 +
| and if v = p, then the number of Vice-Presidents is equal to the number of
 +
| Presidents of the Senate;  so that the numbers may always be substituted
 +
| for the terms themselves, in case no signs of operation occur in the
 +
| equations or inequalities.
 +
|
 +
| C.S. Peirce, CP 3.66
 +
 +
NOF 3.
 +
 +
| It is plain that both the regular non-invertible addition
 +
| and the invertible addition satisfy the absolute conditions.
 +
| But the notation has other recommendations.  The conception
 +
| of 'taking together' involved in these processes is strongly
 +
| analogous to that of summation, the sum of 2 and 5, for example,
 +
| being the number of a collection which consists of a collection of
 +
| two and a collection of five.  Any logical equation or inequality
 +
| in which no operation but addition is involved may be converted
 +
| into a numerical equation or inequality by substituting the
 +
| numbers of the several terms for the terms themselves --
 +
| provided all the terms summed are mutually exclusive.
 +
|
 +
| Addition being taken in this sense,
 +
| 'nothing' is to be denoted by 'zero',
 +
| for then:
 +
|
 +
| x +, 0  =  x
 +
|
 +
| whatever is denoted by x;  and this is the definition
 +
| of 'zero'.  This interpretation is given by Boole, and
 +
| is very neat, on account of the resemblance between the
 +
| ordinary conception of 'zero' and that of nothing, and
 +
| because we shall thus have
 +
|
 +
| [0]  =  0.
 +
|
 +
| C.S. Peirce, CP 3.67
 +
 +
NOF 4.
 +
 +
| The conception of multiplication we have adopted is
 +
| that of the application of one relation to another.  ...
 +
|
 +
| Even ordinary numerical multiplication involves the same idea,
 +
| for 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs,
 +
| where "triplet of" and "pair of" are evidently relatives.
 +
|
 +
| If we have an equation of the form:
 +
|
 +
| xy  =  z
 +
|
 +
| and there are just as many x's per y as there are,
 +
| 'per' things, things of the universe, then we have
 +
| also the arithmetical equation:
 +
|
 +
| [x][y]  =  [z].
 +
|
 +
| For instance, if our universe is perfect men, and there
 +
| are as many teeth to a Frenchman (perfect understood)
 +
| as there are to any one of the universe, then:
 +
|
 +
| ['t'][f]  =  ['t'f]
 +
|
 +
| holds arithmetically.
 +
|
 +
| So if men are just as apt to be black as things in general:
 +
|
 +
| [m,][b]  =  [m,b]
 +
|
 +
| where the difference between [m] and [m,] must not be overlooked.
 +
|
 +
| It is to be observed that:
 +
|
 +
| [!1!]  =  `1`.
 +
|
 +
| Boole was the first to show this connection between logic and
 +
| probabilities.  He was restricted, however, to absolute terms.
 +
| I do not remember having seen any extension of probability to
 +
| relatives, except the ordinary theory of 'expectation'.
 +
|
 +
| Our logical multiplication, then, satisfies the essential conditions
 +
| of multiplication, has a unity, has a conception similar to that of
 +
| admitted multiplications, and contains numerical multiplication as
 +
| a case under it.
 +
|
 +
| C.S. Peirce, CP 3.76
 +
 +
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 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 antecedent second,
 +
whereas almost anyone brought up in our present time frame has difficulty
 +
thinking of a function any other way than as a set of ordered pairs where
 +
the order in each pair lists the functional argument, or domain element,
 +
first and the functional value, or codomain element, second.
 +
 +
It is possible to work all this out in a very nice way within a very general context
 +
of flexible conventions, but not without introducing an order of anachronisms into
 +
Peirce's presentation that I am presently trying to avoid as much as possible.
 +
Thus, I will need to experiment with various sorts of compromise formations.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Having spent a fair amount of time in earnest reflection on the issue,
 +
I cannot see a way to continue my interpretation of Peirce's 1870 LOR,
 +
to master the distance between his conventions of presentation and my
 +
present personal perspectives on relations, without introducing a few
 +
interpretive anachronisms and other artifacts in the process, and the
 +
only excuse that I can make for myself is that at least these will be
 +
novel sorts of anachronisms and artifacts in comparison with the ones
 +
that the reeder may alreedy have seen.  A poor excuse, but all I have.
 +
The least that I can do, then, and I'm something of an expert on that,
 +
is to exposit my personal interpretive apparatus on a separate thread,
 +
where it will not distract too much from the intellectual canon, that
 +
is to opine, the "thinking panpipe" that we find in Peirce's 1870 LOR.
 +
 +
Ripped from the pages of my dissertation, then, I will lay out
 +
some samples of background material on "Relations In General",
 +
as spied from a combinatorial point of view, that I hope will
 +
serve in reeding Peirce's text, if we draw on it judiciously.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The task before us now is to get very clear about the relationships
 +
among relative terms, relations, and the special cases of relations
 +
that are constituted by equivalence relations, functions, and so on.
 +
 +
I am optimistic that the some of the tethering material that I spun
 +
along the "Relations In General" (RIG) thread will help us to track
 +
the equivalential and functional properties of special relations in
 +
a way that will not weigh too heavy on the rather capricious lineal
 +
embedding of syntax in 1-dimensional strings on 2-dimensional pages.
 +
But I cannot see far enough ahead to forsee all the consequences of
 +
trying this tack, and so I cannot help but to be a bit experimental.
 +
 +
The first obstacle to get past is the order convention
 +
that Peirce's orientation to relative terms causes him
 +
to use for functions.  By way of making our discussion
 +
concrete, and directing our attentions to an immediate
 +
object example, let us say that we desire to represent
 +
the "number of" function, that Peirce denotes by means
 +
of square brackets, by means of a 2-adic relative term,
 +
say 'v', where 'v'(t) = [t] = the number of the term t.
 +
 +
To set the 2-adic relative term 'v' within a suitable context of interpretation,
 +
let us suppose that 'v' corresponds to a relation V c R x S, where R is the set
 +
of real numbers and S is a suitable syntactic domain, here described as "terms".
 +
Then the 2-adic relation V is evidently a function from S to R.  We might think
 +
to use the plain letter "v" to denote this function, as v : S -> R, but I worry
 +
this may be a chaos waiting to happen.  Also, I think that we should anticipate
 +
the very great likelihood that we cannot always assign numbers to every term in
 +
whatever syntactic domain S that we choose, so it is probably better to account
 +
the 2-adic relation V as a partial function from S to R.  All things considered,
 +
then, let me try out the following impedimentaria of strategies and compromises.
 +
 +
First, I will adapt the functional arrow notation so that it allows us
 +
to detach the functional orientation from the order in which the names
 +
of domains are written on the page.  Second, I will need to change the
 +
notation for "pre-functions", or "partial functions", from one likely
 +
confound to a slightly less likely confound.  This gives the scheme:
 +
 +
  q : X -> Y means that q is functional at X.
 +
 +
  q : X <- Y means that q is functional at Y.
 +
 +
  q : X ~> Y means that q is pre-functional at X.
 +
 +
  q : X <~ Y means that q is pre-functional at Y.
 +
 +
For now, I will pretend that v is a function in R of S, v : R <- S,
 +
amounting to the functional alias of the 2-adic relation V c R x S,
 +
and associated with the 2-adic relative term 'v' whose relate lies
 +
in the set R of real numbers and whose correlate lies in the set S
 +
of syntactic terms.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
It always helps me to draw lots of pictures of stuff,
 +
so let's extract the somewhat overly compressed bits
 +
of the "Relations In General" thread that we'll need
 +
right away for the applications to Peirce's 1870 LOR,
 +
and draw what icons we can within the frame of Ascii.
 +
 +
For the immediate present, we may start with 2-adic relations
 +
and describe the customary species of relations and functions
 +
in terms of their local and numerical incidence properties.
 +
 +
Let P c X x Y be an arbitrary 2-adic relation.
 +
The following properties of P can be defined:
 +
 +
P is "total" at X    iff  P is (>=1)-regular at X.
 +
 +
P is "total" at Y    iff  P is (>=1)-regular at Y.
 +
 +
P is "tubular" at X  iff  P is (=<1)-regular at X.
 +
 +
P is "tubular" at Y  iff  P is (=<1)-regular at Y.
 +
 +
To illustrate these properties, let us fashion
 +
a "generic enough" example of a 2-adic relation,
 +
E c X x Y, where X = Y = {0, 1, ..., 8, 9}, and
 +
where the bigraph picture of E looks like this:
 +
 +
0  1  2  3  4  5  6  7  8  9
 +
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
 +
 +
If we scan along the X dimension we see that the "Y incidence degrees"
 +
of the X nodes 0 through 9 are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0, in order.
 +
 +
If we scan along the Y dimension we see that the "X incidence degrees"
 +
of the Y nodes 0 through 9 are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, in order.
 +
 +
Thus, E is not total at either X or Y,
 +
since there are nodes in both X and Y
 +
having incidence degrees that equal 0.
 +
 +
Also, E is not tubular at either X or Y,
 +
since there exist nodes in both X and Y
 +
having incidence degrees greater than 1.
 +
 +
Clearly, then, E cannot qualify as a pre-function
 +
or a function on either of its relational domains.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let's continue to work our way through the rest of the first
 +
set of definitions, making up appropriate examples as we go.
 +
 +
| Let P c X x Y be an arbitrary 2-adic relation.
 +
| The following properties of P can be defined:
 +
|
 +
| P is "total" at X    iff  P is (>=1)-regular at X.
 +
|
 +
| P is "total" at Y    iff  P is (>=1)-regular at Y.
 +
|
 +
| P is "tubular" at X  iff  P is (=<1)-regular at X.
 +
|
 +
| P is "tubular" at Y  iff  P is (=<1)-regular at Y.
 +
 +
E_1 exemplifies the quality of "totality at X".
 +
 +
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
 +
 +
E_2 exemplifies the quality of "totality at Y".
 +
 +
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
 +
 +
E_3 exemplifies the quality of "tubularity at X".
 +
 +
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
 +
 +
E_4 exemplifies the quality of "tubularity at Y".
 +
 +
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
 +
 +
| If P c X x Y is tubular at X, then P is known as a "partial function"
 +
| or a "pre-function" from X to Y, frequently signalized by renaming P
 +
| with an alternative lower case name, say "p", and writing p : X ~> Y.
 +
|
 +
| Just by way of formalizing the definition:
 +
|
 +
| P is a "pre-function" P : X ~> Y  iff  P is tubular at X.
 +
|
 +
| P is a "pre-function" P : X <~ Y  iff  P is tubular at Y.
 +
 +
So, E_3 is a pre-function e_3 : X ~> Y,
 +
and E_4 is a pre-function e_4 : X <~ Y.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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
 +
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:
 +
 +
L_x@j  =  {<x_1, ..., x_j, ..., x_k> in L : x_j = x}.
 +
 +
In the case of a 2-adic relation L c X_1 x X_2 = X x Y,
 +
we can reap the benefits of a radical simplification in
 +
the definitions of the local flags.  Also in this case,
 +
we tend to denote L_u@1 by "L_u@X" and L_v@2 by "L_v@Y".
 +
 +
In the light of these considerations, the local flags of
 +
a 2-adic relation L c X x Y may be formulated as follows:
 +
 +
L_u@X  =  {<x, y> in L : x = u}
 +
 +
      =  the set of all ordered pairs in L incident with u in X.
 +
 +
L_v@Y  =  {<x, y> in L : y = v}
 +
 +
      =  the set of all ordered pairs in L incident with v in Y.
 +
 +
A sufficient illustration is supplied by the earlier example E.
 +
 +
0  1  2  3  4  5  6  7  8  9
 +
o  o  o  o  o  o  o  o  o  o  X
 +
    \  |\ /|\  \  \  |  |\
 +
      \ | / | \  \  \ |  | \        E
 +
      \|/ \|  \  \  \|  |  \
 +
o  o  o  o  o  o  o  o  o  o  Y
 +
0  1  2  3  4  5  6  7  8  9
 +
 +
The local flag E_3@X is displayed here:
 +
 +
0  1  2  3  4  5  6  7  8  9
 +
o  o  o  o  o  o  o  o  o  o  X
 +
          /|\
 +
          / | \                        E_3@X
 +
        /  |  \
 +
o  o  o  o  o  o  o  o  o  o  Y
 +
0  1  2  3  4  5  6  7  8  9
 +
 +
The local flag E_2@Y is displayed here:
 +
 +
0  1  2  3  4  5  6  7  8  9
 +
o  o  o  o  o  o  o  o  o  o  X
 +
    \  |  /
 +
      \ | /                            E_2@Y
 +
      \|/
 +
o  o  o  o  o  o  o  o  o  o  Y
 +
0  1  2  3  4  5  6  7  8  9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.8
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Now let's re-examine the "numerical incidence properties" of relations,
 +
concentrating on the definitions of the assorted regularity conditions.
 +
 +
| For instance, L is said to be "c-regular at j" if and only if
 +
| the cardinality of the local flag L_x@j is c for all x in X_j,
 +
| coded in symbols, if and only if |L_x@j| = c for all x in X_j.
 +
|
 +
| In a similar fashion, one can define the NIP's "<c-regular at j",
 +
| ">c-regular at j", and so on.  For ease of reference, I record a
 +
| few of these definitions here:
 +
|
 +
| L is  c-regular at j      iff  |L_x@j|  = c for all x in X_j.
 +
|
 +
| L is (<c)-regular at j    iff  |L_x@j|  < c for all x in X_j.
 +
|
 +
| L is (>c)-regular at j    iff  |L_x@j|  > c for all x in X_j.
 +
|
 +
| L is (=<c)-regular at j  iff  |L_x@j|  =< c for all x in X_j.
 +
|
 +
| L is (>=c)-regular at j  iff  |L_x@j|  >= c for all x in X_j.
 +
 +
Clearly, if any relation is (=<c)-regular on one
 +
of its domains X_j and also (>=c)-regular on the
 +
same domain, then it must be (=c)-regular on the
 +
affected domain X_j, in effect, c-regular at j.
 +
 +
For example, let G = {r, s, t} and H = {1, ..., 9},
 +
and consider the 2-adic relation F c G x H that is
 +
bigraphed here:
 +
 +
    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
 +
 +
We observe that F is 3-regular at G and 1-regular at H.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Among the vast variety of conceivable regularities affecting 2-adic relations,
 +
we pay special attention to the c-regularity conditions where c is equal to 1.
 +
 +
| Let P c X x Y be an arbitrary 2-adic relation.
 +
| The following properties of P can be defined:
 +
|
 +
| P is "total" at X    iff  P is (>=1)-regular at X.
 +
|
 +
| P is "total" at Y    iff  P is (>=1)-regular at Y.
 +
|
 +
| P is "tubular" at X  iff  P is (=<1)-regular at X.
 +
|
 +
| P is "tubular" at Y  iff  P is (=<1)-regular at Y.
 +
 +
We have already looked at 2-adic relations that
 +
separately exemplify each of these regularities.
 +
 +
Also, we introduced a few bits of additional terminology and
 +
special-purpose notations for working with tubular relations:
 +
 +
| P is a "pre-function" P : X ~> Y  iff  P is tubular at X.
 +
|
 +
| P is a "pre-function" P : X <~ Y  iff  P is tubular at Y.
 +
 +
Thus, we arrive by way of this winding stair at the very special stamps
 +
of 2-adic relations P c X x Y that are "total prefunctions" at X (or Y),
 +
"total and tubular" at X (or Y), or "1-regular" at X (or Y), more often
 +
celebrated as "functions" at X (or Y).
 +
 +
| If P is a pre-function P : X ~> Y that happens to be total at X, then P
 +
| is known as a "function" from X to Y, typically indicated as P : X -> Y.
 +
|
 +
| To say that a relation P c X x Y is totally tubular at X is to say that
 +
| it is 1-regular at X.  Thus, we may formalize the following definitions:
 +
|
 +
| P is a "function" p : X -> Y  iff  P is 1-regular at X.
 +
|
 +
| P is a "function" p : X <- Y  iff  P is 1-regular at Y.
 +
 +
For example, let X = Y = {0, ..., 9} and let F c X x Y be
 +
the 2-adic relation that is depicted in the bigraph below:
 +
 +
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
 +
 +
We observe that F is a function at Y,
 +
and we record this fact in either of
 +
the manners F : X <- Y or F : Y -> X.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.10
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In the case of a 2-adic relation F c X x Y that has
 +
the qualifications of a function f : X -> Y, there
 +
are a number of further differentia that arise:
 +
 +
| f is "surjective"  iff  f is total at Y.
 +
|
 +
| f is "injective"    iff  f is tubular at Y.
 +
|
 +
| f is "bijective"    iff  f is 1-regular at Y.
 +
 +
For example, or more precisely, contra example,
 +
the function f : X -> Y that is depicted below
 +
is neither total at Y nor tubular at Y, and so
 +
it cannot enjoy any of the properties of being
 +
sur-, or in-, or bi-jective.
 +
 +
0  1  2  3  4  5  6  7  8  9
 +
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
 +
 +
A cheap way of getting a surjective function out of any function
 +
is to reset its codomain to its range.  For example, the range
 +
of the function f above is Y'=  {0, 2, 5, 6, 7, 8, 9}.  Thus,
 +
if we form a new function g : X -> Y' that looks just like
 +
f on the domain X but is assigned the codomain Y', then
 +
g is surjective, and is described as mapping "onto" Y'.
 +
 +
0  1  2  3  4  5  6  7  8  9
 +
o  o  o  o  o  o  o  o  o  o  X
 +
|    \  |  /    \  \  |  |    \ /
 +
|    \ | /      \  \ |  |    \    g
 +
|      \|/        \  \|  |    / \
 +
o      o          o  o  o  o  o  Y'
 +
0      2          5  6  7  8  9
 +
 +
The function h : Y' -> Y is injective.
 +
 +
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
 +
 +
The function m : X -> Y is bijective.
 +
 +
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
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.11
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
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 to ferret out the local functionalites that may
 +
be immanent in relative terms no matter where they
 +
locate themselves within the domains of relations.
 +
I am hopeful that these skills will serve us in
 +
good stead as we work to build a catwalk from
 +
Peirce's platform to contemporary scenes on
 +
the logic of relatives, and back again.
 +
 +
By way of extending a few very tentative plancks,
 +
let us experiment with the following definitions:
 +
 +
1.  A relative term 'p' and the corresponding relation P c X x Y are both
 +
    called "functional on relates" if and only if P is a function at X,
 +
    in symbols, P : X -> Y.
 +
 +
2.  A relative term 'p' and the corresponding relation P c X x Y are both
 +
    called "functional on correlates" if and only if P is function at Y,
 +
    in symbols, P : X <- Y.
 +
 +
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 P : X x Y, P : X -> Y, P : X <- Y,
 +
as the case may be, when and if it serves to clarify matters.
 +
 +
From this current, perhaps transient, perspective, it appears that
 +
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.
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                          P o Q                          |
 +
|                ____________^____________                |
 +
|                /                        \                |
 +
|              /      P            Q      \              |
 +
|              /      @            @      \              |
 +
|            /      / \          / \      \            |
 +
|            /      /  \        /  \      \            |
 +
|          o      o    o      o    o      o          |
 +
|          X      X    Y      Y    Z      Z          |
 +
|      1,__#      #'p'__$      $'q'__%      %1          |
 +
|          o      o    o      o    o      o          |
 +
|            \    /      \    /      \    /            |
 +
|            \  /        \  /        \  /            |
 +
|              \ /          \ /          \ /              |
 +
|              @            @            @              |
 +
|              !1!          !1!          !1!              |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 16.  Anything that is a 'p' of a 'q' of Anything
 +
 +
From this we extract the "hypergraph picture" of relational composition:
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                P        P o Q        Q                |
 +
|                @          @          @                |
 +
|                / \        / \        / \                |
 +
|              /  \      /  \      /  \              |
 +
|              o    o    o    o    o    o              |
 +
|              X    Y    X    Z    Y    Z              |
 +
|              o    o    o    o    o    o              |
 +
|              \    \  /      \  /    /              |
 +
|                \    \ /        \ /    /                |
 +
|                \    /          \    /                |
 +
|                  \  / \        / \  /                  |
 +
|                  \ /  \___ ___/  \ /                  |
 +
|                    @        @        @                    |
 +
|                  !1!      !1!      !1!                  |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 17.  Relational Composition P o Q
 +
 +
All of the relevant information of these Figures can be compressed
 +
into the form of a "spreadsheet", or constraint satisfaction table:
 +
 +
Table 18.  Relational Composition P o Q
 +
o---------o---------o---------o---------o
 +
|        #  !1!  |  !1!  |  !1!  |
 +
o=========o=========o=========o=========o
 +
|    P    #    X    |    Y    |        |
 +
o---------o---------o---------o---------o
 +
|    Q    #        |    Y    |    Z    |
 +
o---------o---------o---------o---------o
 +
|  P o Q  #    X    |        |    Z    |
 +
o---------o---------o---------o---------o
 +
 +
So the following presents itself as a reasonable plan of study:
 +
Let's see how much easy mileage we can get in our exploration
 +
of functions by adopting the above templates as a paradigm.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.12
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Since functions are special cases of 2-adic relations, and since the space
 +
of 2-adic relations is closed under relational composition, in other words,
 +
the composition of a couple of 2-adic relations is again a 2-adic relation,
 +
we know that the relational composition of a couple of functions has to be
 +
a 2-adic relation.  If it is also necessarily a function, then we would be
 +
justified in speaking of "functional composition", and also of saying that
 +
the space of functions is closed under this functional form of composition.
 +
 +
Just for novelty's sake, let's try to prove this
 +
for relations that are functional on correlates.
 +
 +
So our task is this:  Given a couple of 2-adic relations,
 +
P c X x Y and Q c Y x Z, that are functional on correlates,
 +
P : X <- Y and Q : Y <- Z, we need to determine whether the
 +
relational composition P o Q c X x Z is also P o Q : X <- Z,
 +
or not.
 +
 +
It always helps to begin by recalling the pertinent definitions.
 +
 +
For a 2-adic relation L c X x Y, we have:
 +
 +
L is a "function" L : X <- Y  iff  L is 1-regular at Y.
 +
 +
As for the definition of relational composition,
 +
it is enough to consider the coefficient of the
 +
composite on an arbitrary ordered pair like i:j.
 +
 +
(P o Q)_ij  =  Sum_k (P_ik Q_kj).
 +
 +
So let us begin.
 +
 +
P : X <- Y, or P being 1-regular at Y, means that there
 +
is exactly one ordered pair i:k in P for each k in Y.
 +
 +
Q : Y <- Z, or Q being 1-regular at Z, means that there
 +
is exactly one ordered pair k:j in Q for each j in Z.
 +
 +
Thus, there is exactly one ordered pair i:j in P o Q
 +
for each j in Z, which means that P o Q is 1-regular
 +
at Z, and so we have the function P o Q : X <- Z.
 +
 +
And we are done.
 +
 +
Bur proofs after midnight must be checked the next day.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.13
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
As we make our way toward the foothills of Peirce's 1870 LOR, there
 +
is one piece of equipment that we dare not leave the plains without --
 +
for there is little hope that "l'or dans les montagnes là" will lie
 +
among our prospects without the ready use of its leverage and lifts --
 +
and that is a facility with the utilities that are variously called
 +
"arrows", "morphisms", "homomorphisms", "structure-preserving maps",
 +
and several other names, in accord with the altitude of abstraction
 +
at which one happens to be working, at the given moment in question.
 +
 +
As a middle but not too beaten track, I will lay out the definition
 +
of a morphism in the forms that we will need right off, in a slight
 +
excess of formality at first, but quickly bringing the bird home to
 +
roost on more familiar perches.
 +
 +
Let's say that we have three functions J, K, L
 +
that have the following types and that satisfy
 +
the equation that follows:
 +
 +
| J : X <- Y
 +
|
 +
| K : X <- X x X
 +
|
 +
| L : Y <- Y x Y
 +
|
 +
| J(L(u, v))  =  K(Ju, Jv)
 +
 +
Our sagittarian leitmotif can be rubricized in the following slogan:
 +
 +
>->  The image of the ligature is the compound of the images.  <-<
 +
 +
Where J is the "image", K is the "compound", and L is the "ligature".
 +
 +
Figure 19 presents us with a picture of the situation in question.
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                      K          L                      |
 +
|                      @          @                      |
 +
|                      /|\        /|\                      |
 +
|                    / | \      / | \                    |
 +
|                    v  |  \    v  |  \                    |
 +
|                  o  o  o  o  o  o                  |
 +
|                  X  X  X  Y  Y  Y                  |
 +
|                  o  o  o  o  o  o                  |
 +
|                    ^  ^  ^ /  /  /                    |
 +
|                    \  \  \  /  /                    |
 +
|                      \  \ / \ /  /                      |
 +
|                      \  \  \  /                      |
 +
|                        \ / \ / \ /                        |
 +
|                        @  @  @                        |
 +
|                        J  J  J                        |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 19.  Structure Preserving Transformation J : K <- L
 +
 +
Here, I have used arrowheads to indicate the relational domains
 +
at which each of the relations J, K, L happens to be functional.
 +
 +
Table 20 gives the constraint matrix version of the same thing.
 +
 +
Table 20.  Arrow:  J(L(u, v)) = K(Ju, Jv)
 +
o---------o---------o---------o---------o
 +
|        #    J    |    J    |    J    |
 +
o=========o=========o=========o=========o
 +
|    K    #    X    |    X    |    X    |
 +
o---------o---------o---------o---------o
 +
|    L    #    Y    |    Y    |    Y    |
 +
o---------o---------o---------o---------o
 +
 +
One way to read this Table is in terms of the informational redundancies
 +
that it schematizes.  In particular, it can be read to say that when one
 +
satisfies the constraint in the L row, along with all of the constraints
 +
in the J columns, then the constraint in the K row is automatically true.
 +
That is one way of understanding the equation:  J(L(u, v)) = K(Ju, Jv).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.14
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
First, a correction.  Ignore for now the
 +
gloss that I gave in regard to Figure 19:
 +
 +
| Here, I have used arrowheads to indicate the relational domains
 +
| at which each of the relations J, K, L happens to be functional.
 +
 +
It is more like the feathers of the arrows that serve to mark the
 +
relational domains at which the relations J, K, L are functional,
 +
but it would take yet another construction to make this precise,
 +
as the feathers are not uniquely appointed but many splintered.
 +
 +
Now, as promised, let's look at a more homely example of a morphism,
 +
say, any one of the mappings J : R -> R (roughly speaking) that are
 +
commonly known as "logarithm functions", where you get to pick your
 +
favorite base.  In this case, K(r, s) = r + s and L(u, v) = u . v,
 +
and the defining formula J(L(u, v)) = K(Ju, Jv) comes out looking
 +
like J(u . v) = J(u) + J(v), writing a dot (.) and a plus sign (+)
 +
for the ordinary 2-ary operations of arithmetical multiplication
 +
and arithmetical summation, respectively.
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                      {+}        {.}                      |
 +
|                      @          @                      |
 +
|                      /|\        /|\                      |
 +
|                    / | \      / | \                    |
 +
|                    v  |  \    v  |  \                    |
 +
|                  o  o  o  o  o  o                  |
 +
|                  X  X  X  Y  Y  Y                  |
 +
|                  o  o  o  o  o  o                  |
 +
|                    ^  ^  ^ /  /  /                    |
 +
|                    \  \  \  /  /                    |
 +
|                      \  \ / \ /  /                      |
 +
|                      \  \  \  /                      |
 +
|                        \ / \ / \ /                        |
 +
|                        @  @  @                        |
 +
|                        J  J  J                        |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 21.  Logarithm Arrow J : {+} <- {.}
 +
 +
Thus, where the "image" J is the logarithm map,
 +
the "compound" K is the numerical sum, and the
 +
the "ligature" L is the numerical product, one
 +
obtains the immemorial mnemonic motto:
 +
 +
| The image of the product is the sum of the images.
 +
|
 +
| J(u . v)  =  J(u) + J(v)
 +
|
 +
| J(L(u, v))  =  K(Ju, Jv)
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.15
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I'm going to elaborate a little further on the subject
 +
of arrows, morphisms, or structure-preserving maps, as
 +
a modest amount of extra work at this point will repay
 +
ample dividends when it comes time to revisit Peirce's
 +
"number of" function on logical terms.
 +
 +
The "structure" that is being preserved by a structure-preserving map
 +
is just the structure that we all know and love as a 3-adic relation.
 +
Very typically, it will be the type of 3-adic relation that defines
 +
the type of 2-ary operation that obeys the rules of a mathematical
 +
structure that is known as a "group", that is, a structure that
 +
satisfies the axioms for closure, associativity, identities,
 +
and inverses.
 +
 +
For example, in the previous case of the logarithm map J, we have the data:
 +
 +
| J : R <- R (properly restricted)
 +
|
 +
| K : R <- R x R, where K(r, s) = r + s
 +
|
 +
| L : R <- R x R, where L(u, v) = u . v
 +
 +
Real number addition and real number multiplication (suitably restricted)
 +
are examples of group operations.  If we write the sign of each operation
 +
in braces as a name for the 3-adic relation that constitutes or defines
 +
the corresponding group, then we have the following set-up:
 +
 +
| J : {+} <- {.}
 +
|
 +
| {+} c R x R x R
 +
|
 +
| {.} c R x R x R
 +
 +
In many cases, one finds that both groups are written with the same
 +
sign of operation, typically ".", "+", "*", or simple concatenation,
 +
but they remain in general distinct whether considered as operations
 +
or as relations, no matter what signs of operation are used.  In such
 +
a setting, our chiasmatic theme may run a bit like these two variants:
 +
 +
| The image of the sum is the sum of the images.
 +
|
 +
| The image of the product is the product of the images.
 +
 +
Figure 22 presents a generic picture for groups G and H.
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                      G          H                      |
 +
|                      @          @                      |
 +
|                      /|\        /|\                      |
 +
|                    / | \      / | \                    |
 +
|                    v  |  \    v  |  \                    |
 +
|                  o  o  o  o  o  o                  |
 +
|                  X  X  X  Y  Y  Y                  |
 +
|                  o  o  o  o  o  o                  |
 +
|                    ^  ^  ^ /  /  /                    |
 +
|                    \  \  \  /  /                    |
 +
|                      \  \ / \ /  /                      |
 +
|                      \  \  \  /                      |
 +
|                        \ / \ / \ /                        |
 +
|                        @  @  @                        |
 +
|                        J  J  J                        |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 22.  Group Homomorphism J : G <- H
 +
 +
In a setting where both groups are written with a plus sign,
 +
perhaps even constituting the very same group, the defining
 +
formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the
 +
shape J(u + v) = Ju + Jv, which looks very analogous to the
 +
distributive multiplication of a sum (u + v) by a factor J.
 +
Hence another popular name for a morphism:  a "linear" map.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.16
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I think that we have enough material on morphisms now
 +
to go back and cast a more studied eye on what Peirce
 +
is doing with that "number of" function, the one that
 +
we apply to a logical term 't', absolute or relative
 +
of any number of correlates, by writing it in square
 +
brackets, as ['t'].  It is frequently convenient to
 +
have a prefix notation for this function, and since
 +
Peirce reserves 'n' to signify 'not', I will try to
 +
use 'v', personally thinking of it as a Greek 'nu',
 +
which stands for frequency in physics, and which
 +
kind of makes sense if we think of frequency as
 +
it's habitual in statistics.  End of mnemonics.
 +
 +
My plan will be nothing less plodding than to work through
 +
all of the principal statements that Peirce has made about
 +
the "number of" function up to our present stopping place
 +
in the paper, namely, those that I collected once before
 +
and placed at this location:
 +
 +
LOR.COM 11.2.  http://stderr.org/pipermail/inquiry/2004-November/001814.html
 +
 +
NOF 1.
 +
 +
| I propose to assign to all logical terms, numbers;
 +
| to an absolute term, the number of individuals it denotes;
 +
| to a relative term, the average number of things so related
 +
| to one individual.
 +
|
 +
| Thus in a universe of perfect men ('men'),
 +
| the number of "tooth of" would be 32.
 +
|
 +
| The number of a relative with two correlates would be the
 +
| average number of things so related to a pair of individuals;
 +
| and so on for relatives of higher numbers of correlates.
 +
|
 +
| I propose to denote the number of a logical term by
 +
| enclosing the term in square brackets, thus ['t'].
 +
|
 +
| C.S. Peirce, CP 3.65
 +
 +
We may formalize the role of the "number of" function by assigning it
 +
a local habitation and a name 'v' : S -> R, where S is a suitable set
 +
of signs, called the "syntactic domain", that is ample enough to hold
 +
all of the terms that we might wish to number in a given discussion,
 +
and where R is the real number domain.
 +
 +
Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".
 +
Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect
 +
human dentition, the number of the relative term "tooth of ---" is equal to
 +
the number of teeth of humans divided by the number of humans, that is, 32.
 +
 +
The 2-adic relative term 't' determines a 2-adic relation T c U x V,
 +
where U and V are two universes of discourse, possibly the same one,
 +
that hold among other things all of the teeth and all of the people
 +
that happen to be under discussion, respectively.
 +
 +
A rough indication of the bigraph for T
 +
might be drawn as follows, where I have
 +
tried to sketch in just the toothy part
 +
of U and the peoply part of V.
 +
 +
t_1    t_32  t_33    t_64  t_65    t_96  ...    ...
 +
o  ...  o    o  ...  o    o  ...  o    o  ...  o    U
 +
  \  |  /      \  |  /      \  |  /      \  |  /
 +
  \ | /        \ | /        \ | /        \ | /      T
 +
    \|/          \|/          \|/          \|/
 +
    o            o            o            o        V
 +
    m_1          m_2          m_3          ...
 +
 +
Notice that the "number of" function 'v' : S -> R
 +
needs the data that is represented by this entire
 +
bigraph for T in order to compute the value ['t'].
 +
 +
Finally, one observes that this component of T is a function
 +
in the direction T : U -> V, since we are counting only those
 +
teeth that ideally occupy one and only one mouth of a creature.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.17
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I think that the reader is beginning to get an inkling of the crucial importance of
 +
the "number of" map in Peirce's way of looking at logic, for it's one of the plancks
 +
in the bridge from logic to the theories of probability, statistics, and information,
 +
in which logic forms but a limiting case at one scenic turnout on the expanding vista.
 +
It is, as a matter of necessity and a matter of fact, practically speaking, at any rate,
 +
one way that Peirce forges a link between the "eternal", logical, or rational realm and
 +
the "secular", empirical, or real domain.
 +
 +
With that little bit of encouragement and exhortation,
 +
let us return to the nitty gritty details of the text.
 +
 +
NOF 2.
 +
 +
| But not only do the significations of '=' and '<' here adopted fulfill all
 +
| absolute requirements, but they have the supererogatory virtue of being very
 +
| nearly the same as the common significations.  Equality is, in fact, nothing
 +
| but the identity of two numbers;  numbers that are equal are those which are
 +
| predicable of the same collections, just as terms that are identical are those
 +
| which are predicable of the same classes.  So, to write 5 < 7 is to say that 5
 +
| is part of 7, just as to write f < m is to say that Frenchmen are part of men.
 +
| Indeed, if f < m, then the number of Frenchmen is less than the number of men,
 +
| and if v = p, then the number of Vice-Presidents is equal to the number of
 +
| Presidents of the Senate;  so that the numbers may always be substituted
 +
| for the terms themselves, in case no signs of operation occur in the
 +
| equations or inequalities.
 +
|
 +
| C.S. Peirce, CP 3.66
 +
 +
Peirce is here remarking on the principle that the
 +
measure 'v' on terms "preserves" or "respects" the
 +
prevailing implication, inclusion, or subsumption
 +
relations that impose an ordering on those terms.
 +
 +
In these initiatory passages of the text, Peirce is using a single symbol "<"
 +
to denote the usual linear ordering on numbers, but also what amounts to the
 +
implication ordering on logical terms and the inclusion ordering on classes.
 +
Later, of course, he will introduce distinctive symbols for logical orders.
 +
 +
Now, the links among terms, sets, and numbers can be pursued in all directions,
 +
and Peirce has already indicated in an earlier paper how he would "construct"
 +
the integers from sets, that is, from the aggregate denotations of terms.
 +
 +
We will get back to that at another time.
 +
 +
In the immediate example, we have this sort of statement:
 +
 +
"if f < m, then the number of Frenchmen is less than the number of men"
 +
 +
In symbolic form, this would be written:
 +
 +
f < m  =>  [f] < [m]
 +
 +
Here, the "<" on the left is a logical ordering on syntactic terms
 +
while the "<" on the right is an arithmetic ordering on real numbers.
 +
 +
The type of principle that comes up here is usually discussed
 +
under the question of whether a map between two ordered sets
 +
is "order-preserving" or not.  The general type of question
 +
may be formalized in the following way.
 +
 +
Let X_1 be a set with an ordering denoted by "<_1".
 +
Let X_2 be a set with an ordering denoted by "<_2".
 +
 +
What makes an ordering what it is will commonly be
 +
a set of axioms that defines the properties of the
 +
order relation in question.  Since one frequently
 +
has occasion to view the same set in the light of
 +
several different order relations, one will often
 +
resort to explicit forms like (X, <_1), (X, <_2),
 +
and so on, to invoke a set with a given ordering.
 +
 +
A map F : (X_1, <_1) -> (X_2, <_2) is "order-preserving"
 +
if and only if a statement of a particular form holds
 +
for all x and y in (X_1, <_1), specifically, this:
 +
 +
x <_1 y  =>  Fx <_2 Fy
 +
 +
The action of the "number of" map 'v' : (S, <_1) -> (R, <_2)
 +
has just this character, as exemplified by its application to
 +
the case where x = f = "frenchman" and y = m = "man", like so:
 +
 +
| f < m  =>  [f] < [m]
 +
|
 +
| f < m  =>  'v'f < 'v'm
 +
 +
Here, to be more exacting, we may interpret the "<" on the left
 +
as "proper subsumption", that is, excluding the equality case,
 +
while we read the "<" on the right as the usual "less than".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.18
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
There is a comment that I ought to make on the concept of
 +
a "structure preserving map", including as a special case
 +
the idea of an "order-preserving map".  It seems to be a
 +
peculiarity of mathematical usage in general -- at least,
 +
I don't think it's just me -- that "preserving structure"
 +
always means "preserving 'some', not of necessity 'all',
 +
of the structure in question".  People sometimes express
 +
this by speaking of "structure preservation in measure",
 +
the implication being that any property that is amenable
 +
to being qualified in manner is potentially amenable to
 +
being quantified in degree, perhaps in such a way as to
 +
answer questions like "How structure-preserving is it?".
 +
 +
Let's see how this remark applies to the order-preserving property of
 +
the "number of" mapping 'v' : S -> R.  For any pair of absolute terms
 +
x and y in the syntactic domain S, we have the following implications,
 +
where "-<" denotes the logical subsumption relation on terms and "=<"
 +
is the "less than or equal to" relation on the real number domain R.
 +
 +
x -< y  =>  'v'x =< 'v'y
 +
 +
Equivalently:
 +
 +
x -< y  =>  [x] =< [y]
 +
 +
It is easy to see that nowhere near all of the distinctions that make up
 +
the structure of the ordering on the left hand side will be preserved as
 +
one passes to the right hand side of these implication statements, but
 +
that is not required in order to call the map 'v' "order-preserving",
 +
or what is also known as an "order morphism".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.19
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Up to this point in the LOR of 1870, Peirce has introduced the
 +
"number of" measure on logical terms and discussed the extent
 +
to which this measure, 'v' : S -> R such that 'v' : s ~> [s],
 +
exhibits a couple of important measure-theoretic principles:
 +
 +
1.  The "number of" map exhibits a certain type of "uniformity property",
 +
    whereby the value of the measure on a uniformly qualified population
 +
    is in fact actualized by each member of the population.
 +
 +
2.  The "number of" map satisfies an "order morphism principle", whereby
 +
    the illative partial ordering of logical terms is reflected up to a
 +
    partial extent by the arithmetical linear ordering of their measures.
 +
 +
Peirce next takes up the action of the "number of" map on the two types of,
 +
loosely speaking, "additive" operations that we normally consider in logic.
 +
 +
NOF 3.
 +
 +
| It is plain that both the regular non-invertible addition and the
 +
| invertible addition satisfy the absolute conditions.  (CP 3.67).
 +
 +
The "regular non-invertible addition" is signified by "+,",
 +
corresponding to what we'd call the inclusive disjunction
 +
of logical terms or the union of their extensions as sets.
 +
 +
The "invertible addition" is signified in algebra by "+",
 +
corresponding to what we'd call the exclusive disjunction
 +
of logical terms or the symmetric difference of their sets,
 +
ignoring many details and nuances that are often important,
 +
of course.
 +
 +
| But the notation has other recommendations.  The conception of 'taking together'
 +
| involved in these processes is strongly analogous to that of summation, the sum
 +
| of 2 and 5, for example, being the number of a collection which consists of a
 +
| collection of two and a collection of five.  (CP 3.67).
 +
 +
A full interpretation of this remark will require us to pick up the precise
 +
technical sense in which Peirce is using the word "collection", and that will
 +
take us back to his logical reconstruction of certain aspects of number theory,
 +
all of which I am putting off to another time, but it is still possible to get
 +
a rough sense of what he's saying relative to the present frame of discussion.
 +
 +
The "number of" map 'v' : S -> R evidently induces
 +
some sort of morphism with respect to logical sums.
 +
If this were straightforwardly true, we could write:
 +
 +
|?| 'v'(x +, y)  =  'v'x + 'v'y
 +
|?|
 +
|?| Equivalently:
 +
|?|
 +
|?| [x +, y]  =  [x] + [y]
 +
 +
Of course, things are just not that simple in the case
 +
of inclusive disjunction and set-theoretic unions, so
 +
we'd "probably" invent a word like "sub-additive" to
 +
describe the principle that does hold here, namely:
 +
 +
| 'v'(x +, y)  =<  'v'x + 'v'y
 +
|
 +
| Equivalently:
 +
|
 +
| [x +, y]  =<  [x] + [y]
 +
 +
This is why Peirce trims his discussion of this point with the following hedge:
 +
 +
| Any logical equation or inequality in which no operation but addition
 +
| is involved may be converted into a numerical equation or inequality by
 +
| substituting the numbers of the several terms for the terms themselves --
 +
| provided all the terms summed are mutually exclusive.  (CP 3.67).
 +
 +
Finally, a morphism with respect to addition,
 +
even a contingently qualified one, must do the
 +
right stuff on behalf of the additive identity:
 +
 +
| Addition being taken in this sense,
 +
|'nothing' is to be denoted by 'zero',
 +
| for then:
 +
|
 +
| x +, 0  =  x
 +
|
 +
| whatever is denoted by x;  and this is the definition
 +
| of 'zero'.  This interpretation is given by Boole, and
 +
| is very neat, on account of the resemblance between the
 +
| ordinary conception of 'zero' and that of nothing, and
 +
| because we shall thus have
 +
|
 +
| [0]  =  0.
 +
|
 +
| C.S. Peirce, CP 3.67
 +
 +
With respect to the nullity 0 in S and the number 0 in R, we have:
 +
 +
'v'0  =  [0]  =  0.
 +
 +
In sum, therefor, it also serves that only preserves
 +
a due respect for the function of a vacuum in nature.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.20
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We arrive at the last, for the time being, of
 +
Peirce's statements about the "number of" map.
 +
 +
NOF 4.
 +
 +
| The conception of multiplication we have adopted is
 +
| that of the application of one relation to another.  ...
 +
|
 +
| Even ordinary numerical multiplication involves the same idea,
 +
| for 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs,
 +
| where "triplet of" and "pair of" are evidently relatives.
 +
|
 +
| If we have an equation of the form:
 +
|
 +
| xy  =  z
 +
|
 +
| and there are just as many x's per y as there are
 +
|'per' things, things of the universe, then we have
 +
| also the arithmetical equation:
 +
|
 +
| [x][y]  =  [z].
 +
|
 +
| C.S. Peirce, CP 3.76
 +
 +
Peirce is here observing what we might dub a "contingent morphism"
 +
or a "skeptraphotic arrow", if you will.  Provided that a certain
 +
condition, to be named and, what is more hopeful, to be clarified
 +
in short order, happens to be satisfied, we would find it holding
 +
that the "number of" map 'v' : S -> R such that 'v's = [s] serves
 +
to preserve the multiplication of relative terms, that is as much
 +
to say, the composition of relations, in the form:  [xy] = [x][y].
 +
 +
So let us try to uncross Peirce's manifestly chiasmatic encryption
 +
of the condition that is called on in support of this preservation.
 +
 +
Proviso for [xy] = [x][y] --
 +
 +
| there are just as many x's per y
 +
| as there are 'per' things<,>
 +
| things of the universe ...
 +
 +
I have placed angle brackets around
 +
a comma that CP shows but CE omits,
 +
not that it helps much either way.
 +
So let us resort to the example:
 +
 +
| For instance, if our universe is perfect men, and there
 +
| are as many teeth to a Frenchman (perfect understood)
 +
| as there are to any one of the universe, then:
 +
|
 +
| ['t'][f]  =  ['t'f]
 +
|
 +
| holds arithmetically.  (CP 3.76).
 +
 +
Now that is something that we can sink our teeth into,
 +
and trace the bigraph representation of the situation.
 +
In order to do this, it will help to recall our first
 +
examination of the "tooth of" relation, and to adjust
 +
the picture that we sketched of it on that occasion.
 +
 +
Transcribing Peirce's example, we may let m = "man" and 't' = "tooth of ---".
 +
Then 'v'('t') = ['t'] = ['t'm]/[m], that is to say, in a universe of perfect
 +
human dentition, the number of the relative term "tooth of ---" is equal to
 +
the number of teeth of humans divided by the number of humans, that is, 32.
 +
 +
The 2-adic relative term 't' determines a 2-adic relation T c U x V,
 +
where U and V are two universes of discourse, possibly the same one,
 +
that hold among other things all of the teeth and all of the people
 +
that happen to be under discussion, respectively.  To make the case
 +
as simple as we can and still cover the point, let's say that there
 +
are just four people in our initial universe of discourse, and that
 +
just two of them are French.  The bigraphic composition below shows
 +
all of the pertinent facts of the case.
 +
 +
T_1    T_32  T_33    T_64  T_65    T_96  T_97    T_128
 +
o  ...  o    o  ...  o    o  ...  o    o  ...  o      U
 +
  \  |  /      \  |  /      \  |  /      \  |  /
 +
  \ | /        \ | /        \ | /        \ | /      't'
 +
    \|/          \|/          \|/          \|/
 +
    o            o            o            o          V = m = 1
 +
                  |                          |
 +
                  |                          |        'f'
 +
                  |                          |
 +
    o            o            o            o          V = m = 1
 +
    J            K            L            M
 +
 +
Here, the order of relational composition flows up the page.
 +
For convenience, the absolute term f = "frenchman" has been
 +
converted by using the comma functor to give the idempotent
 +
representation 'f' = f, = "frenchman that is ---", and thus
 +
it can be taken as a selective from the universe of mankind.
 +
 +
By way of a legend for the figure, we have the following data:
 +
 +
| m  =  J +, K +, L +, M  =  1
 +
|
 +
| f  =  K +, M
 +
|
 +
|'f'  =  K:K +, M:M
 +
|
 +
|'t'  =  (T_001 +, ... +, T_032):J  +,
 +
|        (T_033 +, ... +, T_064):K  +,
 +
|        (T_065 +, ... +, T_096):L  +,
 +
|        (T_097 +, ... +, T_128):M
 +
 +
Now let's see if we can use this picture
 +
to make sense of the following statement:
 +
 +
| For instance, if our universe is perfect men, and there
 +
| are as many teeth to a Frenchman (perfect understood)
 +
| as there are to any one of the universe, then:
 +
|
 +
| ['t'][f]  =  ['t'f]
 +
|
 +
| holds arithmetically.  (CP 3.76).
 +
 +
In the lingua franca of statistics, Peirce is saying this:
 +
That if the population of Frenchmen is a "fair sample" of
 +
the general population with regard to dentition, then the
 +
morphic equation ['t'f] = ['t'][f], whose transpose gives
 +
['t'] = ['t'f]/[f], is every bite as true as the defining
 +
equation in this circumstance, namely, ['t'] = ['t'm]/[m].
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.21
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
One more example and one more general observation, and then we will
 +
be all caught up with our homework on Peirce's "number of" function.
 +
 +
| So if men are just as apt to be black as things in general:
 +
|
 +
| [m,][b]  =  [m,b]
 +
|
 +
| where the difference between [m] and [m,] must not be overlooked.
 +
|
 +
| C.S. Peirce, CP 3.76
 +
 +
The protasis, "men are just as apt to be black as things in general",
 +
is elliptic in structure, and presents us with a potential ambiguity.
 +
If we had no further clue to its meaning, it might be read as either:
 +
 +
1.  Men are just as apt to be black as things in general are apt to be black.
 +
 +
2.  Men are just as apt to be black as men are apt to be things in general.
 +
 +
The second interpretation, if grammatical, is pointless to state,
 +
since it equates a proper contingency with an absolute certainty.
 +
 +
So I think it is safe to assume this paraphrase of what Peirce intends:
 +
 +
3.  Men are just as likely to be black as things in general are likely to be black.
 +
 +
Stated in terms of the conditional probability:
 +
 +
4.  P(b|m)  =  P(b)
 +
 +
From the definition of conditional probability:
 +
 +
5.  P(b|m)  =  P(b m)/P(m)
 +
 +
Equivalently:
 +
 +
6.  P(b m)  =  P(b|m)P(m)
 +
 +
Thus we may derive the equivalent statement:
 +
 +
7.  P(b m)  =  P(b|m)P(m)  =  P(b)P(m)
 +
 +
And this, of course, is the definition of independent events, as
 +
applied to the event of being Black and the event of being a Man.
 +
 +
It seems like a likely guess, then, that this is the content of Peirce's
 +
statement about frequencies, [m,b] = [m,][b], in this case normalized to
 +
produce the equivalent statement about probabilities:  P(m b) = P(m)P(b).
 +
 +
Let's see if this checks out.
 +
 +
Let n be the number of things in general, in Peirce's lingo, n = [1].
 +
On the assumption that m and b are associated with independent events,
 +
we get [m,b] = P(m b)n = P(m)P(b)n = P(m)[b] = [m,][b], so we have to
 +
interpret [m,] = "the average number of men per things in general" as
 +
P(m) = the probability of a thing in general being a man.  Seems okay.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.22
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let's look at that last example from a different angle.
 +
 +
| So if men are just as apt to be black as things in general:
 +
|
 +
| [m,][b]  =  [m,b]
 +
|
 +
| where the difference between [m] and [m,] must not be overlooked.
 +
|
 +
| C.S. Peirce, CP 3.76
 +
 +
In different lights the formula [m,b] = [m,][b] presents itself
 +
as an "aimed arrow", "fair sample", or "independence" condition.
 +
 +
The example apparently assumes a universe of "things in general",
 +
encompassing among other things the denotations of the absolute
 +
terms m = "man" and b = "black".  That suggests to me that we
 +
might well illustrate this case in relief, by returning to
 +
our earlier staging of 'Othello' and seeing how well that
 +
universe of dramatic discourse observes the premiss that
 +
"men are just as apt to be black as things in general".
 +
 +
Here is the relevant data:
 +
 +
| 1  =  B +, C +, D +, E +, I +, J +, O
 +
|
 +
| b  =  O
 +
|
 +
| m  =  C +, I +, J +, O
 +
|
 +
| 1,  =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O
 +
|
 +
| b,  =  O:O
 +
|
 +
| m,  =  C:C +, I:I +, J:J +, O:O
 +
 +
The "fair sampling" or "episkeptral arrow" condition is tantamount to this:
 +
"Men are just as apt to be black as things in general are apt to be black".
 +
In other words, men are a fair sample of things in general with respect to
 +
the factor of being black.
 +
 +
Should this hold, the consequence would be:
 +
 +
[m,b]  =  [m,][b].
 +
 +
When [b] is not zero, we obtain the result:
 +
 +
[m,]  =  [m,b]/[b].
 +
 +
Once again, the absolute term b = "black" is most felicitously depicted
 +
by way of its idempotent representation 'b' = b, = "black that is ---",
 +
and thus it can be taken as a selective from the universe of discourse.
 +
 +
Here is the bigraph for the composition:
 +
 +
m,b  =  "man that is black",
 +
 +
here represented in the equivalent form:
 +
 +
m,b,  =  "man that is black that is ---".
 +
 +
B  C  D  E  I  J  O
 +
o  o  o  o  o  o  o  1
 +
    |          |  |  |
 +
    |          |  |  |  m,
 +
    |          |  |  |
 +
o  o  o  o  o  o  o  1
 +
                        |
 +
                        |  b,
 +
                        |
 +
o  o  o  o  o  o  o  1
 +
B  C  D  E  I  J  O
 +
 +
Thus we observe one of the more factitious facts
 +
that hold in this universe of discourse, namely:
 +
 +
m,b  =  b.
 +
 +
Another way of saying that is:
 +
 +
b  -<  m.
 +
 +
That in itself is enough to puncture any notion
 +
that b and m are statistically independent, but
 +
let us continue to develop the plot a bit more.
 +
 +
Putting all of the general formulas and particular facts together,
 +
we arrive at following summation of situation in the Othello case:
 +
 +
If the fair sampling condition holds:
 +
 +
[m,]  =  [m,b]/[b]  =  [b]/[b]  =  `1`,
 +
 +
In fact, however, it is the case that:
 +
 +
[m,]  =  [m,1]/[1]  =  [m]/[1]  =  4/7.
 +
 +
In sum, it is not the case in the Othello example that
 +
"men are just as apt to be black as things in general".
 +
 +
Expressed in terms of probabilities:  P(m) = 4/7 and P(b) = 1/7.
 +
 +
If these were independent we'd have:  P(mb) = 4/49.
 +
 +
On the contrary, P(mb) = P(b) = 1/7.
 +
 +
Another way to see it is as follows:  P(b|m) = 1/4 while P(b) = 1/7.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.23
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let me try to sum up as succinctly as possible the lesson
 +
that we ought to take away from Peirce's last "number of"
 +
example, since I know that the account that I have given
 +
of it so far may appear to have wandered rather widely.
 +
 +
| So if men are just as apt to be black as things in general:
 +
|
 +
| [m,][b]  =  [m,b]
 +
|
 +
| where the difference between [m] and [m,] must not be overlooked.
 +
|
 +
| C.S. Peirce, CP 3.76
 +
 +
In different lights the formula [m,b] = [m,][b] presents itself
 +
as an "aimed arrow", "fair sample", or "independence" condition.
 +
I had taken the tack of illustrating this polymorphous theme in
 +
bas relief, that is, via detour through a universe of discourse
 +
where it fails.  Here's a brief reminder of the Othello example:
 +
 +
B  C  D  E  I  J  O
 +
o  o  o  o  o  o  o  1
 +
    |          |  |  |
 +
    |          |  |  |  m,
 +
    |          |  |  |
 +
o  o  o  o  o  o  o  1
 +
                        |
 +
                        |  b,
 +
                        |
 +
o  o  o  o  o  o  o  1
 +
B  C  D  E  I  J  O
 +
 +
The condition, "men are just as apt to be black as things in general",
 +
is expressible in terms of conditional probabilities as P(b|m) = P(b),
 +
written out, the probability of the event Black given the event Male
 +
is exactly equal to the unconditional probability of the event Black.
 +
 +
Thus, for example, it is sufficient to observe in the Othello setting
 +
that P(b|m) = 1/4 while P(b) = 1/7 in order to cognize the dependency,
 +
and thereby to tell that the ostensible arrow is anaclinically biased.
 +
 +
This reduction of a conditional probability to an absolute probability,
 +
in the form P(A|Z) = P(A), is a familiar disguise, and yet in practice
 +
one of the ways that we most commonly come to recognize the condition
 +
of independence P(AZ) = P(A)P(Z), via the definition of a conditional
 +
probability according to the rule P(A|Z) = P(AZ)/P(Z).  To recall the
 +
familiar consequences, the definition of conditional probability plus
 +
the independence condition yields P(A|Z) = P(AZ)/P(Z) = P(A)P(Z)/P(Z),
 +
to wit, P(A|Z) = P(A).
 +
 +
As Hamlet discovered, there's a lot to be learned from turning a crank.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 11.24
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
And so we come to the end of the "number of" examples
 +
that we found on our agenda at this point in the text:
 +
 +
| It is to be observed that:
 +
|
 +
| [!1!]  =  `1`.
 +
|
 +
| Boole was the first to show this connection between logic and
 +
| probabilities.  He was restricted, however, to absolute terms.
 +
| I do not remember having seen any extension of probability to
 +
| relatives, except the ordinary theory of 'expectation'.
 +
|
 +
| Our logical multiplication, then, satisfies the essential conditions
 +
| of multiplication, has a unity, has a conception similar to that of
 +
| admitted multiplications, and contains numerical multiplication as
 +
| a case under it.
 +
|
 +
| C.S. Peirce, CP 3.76
 +
 +
There appears to be a problem with the printing of the text at this point.
 +
Let us first recall the conventions that I am using in this transcription:
 +
`1` for the "antique 1" that Peirce defines as !1!_oo = "something", and
 +
!1! for the "bold 1" that signifies the ordinary 2-identity relation.
 +
 +
CP 3 gives [!1!] = `1`, which I cannot make any sense of.
 +
CE 2 gives [!1!] =  1 , which makes sense on the reading
 +
of "1" as denoting the natural number 1, and not as the
 +
absolute term "1" that denotes the universe of discourse.
 +
On this reading, [!1!] is the average number of things
 +
related by the identity relation !1! to one individual,
 +
and so it makes sense that [!1!] = 1 : N, where "N" is
 +
the set or type of the natural numbers {0, 1, 2, ...}.
 +
 +
With respect to the 2-identity !1! in the syntactic domain S
 +
and the number 1 in the non-negative integers N c R, we have:
 +
 +
'v'!1!  =  [!1!]  =  1.
 +
 +
And so the "number of" mapping 'v' : S -> R has another one
 +
of the properties that would be required of an arrow S -> R.
 +
 +
The manner in which these arrows and qualified arrows help us
 +
to construct a suspension bridge that unifies logic, semiotics,
 +
statistics, stochastics, and information theory will be one of
 +
the main themes that I aim to elaborate throughout the rest of
 +
this inquiry.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 12
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| The Sign of Involution
 +
|
 +
| I shall take involution in such a sense that x^y
 +
| will denote everything which is an x for every
 +
| individual of y.
 +
|
 +
| Thus
 +
|
 +
| 'l'^w
 +
|
 +
| will be a lover of every woman.
 +
|
 +
| Then
 +
|
 +
| ('s'^'l')^w
 +
|
 +
| will denote whatever stands to every woman in
 +
| the relation of servant of every lover of hers;
 +
|
 +
| and
 +
|
 +
| 's'^('l'w)
 +
|
 +
| will denote whatever is a servant of
 +
| everything that is lover of a woman.
 +
|
 +
| So that
 +
|
 +
| ('s'^'l')^w  =  's'^('l'w).
 +
|
 +
| C.S. Peirce, CP 3.77
 +
|
 +
| Charles Sanders Peirce,
 +
|"Description of a Notation for the Logic of Relatives,
 +
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
 +
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
 +
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Commentary Note 12
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Let us make a few preliminary observations about the
 +
"logical sign of involution", as Peirce uses it here:
 +
 +
| The Sign of Involution
 +
|
 +
| I shall take involution in such a sense that x^y
 +
| will denote everything which is an x for every
 +
| individual of y.
 +
|
 +
| Thus
 +
|
 +
| 'l'^w
 +
|
 +
| will be a lover of every woman.
 +
|
 +
| C.S. Peirce, CP 3.77
 +
 +
In arithmetic, the "involution" x^y, or the "exponentiation" of x
 +
to the power of y, is the iterated multiplication of the factor x,
 +
repeated as many times as there are ones making up the exponent y.
 +
 +
In analogous fashion, 'l'^w is the iterated multiplication of 'l',
 +
repeated as many times as there are individuals under the term w.
 +
 +
For example, suppose that the universe of discourse has,
 +
among other things, just the three women, W_1, W_2, W_3.
 +
This could be expressed in Peirce's notation by writing:
 +
 +
w  =  W_1 +, W_2 +, W_3.
 +
 +
In this setting, we would have:
 +
 +
'l'^w  =  'l'^(W_1 +, W_2 +, W_3)  =  'l'W_1 , 'l'W_2 , 'l'W_3.
 +
 +
That is, a lover of every woman in the universe of discourse
 +
would be a lover of W_1 and a lover of W_2 and lover of W_3.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Note 13
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work Area
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Up to this point in the discussion, we have observed that
 +
the "number of" map 'v' : S -> R such that 'v's = [s] has
 +
the following morphic properties:
 +
 +
0.  [0]  =  0
 +
 +
1.  'v'
 +
 +
2.  x -< y  =>  [x] =< [y]
 +
 +
3.  [x +, y]  =<  [x] + [y]
 +
 +
contingent:
 +
 +
4.  [xy]  =  [x][y]
 +
 +
view relation P c X x Y x Z as related to three functions:
 +
 +
`p_1` c
 +
`p_3` c X x Y x Pow(Z)
 +
 +
 +
f(x)
 +
 +
f(x+y) = f(x) + f(y)
 +
 +
f(p(x, y))  =  q(f(x), f(y))
 +
 +
P(x, y, z)
 +
 +
(f^-1)(y)
 +
 +
f(z(x, y))  =  z'(f(x), f(y))
 +
 +
Definition.  f(x:y:z)  =  (fx:fy:fz).
 +
 +
f(x:y:z)  =  (fx:fy:
 +
 +
x:y:z in R => fx:fy:fz in fR
 +
 +
R(x, y, z) => (fR)(fx, fy, fz)
 +
 +
(L, x, y, z) => (fL, fx, fy, fz)
 +
 +
(x, y, z, L) => (xf, yf, zf, Lf)
 +
 +
(x, y, z, b) => (xf, yf, zf, bf)
 +
 +
 +
fzxy = z'(fx)(fy)
 +
 +
 +
        F
 +
        o
 +
        |
 +
        o
 +
        / \
 +
      o  o
 +
                      o
 +
                  .  |  .
 +
                .    |    .
 +
            .        |        .
 +
          .          o          .
 +
                  . / \ .
 +
                .  /  \  .
 +
            .    /    \    .
 +
          .      o      o      .
 +
                    . .    .
 +
                    .  .      .
 +
                                  .
 +
 +
                     
 +
  C o        . / \ .        o
 +
    |    .  /  \  .    | CF
 +
    |  .    o    o    .  |
 +
  f o    .    .    .    o fF
 +
    / \ .    .    .      / \
 +
  / . \  .              o  o
 +
X o    o Y              XF  YF
 +
 +
<u, v, w> in P ->
 +
 +
o---------o---------o---------o---------o
 +
|        #    h    |    h    |    f    |
 +
o=========o=========o=========o=========o
 +
|    P    #    X    |    Y    |    Z    |
 +
o---------o---------o---------o---------o
 +
|    Q    #    U    |    V    |    W    |
 +
o---------o---------o---------o---------o
 +
 +
Products of diagonal extensions:
 +
 +
1,1,  =  !1!!1!
 +
 +
      =  "anything that is anything that is ---"
 +
 +
      =  "anything that is ---"
 +
 +
      =  !1!
 +
 +
m,n  =  "man that is noble" 
 +
 +
    =  (C:C +, I:I +, J:J +, O:O)(C +, D +, O)
 +
 +
    =  C +, O
 +
 +
n,m  =  "noble that is man"
 +
 +
    =  (C:C +, D:D +, O:O)(C +, I +, J +, O)
 +
 +
    =  C +, O
 +
 +
n,w  =  "noble that is woman"
 +
 +
    =  (C:C +, D:D +, O:O)(B +, D +, E)
 +
 +
    =  D
 +
 +
w,n  =  "woman that is noble"
 +
 +
    =  (B:B +, D:D +, E:E)(C +, D +, O)
 +
 +
    =  D
 +
 +
Given a set X and a subset M c X, define e_M,
 +
the "idempotent representation" of M over X,
 +
as the 2-adic relation e_M c X x X which is
 +
the identity relation on M.  In other words,
 +
e_M = {<x, x> : x in M}.
 +
 +
Transposing this by steps into Peirce's notation:
 +
 +
e_M  =  {<x, x> : x in M}
 +
 +
    =  {x:x : x in M}
 +
 +
    =  Sum_X |x in M| x:x
 +
 +
'l'  =  "lover of ---"
 +
 +
's'  =  "servant of ---"
 +
 +
'l',  =  "lover that is --- of ---"
 +
 +
's',  =  "servant that is --- of ---"
 +
 +
| But not only may any absolute term be thus regarded as a relative term,
 +
| but any relative term may in the same way be regarded as a relative with
 +
| one correlate more.  It is convenient to take this additional correlate
 +
| as the first one.
 +
|
 +
| Then:
 +
|
 +
| 'l','s'w
 +
|
 +
| will denote a lover of a woman that is a servant of that woman.
 +
|
 +
| C.S. Peirce, CP 3.73
 +
 +
o~~~~~~~~~o~~~~+~~~~o~~~~~~~~~o~~~~~~~~~o~~~~+~~~~o~~~~~~~~~o
 +
o-----------------------------o-----------------------------o
 +
|  Objective Framework (OF)  | Interpretive Framework (IF) |
 +
o-----------------------------o-----------------------------o
 +
|          Objects          |            Signs            |
 +
o-----------------------------o-----------------------------o
 +
|                                                          |
 +
|          C  o---------------                            |
 +
|                                                          |
 +
|          F  o---------------                            |
 +
|                                                          |
 +
|          I  o---------------                            |
 +
|                                                          |
 +
|          O  o---------------                            |
 +
|                                                          |
 +
|          B  o---------------                            |
 +
|                                                          |
 +
|          D  o---------------                            |
 +
|                                                          |
 +
|          E  o---------------                            |
 +
|                                o "m"                    |
 +
|                                /                          |
 +
|                              /                          |
 +
|                              /                            |
 +
|          o  o  o-----------@                            |
 +
|                              \                            |
 +
|                              \                          |
 +
|                                \                          |
 +
|                                o                        |
 +
|                                                          |
 +
o-----------------------------o-----------------------------o
 +
 +
†‡||§¶
 +
@#||$%
 +
 +
quality, reflection, synecdoche
 +
 +
1.  neglect of
 +
2.  neglect of
 +
3.  neglect of nil?
 +
 +
Now, it's not the end of the story, of course, but it's a start.
 +
The significant thing is what is usually the significant thing
 +
in mathematics, at least, that two distinct descriptions refer
 +
to the same things.  Incidentally, Peirce is not really being
 +
as indifferent to the distinctions between signs and things
 +
as this ascii text makes him look, but uses a host of other
 +
type-faces to distinguish the types and the uses of signs.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
GR = Gary Richmond
 +
 +
GR: I wonder if the necessary "elementary triad" spoken of
 +
    below isn't somehow implicated in those discussions
 +
    "invoking a 'closure principle'".
 +
 +
GR, quoting CSP:
 +
 +
    | CP 1.292.  It can further be said in advance, not, indeed,
 +
    | purely a priori but with the degree of apriority that is
 +
    | proper to logic, namely, as a necessary deduction from
 +
    | the fact that there are signs, that there must be an
 +
    | elementary triad.  For were every element of the
 +
    | phaneron a monad or a dyad, without the relative
 +
    | of teridentity (which is, of course, a triad),
 +
    | it is evident that no triad could ever be
 +
    | built up.  Now the relation of every sign
 +
    | to its object and interpretant is plainly
 +
    | a triad.  A triad might be built up of
 +
    | pentads or of any higher perissad
 +
    | elements in many ways.  But it
 +
    | can be proved -- and really
 +
    | with extreme simplicity,
 +
    | though the statement of
 +
    | the general proof is
 +
    | confusing -- that no
 +
    | element can have
 +
    | a higher valency
 +
    | than three.
 +
 +
GR: (Of course this passage also directly relates
 +
    to the recent thread on Identity and Teridentity.)
 +
 +
Yes, generally speaking, I think that there are deep formal principles here
 +
that manifest themselves in these various guises:  the levels of intention
 +
or the orders of reflection, the sign relation, pragmatic conceivability,
 +
the generative sufficiency of 3-adic relations for all practical intents,
 +
and the irreducibility of continuous relations.  I have run into themes
 +
in combinatorics, group theory, and Lie algebras that are tantalizingly
 +
reminiscent of the things that Peirce says here, but it will take me
 +
some time to investigate them far enough to see what's going on.
 +
 +
GR: PS.  I came upon the above passage last night reading through
 +
    the Peirce selections in John J. Stuhr's 'Classical American
 +
    Philosophy:  Essential Readings and Interpretive Essays',
 +
    Oxford University, 1987 (the passage above is found on
 +
    pp 61-62), readily available in paperback in a new
 +
    edition, I believe.
 +
 +
GR: An aside:  These excerpts in Sturh include versions of a fascinating
 +
    "Intellectual Autobiography", Peirce's summary of his scientific,
 +
    especially, philosophic accomplishments.  I've seen them published
 +
    nowhere else.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BU = Ben Udell
 +
JA = Jon Awbrey
 +
 +
BU: I'm in the process of moving back to NYC and have had little opportunity
 +
    to do more than glance through posts during the past few weeks, but this
 +
    struck me because it sounds something I really would like to know about,
 +
    but I didn't understand it:
 +
 +
JA: Notice that Peirce follows the mathematician's usual practice,
 +
    then and now, of making the status of being an "individual" or
 +
    a "universal" relative to a discourse in progress.  I have come
 +
    to appreciate more and more of late how radically different this
 +
    "patchwork" or "piecewise" approach to things is from the way of
 +
    some philosophers who seem to be content with nothing less than
 +
    many worlds domination, which means that they are never content
 +
    and rarely get started toward the solution of any real problem.
 +
    Just my observation, I hope you understand.
 +
 +
BU: "Many worlds domination", "nothing less than many worlds domination" --
 +
    as opposed to the patchwork or piecewise approach.  What is many worlds
 +
    domination?  When I hear "many worlds" I think of Everett's Many Worlds
 +
    interpretation of quantum mechanics.
 +
 +
Yes, it is a resonance of Edward, Everett, and All the Other Whos in Whoville,
 +
but that whole microcosm is itself but the frumious reverberation of Leibniz's
 +
Maenadolatry.
 +
 +
More sequitur, though, this is an issue that has simmered beneath
 +
the surface of my consciousness for several decades now and only
 +
periodically percolates itself over the hyper-critical thrashold
 +
of expression.  Let me see if I can a better job of it this time.
 +
 +
The topic is itself a patchwork of infernally recurrent patterns.
 +
Here are a few pieces of it that I can remember arising recently:
 +
 +
| Zeroth Law Of Semantics
 +
|
 +
| Meaning is a privilege not a right.
 +
| Not all pictures depict.
 +
| Not all signs denote.
 +
|
 +
| Never confuse a property of a sign,
 +
| for instance, existence,
 +
| with a sign of a property,
 +
| for instance, existence.
 +
|
 +
| Taking a property of a sign,
 +
| for a sign of a property,
 +
| is the zeroth sign of
 +
| nominal thinking,
 +
| and the first
 +
| mistake.
 +
|
 +
| Also Sprach Zero*
 +
 +
A less catchy way of saying "meaning is a privilege not a right"
 +
would most likely be "meaning is a contingency not a necessity".
 +
But if I reflect on that phrase, it does not quite satisfy me,
 +
since a deeper lying truth is that contingency and necessity,
 +
connections in fact and connections beyond the reach of fact,
 +
depend on a line of distinction that is itself drawn on the
 +
scene of observation from the embodied, material, physical,
 +
non-point massive, non-purely-spectrelative point of view
 +
of an agent or community of interpretation, a discursive
 +
universe, an engauged interpretant, a frame of at least
 +
partial self-reverence, a hermeneutics in progress, or
 +
a participant observer.  In short, this distinction
 +
between the contingent and the necessary is itself
 +
contingent, which means, among other things, that
 +
signs are always indexical at some least quantum.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
JR = Joe Ransdell
 +
 +
JR: Would the Kripke conception of the "rigid designator" be an instance
 +
    of the "many worlds domination"?  I was struck by your speaking of
 +
    the "patchwork or piecewise" approach as well in that it seemed to
 +
    me you might be expressing the same general idea that I have usually
 +
    thought of in terms of contextualism instead:  I mean the limits it
 +
    puts upon what you can say a priori if you really take contextualism
 +
    seriously, which is the same as recognizing indexicality as incapable
 +
    of elimination, I think.
 +
 +
Yes, I think this is the same ballpark of topics.
 +
I can't really speak for what Kripke had in mind,
 +
but I have a practical acquaintance with the way
 +
that some people have been trying to put notions
 +
like this to work on the applied ontology scene,
 +
and it strikes me as a lot of nonsense.  I love
 +
a good parallel worlds story as much as anybody,
 +
but it strikes me that many worlds philosophers
 +
have the least imagination of anybody as to what
 +
an alternative universe might really be like and
 +
so I prefer to read more creative writers when it
 +
comes to that.  But serially, folks, I think that
 +
the reason why some people evidently feel the need
 +
for such outlandish schemes -- and the vast majority
 +
of the literature on counterfactual conditionals falls
 +
into the same spaceboat as this -- is simply that they
 +
have failed to absorb, through the fault of Principian
 +
filters, a quality that Peirce's logic is thoroughly
 +
steeped in, namely, the functional interpretation
 +
of logical terms, that is, as signs referring to
 +
patterns of contingencies.  It is why he speaks
 +
more often, and certainly more sensibly and to
 +
greater effect, of "conditional generals" than
 +
of "modal subjunctives".  This is also bound up
 +
with that element of sensibility that got lost in
 +
the transition from Peircean to Fregean quantifiers.
 +
Peirce's apriorities are always hedged with risky bets.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BU = Benjamin Udell
 +
 +
BU: I wish I had more time to ponder the "many-worlds" issue (& that my books
 +
    were not currently disappearing into heavily taped boxes).  I had thought
 +
    of the piecemeal approach's opposite as the attempt to build a kind of
 +
    monolithic picture, e.g., to worry that there is not an infinite number
 +
    of particles in the physical universe for the infinity integers.  But
 +
    maybe the business with rigid designators & domination of many worlds
 +
    has somehow to do with monolithism.
 +
 +
Yes, that's another way of saying it.  When I look to my own priorities,
 +
my big worry is that logic as a discipline is not fulfilling its promise.
 +
I have worked in too many settings where the qualitative researchers and
 +
the quantitative researchers could barely even talk to one an Other with
 +
any understanding, and this I recognized as a big block to inquiry since
 +
our first notice of salient facts and significant phenomena is usually
 +
in logical, natural language, or qualitative forms, while our eventual
 +
success in resolving anomalies and solving practical problems depends
 +
on our ability to formalize, operationalize, and quantify the issues,
 +
even if only to a very partial degree, as it generally turns out.
 +
 +
When I look to the history of how logic has been deployed in mathematics,
 +
and through those media in science generally, it seems to me that the
 +
Piece Train started to go off track with the 'Principia Mathematica'.
 +
All pokes in the rib aside, however, I tend to regard this event
 +
more as the symptom of a localized cultural phenomenon than as
 +
the root cause of the broader malaise.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
CG = Clark Goble
 +
JA = Jon Awbrey
 +
 +
JA, quoting CSP:
 +
 +
    | For example,
 +
    |
 +
    | f + u
 +
    |
 +
    | means all Frenchmen besides all violinists, and,
 +
    | therefore, considered as a logical term, implies
 +
    | that all French violinists are 'besides themselves'.
 +
 +
CG: Could you clarify your use of "besides"?
 +
 +
CG: I think I am following your thinking in that you
 +
    don't want the logical terms to be considered
 +
    to have any necessary identity between them.
 +
    Is that right?
 +
 +
I use vertical sidebars "|" for long quotations, so this
 +
is me quoting Peirce at CP 3.67 who is explaining in an
 +
idiomatic way Boole's use of the plus sign for a logical
 +
operation that is strictly speaking limited to terms for
 +
mutually exclusive classes.  The operation would normally
 +
be extended to signify the "symmetric difference" operator.
 +
But Peirce is saying that he prefers to use the sign "+,"
 +
for inclusive disjunction, corresponding to the union of
 +
the associated classes.  Peirce calls Boole's operation
 +
"invertible" because it amounts to the sum operation in
 +
a field, whereas the inclusive disjunction or union is
 +
"non-invertible", since knowing that A |_| B = C does
 +
not allow one to say determinately that A = C - B.
 +
I can't recall if Boole uses this 'besides' idiom,
 +
but will check later.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
CG = Clark Goble
 +
JA = Jon Awbrey
 +
 +
JA: I use vertical sidebars "|" for long quotations, so this
 +
    is me quoting Peirce at CP 3.67 who is explaining in an
 +
    idiomatic way Boole's use of the plus sign for a logical
 +
    operation that is strictly speaking limited to terms for
 +
    mutually exclusive classes.
 +
 +
CG: Is that essay related to any of the essays
 +
    in the two volume 'Essential Peirce'?  I'm
 +
    rather interested in how he speaks there.
 +
 +
No, the EP volumes are extremely weak on logical selections.
 +
I see nothing there that deals with the logic of relatives.
 +
 +
JA: But Peirce is saying that he prefers to use the sign "+,"
 +
    for inclusive disjunction, corresponding to the union of
 +
    the associated classes.
 +
 +
CG: The reason I asked was more because it seemed
 +
    somewhat interesting in light of the logic of
 +
    operators in quantum mechanics.  I was curious
 +
    if the use of "beside" might relate to that.
 +
    But from what you say it probably was just me
 +
    reading too much into the quote.  The issue of
 +
    significance was whether the operation entailed
 +
    the necessity of mutual exclusivity or whether
 +
    some relationship between the classes might be
 +
    possible.  I kind of latched on to Peirce's
 +
    odd statement about "all French violinists
 +
    are 'beside themselves'".
 +
 +
CG: Did Peirce have anything to say about
 +
    what we'd call non-commuting operators?
 +
 +
In general, 2-adic relative terms are non-commutative.
 +
For example, a brother of a mother is not identical to
 +
a mother of a brother.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
GR = Gary Richmond
 +
 +
GR: I am very much enjoying, which is to say,
 +
    learning from your interlacing commentary
 +
    on Peirce's 1870 "Logic of Relatives" paper.
 +
 +
GR: What an extraordinary paper the 1870 "LOG" is!  Your notes helped
 +
    me appreciate the importance of the unanticipated proposal of P's
 +
    to "assign to all logical terms, numbers".  On the other hand,
 +
    the excerpts suggested to we why Peirce finally framed his
 +
    Logic of Relatives into graphical form.  Still, I think
 +
    that a thorough examination of the 1970 paper might
 +
    serve as propaedeutic (and of course, much more)
 +
    for the study of the alpha and beta graphs.
 +
 +
Yes, there's gold in them thar early logic papers that has been "panned"
 +
but nowhere near mined in depth yet.  The whole quiver of arrows between
 +
terms and numbers harks back to the 'numeri characteristici' of  Leibniz,
 +
of course, but Leibniz attended more on the intensional chains of being
 +
while Peirce will here start to "escavate" the extensional hierarchies.
 +
 +
I consider myself rewarded that you see the incipient impulse toward
 +
logical graphs, as one of the most striking things to me about this
 +
paper is to see these precursory seeds already planted here within
 +
it and yet to know how long it will take them to sprout and bloom.
 +
 +
Peirce is obviously struggling to stay within the linotyper's art --
 +
a thing that we, for all our exorbitant hype about markable text,
 +
are still curiously saddled with -- but I do not believe that it
 +
is possible for any mind equipped with a geometrical imagination
 +
to entertain these schemes for connecting up terminological hubs
 +
with their terminological terminals without perforce stretching
 +
imaginary strings between the imaginary gumdrops.
 +
 +
GR: I must say though that the pace at which you've been throwing this at us
 +
    is not to be kept up with by anyone I know "in person or by reputation".
 +
    I took notes on the first 5 or 6 Notes, but can now just barely find
 +
    time to read through your posts.
 +
 +
Oh, I was trying to burrow as fast as I could toward the more untapped veins --
 +
I am guessing that things will probably "descalate" a bit over the next week,
 +
but then, so will our attention spans ...
 +
 +
Speaking of which, I will have to break here, and pick up the rest later ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 8
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
GR = Gary Richmond
 +
 +
GR: In any event, I wish that you'd comment on Note 5 more directly (though
 +
    you do obliquely in your own diagramming of "every [US] Vice-President(s) ...
 +
    [who is] every President(s) of the US Senate".
 +
 +
There are several layers of things to say about that,
 +
and I think that it would be better to illustrate the
 +
issues by way of the examples that Peirce will soon be
 +
getting to, but I will see what I can speak to for now.
 +
 +
GR: But what interested me even more in LOR, Note 5, was the sign < ("less than"
 +
    joined to the sign of identity = to yield P's famous sign -< (or more clearly,
 +
    =<) of inference, which combines the two (so that -< (literally, "as small as")
 +
    means "is".  I must say I both "get" this and don't quite (Peirce's example(s) of
 +
    the frenchman helped a little).  Perhaps your considerably more mathematical mind
 +
    can help clarify this for a non-mathematician such as myself.  (My sense is that
 +
    "as small as" narrows the terms so that "everything that occurs in the conclusion
 +
    is already contained in the premise.)  I hope I'm not being obtuse here.  I'm sure
 +
    it's "all too simple for words".
 +
 +
Then let us draw a picture.
 +
 +
"(F (G))", read "not F without G", means that F (G), that is, F and not G,
 +
is the only region exempted from the occupation of being in this universe:
 +
 +
o-----------------------------------------------------------o
 +
|`X`````````````````````````````````````````````````````````|
 +
|```````````````````````````````````````````````````````````|
 +
|`````````````o-------------o```o-------------o`````````````|
 +
|````````````/              \`/```````````````\````````````|
 +
|```````````/                o`````````````````\```````````|
 +
|``````````/                /`\`````````````````\``````````|
 +
|`````````/                /```\`````````````````\`````````|
 +
|````````/                /`````\`````````````````\````````|
 +
|```````o                o```````o`````````````````o```````|
 +
|```````|                |```````|`````````````````|```````|
 +
|```````|                |```````|`````````````````|```````|
 +
|```````|        F        |```````|````````G````````|```````|
 +
|```````|                |```````|`````````````````|```````|
 +
|```````|                |```````|`````````````````|```````|
 +
|```````o                o```````o`````````````````o```````|
 +
|````````\                \`````/`````````````````/````````|
 +
|`````````\                \```/`````````````````/`````````|
 +
|``````````\                \`/`````````````````/``````````|
 +
|```````````\                o`````````````````/```````````|
 +
|````````````\              /`\```````````````/````````````|
 +
|`````````````o-------------o```o-------------o`````````````|
 +
|```````````````````````````````````````````````````````````|
 +
|```````````````````````````````````````````````````````````|
 +
o-----------------------------------------------------------o
 +
 +
Collapsing the vacuous region like soapfilm popping on a wire frame,
 +
we draw the constraint (F (G)) in the following alternative fashion:
 +
 +
o-----------------------------------------------------------o
 +
|`X`````````````````````````````````````````````````````````|
 +
|```````````````````````````````````````````````````````````|
 +
|```````````````````````````````o-------------o`````````````|
 +
|``````````````````````````````/```````````````\````````````|
 +
|`````````````````````````````o`````````````````\```````````|
 +
|````````````````````````````/`\`````````````````\``````````|
 +
|```````````````````````````/```\`````````````````\`````````|
 +
|``````````````````````````/`````\`````````````````\````````|
 +
|`````````````````````````o```````o`````````````````o```````|
 +
|`````````````````````````|```````|`````````````````|```````|
 +
|`````````````````````````|```````|`````````````````|```````|
 +
|`````````````````````````|```F```|````````G````````|```````|
 +
|`````````````````````````|```````|`````````````````|```````|
 +
|`````````````````````````|```````|`````````````````|```````|
 +
|`````````````````````````o```````o`````````````````o```````|
 +
|``````````````````````````\`````/`````````````````/````````|
 +
|```````````````````````````\```/`````````````````/`````````|
 +
|````````````````````````````\`/`````````````````/``````````|
 +
|`````````````````````````````o`````````````````/```````````|
 +
|``````````````````````````````\```````````````/````````````|
 +
|```````````````````````````````o-------------o`````````````|
 +
|```````````````````````````````````````````````````````````|
 +
|```````````````````````````````````````````````````````````|
 +
o-----------------------------------------------------------o
 +
 +
So, "(F (G))", "F => G", "F =< G", "F -< G", "F c G",
 +
under suitable mutations of interpretation, are just
 +
so many ways of saying that the denotation of "F" is
 +
contained within the denotation of "G".
 +
 +
Now, let us look to the "characteristic functions" or "indicator functions"
 +
of the various regions of being.  It is frequently convenient to ab-use the
 +
same letters for them and merely keep a variant interpretation "en thy meme",
 +
but let us be more meticulous here, and reserve the corresponding lower case
 +
letters "f" and "g" to denote the indicator functions of the regions F and G,
 +
respectively.
 +
 +
Taking B = {0, 1} as the boolean domain, we have:
 +
 +
f, g : X -> B
 +
 +
(f^(-1))(1)  =  F
 +
 +
(g^(-1))(1)  =  G
 +
 +
In general, for h : X -> B, an expression like "(h^(-1))(1)"
 +
can be read as "the inverse of h evaluated at 1", in effect,
 +
denoting the set of points in X where h evaluates to "true".
 +
This is called the "fiber of truth" in h, and I have gotten
 +
where I like to abbreviate it as "[|h|]".
 +
 +
Accordingly, we have:
 +
 +
F  =  [|f|]  =  (f^(-1))(1)  c  X
 +
 +
G  =  [|g|]  =  (g^(-1))(1)  c  X
 +
 +
This brings us to the question, what sort
 +
of "functional equation" between f and g
 +
goes with the regional constraint (F (G))?
 +
 +
Just this, that f(x) =< g(x) for all x in X,
 +
where the '=<' relation on the values in B
 +
has the following operational table for
 +
the pairing "row head =< column head".
 +
 +
o---------o---------o---------o
 +
|  =<    #    0    |    1    |
 +
o=========o=========o=========o
 +
|    0    #    1    |    1    |
 +
o---------o---------o---------o
 +
|    1    #    0    |    1    |
 +
o---------o---------o---------o
 +
 +
And this, of course, is the same thing as the truth table
 +
for the conditional connective or the implication relation.
 +
 +
GR: By the way, in the semiosis implied by the modal gamma graphs,
 +
    could -< (were it used there, which of course it is not) ever
 +
    be taken to mean,"leads to" or "becomes" or "evolves into"?
 +
    I informally use it that way myself, using the ordinary
 +
    arrow for implication.
 +
 +
I am a bit insensitive to the need for modal logic,
 +
since necessity in mathematics always seems to come
 +
down to being a matter of truth for all actual cases,
 +
if under an expanded sense of actuality that makes it
 +
indiscernible from possibility, so I must beg off here.
 +
But there are places where Peirce makes a big deal about
 +
the advisability of drawing the '-<' symbol in one fell
 +
stroke of the pen, kind of like a "lazy gamma" -- an old
 +
texican cattle brand -- and I have seen another place where
 +
he reads "A -< B" as "A, in every way that it can be, is B",
 +
as if this '-<' fork in the road led into a veritable garden
 +
of branching paths.
 +
 +
And out again ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
GR = Gary Richmond
 +
JA = Jon Awbrey
 +
 +
JA: I am a bit insensitive to the need for modal logic,
 +
    since necessity in mathematics always seems to come
 +
    down to being a matter of truth for all actual cases,
 +
    if under an expanded sense of actuality that makes it
 +
    indiscernible from possibility, so I must beg off here.
 +
 +
GR: I cannot agree with you regarding modal logic.  Personally
 +
    I feel that the gamma part of the EG's is of the greatest
 +
    interest and potential importance, and as Jay Zeman has
 +
    made clear in his dissertation, Peirce certainly thought
 +
    this as well.
 +
 +
You disagree that I am insensitive?  Well, certainly nobody has ever done that before!
 +
No, I phrased it that way to emphasize the circumstance that it ever hardly comes up
 +
as an issue within the limited purview of my experience, and when it does -- as in
 +
topo-logical boundary situations -- it seems to require a sort of analysis that
 +
doesn't comport all that well with the classical modes and natural figures of
 +
speech about it.  Then again, I spent thirty years trying to motorize Alpha,
 +
have only a few good clues how I would go about Beta, and so Gamma doesn't
 +
look like one of those items on my plate.
 +
 +
Speeching Of Which ---
 +
Best Of The Season ...
 +
And Happy Trailing ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 10
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
JA = Jon Awbrey
 +
 +
BM: Thanks for your very informative talk.  There
 +
    is a point that I did not understand in note 35:
 +
 +
JA: If we operate in accordance with Peirce's example of `g`'o'h
 +
    as the "giver of a horse to an owner of that horse", then we
 +
    may assume that the associative law and the distributive law
 +
    are by default in force, allowing us to derive this equation:
 +
 +
JA: 'l','s'w  =  'l','s'(B +, D +, E)  =  'l','s'B +, 'l','s'D +, 'l','s'E
 +
 +
BM: May be because language or more probably my lack of training in logic, what
 +
    does mean that "associative law and distributive law are by default in force"?
 +
 +
Those were some tricky Peirces,
 +
and I was trying to dodge them
 +
as artful as could be, but now
 +
you have fastly apprehended me!
 +
 +
It may be partly that I left out the initial sections of this paper where Peirce
 +
discusses how he will regard the ordinarily applicable principles in the process
 +
of trying to extend and generalize them (CP 3.45-62), but there may be also an
 +
ambiguity in Peirce's use of the phrase "absolute conditions" (CP 3.62-68).
 +
Does he mean "absolutely necessary", "indispensable", "inviolate", or
 +
does he mean "the conditions applying to the logic of absolute terms",
 +
in which latter case we would expect to alter them sooner or later?
 +
 +
We lose the commutative law, xy = yx, as soon as we extend to 2-adic relations,
 +
but keep the associative law, x(yz) = (xy)z, as the multiplication of 2-adics
 +
is the logical analogue of ordinary matrix multiplication, and Peirce like
 +
most mathematicians treats the double distributive law, x(y + z) = xy + xz
 +
and (x + y)z = xz + yz, and as something that must be striven to preserve
 +
as far as possible.
 +
 +
Strictly speaking, Peirce is already using a principle that goes beyond
 +
the ordinary associative law, but that is recognizably analogous to it,
 +
for example, in the modified Othello case, where (J:J:D)(J:D)(D) = J.
 +
If it were strictly associative, then we would have the following:
 +
 +
1.  (J:J:D)((J:D)(D))  =  (J:J:D)(J)  =  0?
 +
 +
2.  ((J:J:D)(J:D))(D)  =  (J)(D)  =  0?
 +
 +
In other words, the intended relational linkage would be broken.
 +
However, the type of product that Peirce is taking for granted
 +
in this situation often occurs in mathematics in just this way.
 +
There is another location where he comments more fully on this,
 +
but I have the sense that it was a late retrospective remark,
 +
and I do not recall if it was in CP or in the microfilm MS's
 +
that I read it.
 +
 +
By "default" conditions I am referring more or less to what
 +
Peirce says at the end of CP 3.69, where he use an argument
 +
based on the distributive principle to rationalize the idea
 +
that 'A term multiplied by two relatives shows that the same
 +
individual is in the two relations'.  This means, for example,
 +
that one can let "`g`'o'h", without subjacent marks or numbers,
 +
be interpreted on the default convention of "overlapping scopes",
 +
where the two correlates of `g` are given by the next two terms
 +
in line, namely, 'o' and h, and the single correlate of 'o' is
 +
given by the very next term in line, namely, h.  Thus, it is
 +
only when this natural scoping cannot convey the intended
 +
sense that we have to use more explicit mark-up devices.
 +
 +
BM: About another point:  do you think that the LOR could be of some help to solve
 +
    the puzzle of the "second way of dividing signs" where CSP concludes that 66
 +
    classes could be made out of the 10 divisions (Letters to lady Welby)?
 +
    (As I see them, the ten divisions involve a mix of relative terms,
 +
    dyadic relations and a triadic one.  In order to make 66 classes
 +
    it is clear that these 10 divisions have to be stated under some
 +
    linear order.  The nature of this order is at the bottom of the
 +
    disagreements on the subject).
 +
 +
This topic requires a longer excuse from me
 +
than I am able to make right now, but maybe
 +
I'll get back to it later today or tomorrow.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 11
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
 +
BM: About another point:  do you think that the LOR could be of some help
 +
    to solve the puzzle of the "second way of dividing signs" where CSP
 +
    concludes that 66 classes could be made out of the 10 divisions
 +
    (Letters to lady Welby)?  (As I see them, the ten divisions
 +
    involve a mix of relative terms, dyadic relations and
 +
    a triadic one.  In order to make 66 classes it is
 +
    clear that these 10 divisions have to be stated
 +
    under some linear order.  The nature of this
 +
    order is at the bottom of the disagreements
 +
    on the subject).
 +
 +
Yes.  At any rate, I have a pretty clear sense from reading Peirce's work
 +
in the period 1865-1870 that the need to understand the function of signs
 +
in scientific inquiry is one of the main reasons he found himself forced
 +
to develop both the theory of information and the logic of relatives.
 +
 +
Peirce's work of this period is evenly distributed across the extensional
 +
and intensional pans of the balance in a way that is very difficult for us
 +
to follow anymore.  I remember when I started looking into this I thought of
 +
myself as more of an "intensional, synthetic" than an "extensional, analytic"
 +
type of thinker, but that seems like a long time ago, as it soon became clear
 +
that much less work had been done in the Peirce community on the extensional
 +
side of things, while that was the very facet that needed to be polished up
 +
in order to reconnect logic with empirical research and mathematical models.
 +
So I fear that I must be content that other able people are working on the
 +
intensional classification of sign relations.
 +
 +
Still, the way that you pose the question is very enticing,
 +
so maybe it is time for me to start thinking about this
 +
aspect of sign relations again, if you could say more
 +
about it.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 12
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
 +
BM: The pairing "intensional, synthetic" against the other "extensional, analytic"
 +
    is not one that I would have thought so.  I would have paired synthetic with
 +
    extensional because synthesis consists in adding new facts to an already made
 +
    conception.  On the other side analysis looks to be the determination of
 +
    features while neglecting facts.  But may be there is something like
 +
    a symmetry effect leading to the same view from two different points.
 +
 +
Oh, it's not too important, as I don't put a lot of faith in such divisions,
 +
and the problem for me is always how to integrate the facets of the object,
 +
or the faculties of the mind -- but there I go being synthetic again!
 +
 +
I was only thinking of a conventional contrast that used to be drawn
 +
between different styles of thinking in mathematics, typically one
 +
points to Descartes, and the extensionality of analytic geometry,
 +
versus Desargues, and the intensionality of synthetic geometry.
 +
 +
It may appear that one has side-stepped the issue of empiricism
 +
that way, but then all that stuff about the synthetic a priori
 +
raises its head, and we have Peirce's insight that mathematics
 +
is observational and even experimental, and so I must trail off
 +
into uncoordinated elliptical thoughts ...
 +
 +
The rest I have to work at a while, and maybe go back to the Welby letters.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 13
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
 +
BM: I will try to make clear the matter, at least as far as I understand it
 +
    for now.  We can summarize in a table the 10 divisions with their number
 +
    in a first column, their title in current (peircean) language in the second
 +
    and some kind of logical notation in the third.  The sources come mainly from
 +
    the letters to Lady Welby.  While the titles come from CP 8.344, the third column
 +
    comes from my own interpretation.
 +
 +
BM: So we get:
 +
 +
I    - According to the Mode of Apprehension of the Sign itself            - S
 +
II  - According to the Mode of Presentation of the Immediate Object        - Oi
 +
III  - According to the Mode of Being of the Dynamical Object              - Od
 +
IV  - According to the Relation of the Sign to its Dynamical Object        - S-Od
 +
V    - According to the Mode of Presentation of the Immediate Interpretant  - Ii
 +
VI  - According to the Mode of Being of the Dynamical Interpretant        - Id
 +
VII  - According to the relation of the Sign to the Dynamical Interpretant  - S-Id
 +
VIII - According to the Nature of the Normal Interpretant                  - If
 +
IX  - According to the the relation of the Sign to the Normal Interpretant - S-If
 +
X    - According to the Triadic Relation of the Sign to its Dynamical Object
 +
      and to its Normal Interpretant                                      - S-Od-If
 +
 +
For my future study, I will reformat the table in a way that I can muse upon.
 +
I hope the roman numerals have not become canonical, as I cannot abide them.
 +
 +
Table.  Ten Divisions of Signs (Peirce, Morand)
 +
o---o---------------o------------------o------------------o---------------o
 +
|  | According To: | Of:              | To:              |              |
 +
o===o===============o==================o==================o===============o
 +
| 1 | Apprehension  | Sign Itself      |                  | S            |
 +
| 2 | Presentation  | Immediate Object |                  | O_i          |
 +
| 3 | Being        | Dynamical Object |                  | O_d          |
 +
| 4 | Relation      | Sign            | Dynamical Object | S : O_d      |
 +
o---o---------------o------------------o------------------o---------------o
 +
| 5 | Presentation  | Immediate Interp |                  | I_i          |
 +
| 6 | Being        | Dynamical Interp |                  | I_d          |
 +
| 7 | Relation      | Sign            | Dynamical Interp | S : I_d      |
 +
o---o---------------o------------------o------------------o---------------o
 +
| 8 | Nature        | Normal Interp    |                  | I_f          |
 +
| 9 | Relation      | Sign            | Normal Interp    | S : I_f      |
 +
o---o---------------o------------------o------------------o---------------o
 +
| A | Relation      | Sign            | Dynamical Object |              |
 +
|  |              |                  | & Normal Interp  | S : O_d : I_f |
 +
o---o---------------o------------------o------------------o---------------o
 +
 +
Just as I have always feared, this classification mania
 +
appears to be communicable!  But now I must definitely
 +
review the Welby correspondence, as all this stuff was
 +
a blur to my sensibilities the last 10 times I read it.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 14
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
 +
[Table.  Ten Divisions of Signs (Peirce, Morand)]
 +
 +
BM: Yes this is clearer (in particular in expressing relations with :)
 +
 +
This is what Peirce used to form elementary relatives, for example,
 +
o:s:i = <o, s, i>, and I find it utterly ubertous in a wide variety
 +
of syntactic circumstances.
 +
 +
BM: I suggest making a correction to myself if
 +
    the table is destinate to become canonic.
 +
 +
Hah!  Good one!
 +
 +
BM: I probably made a too quick jump from Normal Interpretant to Final Interpretant.
 +
    As we know, the final interpretant, the ultimate one is not a sign for Peirce
 +
    but a habit.  So for the sake of things to come it would be more careful to
 +
    retain I_n in place of I_f for now.
 +
 +
This accords with my understanding of how the word is used in mathematics.
 +
In my own work it has been necessary to distinguish many different species
 +
of expressions along somewhat similar lines, for example:  arbitrary, basic,
 +
canonical, decidable, normal, periodic, persistent, prototypical, recurrent,
 +
representative, stable, typical, and so on.  So I will make the changes below:
 +
 +
Table.  Ten Divisions of Signs (Peirce, Morand)
 +
o---o---------------o------------------o------------------o---------------o
 +
|  | According To: | Of:              | To:              |              |
 +
o===o===============o==================o==================o===============o
 +
| 1 | Apprehension  | Sign Itself      |                  | S            |
 +
| 2 | Presentation  | Immediate Object |                  | O_i          |
 +
| 3 | Being        | Dynamical Object |                  | O_d          |
 +
| 4 | Relation      | Sign            | Dynamical Object | S : O_d      |
 +
o---o---------------o------------------o------------------o---------------o
 +
| 5 | Presentation  | Immediate Interp |                  | I_i          |
 +
| 6 | Being        | Dynamical Interp |                  | I_d          |
 +
| 7 | Relation      | Sign            | Dynamical Interp | S : I_d      |
 +
o---o---------------o------------------o------------------o---------------o
 +
| 8 | Nature        | Normal Interp    |                  | I_n          |
 +
| 9 | Relation      | Sign            | Normal Interp    | S : I_n      |
 +
o---o---------------o------------------o------------------o---------------o
 +
| A | Tri. Relation | Sign            | Dynamical Object |              |
 +
|  |              |                  | & Normal Interp  | S : O_d : I_n |
 +
o---o---------------o------------------o------------------o---------------o
 +
 +
BM: Peirce gives the following definition (CP 8.343):
 +
 +
BM, quoting CSP:
 +
 +
    | It is likewise requisite to distinguish
 +
    | the 'Immediate Interpretant', i.e. the
 +
    | Interpretant represented or signified in
 +
    | the Sign, from the 'Dynamic Interpretant',
 +
    | or effect actually produced on the mind
 +
    | by the Sign;  and both of these from
 +
    | the 'Normal Interpretant', or effect
 +
    | that would be produced on the mind by
 +
    | the Sign after sufficient development
 +
    | of thought.
 +
    |
 +
    | C.S. Peirce, 'Collected Papers', CP 8.343.
 +
 +
Well, you've really tossed me in the middle of the briar patch now!
 +
I must continue with my reading from the 1870 LOR, but now I have
 +
to add to my do-list the problems of comparing the whole variorum
 +
of letters and drafts of letters to Lady Welby.  I only have the
 +
CP 8 and Wiener versions here, so I will depend on you for ample
 +
excerpts from the Lieb volume.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 15
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I will need to go back and pick up the broader contexts of your quotes.
 +
For ease of study I break Peirce's long paragraphs into smaller pieces.
 +
 +
| It seems to me that one of the first useful steps toward a science
 +
| of 'semeiotic' ([Greek 'semeiootike']), or the cenoscopic science
 +
| of signs, must be the accurate definition, or logical analysis,
 +
| of the concepts of the science.
 +
|
 +
| I define a 'Sign' as anything which on the one hand
 +
| is so determined by an Object and on the other hand
 +
| so determines an idea in a person's mind, that this
 +
| latter determination, which I term the 'Interpretant'
 +
| of the sign, is thereby mediately determined by that
 +
| Object.
 +
|
 +
| A sign, therefore, has a triadic relation to
 +
| its Object and to its Interpretant.  But it is
 +
| necessary to distinguish the 'Immediate Object',
 +
| or the Object as the Sign represents it, from
 +
| the 'Dynamical Object', or really efficient
 +
| but not immediately present Object.
 +
|
 +
| It is likewise requisite to distinguish
 +
| the 'Immediate Interpretant', i.e. the
 +
| Interpretant represented or signified in
 +
| the Sign, from the 'Dynamic Interpretant',
 +
| or effect actually produced on the mind
 +
| by the Sign;  and both of these from
 +
| the 'Normal Interpretant', or effect
 +
| that would be produced on the mind by
 +
| the Sign after sufficient development
 +
| of thought.
 +
|
 +
| On these considerations I base a recognition of ten respects in which Signs
 +
| may be divided.  I do not say that these divisions are enough.  But since
 +
| every one of them turns out to be a trichotomy, it follows that in order
 +
| to decide what classes of signs result from them, I have 3^10, or 59049,
 +
| difficult questions to carefully consider;  and therefore I will not
 +
| undertake to carry my systematical division of signs any further,
 +
| but will leave that for future explorers.
 +
|
 +
| C.S. Peirce, 'Collected Papers', CP 8.343.
 +
 +
You never know when the future explorer will be yourself.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 16
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Burks, the editor of CP 8, attaches this footnote
 +
to CP 8.342-379, "On the Classification of Signs":
 +
 +
| From a partial draft of a letter to Lady Welby, bearing
 +
| the dates of 24, 25, and 28 December 1908, Widener IB3a,
 +
| with an added quotation in 368n23.  ...
 +
 +
There is a passage roughly comparable to CP 8.343 in a letter
 +
to Lady Welby dated 23 December 1908, pages 397-409 in Wiener,
 +
which is incidentally the notorious "sop to Cerberus" letter:
 +
 +
| It is usual and proper to distinguish two Objects of a Sign,
 +
| the Mediate without, and the Immediate within the Sign.  Its
 +
| Interpretant is all that the Sign conveys:  acquaintance with
 +
| its Object must be gained by collateral experience.
 +
|
 +
| The Mediate Object is the Object outside of the Sign;  I call
 +
| it the 'Dynamoid' Object.  The Sign must indicate it by a hint;
 +
| and this hint, or its substance, is the 'Immediate' Object.
 +
|
 +
| Each of these two Objects may be said to be capable of either of
 +
| the three Modalities, though in the case of the Immediate Object,
 +
| this is not quite literally true.
 +
|
 +
| Accordingly, the Dynamoid Object may be a Possible;  when I term
 +
| the Sign an 'Abstractive';  such as the word Beauty;  and it will be
 +
| none the less an Abstractive if I speak of "the Beautiful", since it is
 +
| the ultimate reference, and not the grammatical form, that makes the sign
 +
| an 'Abstractive'.
 +
|
 +
| When the Dynamoid Object is an Occurrence (Existent thing or Actual fact
 +
| of past or future), I term the Sign a 'Concretive';  any one barometer
 +
| is an example;  and so is a written narrative of any series of events.
 +
|
 +
| For a 'Sign' whose Dynamoid Object is a Necessitant, I have at present
 +
| no better designation than a 'Collective', which is not quite so bad a
 +
| name as it sounds to be until one studies the matter:  but for a person,
 +
| like me, who thinks in quite a different system of symbols to words, it
 +
| is so awkward and often puzzling to translate one's thought into words!
 +
|
 +
| If the Immediate Object is a "Possible", that is, if the Dynamoid Object
 +
| is indicated (always more or less vaguely) by means of its Qualities, etc.,
 +
| I call the Sign a 'Descriptive';
 +
|
 +
| if the Immediate is an Occurrence, I call the Sign a 'Designative';
 +
|
 +
| and if the Immediate Object is a Necessitant, I call the Sign a
 +
| 'Copulant';  for in that case the Object has to be so identified
 +
| by the Interpreter that the Sign may represent a necessitation.
 +
| My name is certainly a temporary expedient.
 +
|
 +
| It is evident that a possible can determine nothing but a Possible,
 +
| it is equally so that a Necessitant can be determined by nothing but
 +
| a Necessitant.  Hence it follows from the Definition of a Sign that
 +
| since the Dynamoid Object determines the Immediate Object,
 +
|
 +
|    Which determines the Sign itself,
 +
|    which determines the Destinate Interpretant
 +
|    which determines the Effective Interpretant
 +
|    which determines the Explicit Interpretant
 +
|
 +
| the six trichotomies, instead of determining 729 classes of signs,
 +
| as they would if they were independent, only yield 28 classes;
 +
| and if, as I strongly opine (not to say almost prove), there
 +
| are four other trichotomies of signs of the same order of
 +
| importance, instead of making 59,049 classes, these will
 +
| only come to 66.
 +
|
 +
| The additional 4 trichotomies are undoubtedly, first:
 +
|
 +
|    Icons*,  Symbols,  Indices,
 +
|
 +
|*(or Simulacra, Aristotle's 'homoiomata'), caught from Plato, who I guess took it
 +
| from the Mathematical school of logic, for it earliest appears in the 'Phaedrus'
 +
| which marks the beginning of Plato's being decisively influenced by that school.
 +
| Lutoslowski is right in saying that the 'Phaedrus' is later than the 'Republic'
 +
| but his date 379 B.C. is about eight years too early.
 +
|
 +
| and then 3 referring to the Interpretants.  One of these I am pretty confident
 +
| is into:  'Suggestives', 'Imperatives', 'Indicatives', where the Imperatives
 +
| include the Interrogatives.  Of the other two I 'think' that one must be
 +
| into Signs assuring their Interpretants by:
 +
|
 +
|    Instinct,  Experience,  Form.
 +
|
 +
| The other I suppose to be what, in my 'Monist'
 +
| exposition of Existential Graphs, I called:
 +
|
 +
|    Semes,  Phemes,  Delomes.
 +
|
 +
| CSP, 'Selected Writings', pp. 406-408.
 +
|
 +
|'Charles S. Peirce:  Selected Writings (Values in a Universe of Chance)',
 +
| edited with an introduction and notes by Philip P. Wiener, Dover,
 +
| New York, NY, 1966.  Originally published under the subtitle
 +
| in parentheses above, Doubleday & Company, 1958.
 +
 +
But see CP 4.549-550 for a significant distinction between
 +
the categories (or modalities) and the orders of intention.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 17
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
JA = Jon Awbrey
 +
 +
JA: In closing, observe that the teridentity relation has turned up again
 +
    in this context, as the second comma-ing of the universal term itself:
 +
 +
    1,, = B:B:B +, C:C:C +, D:D:D +, E:E:E +, I:I:I +, J:J:J +, O:O:O.
 +
 +
HC: I see that you've come around to a mention of teridentity again, Jon.
 +
    Still, if I recall the prior discussions, then no one doubts that we
 +
    can have a system of notation in which teridentity appears (I don't
 +
    actually see it here).
 +
 +
Perhaps we could get at the root of the misunderstanding
 +
if you tell me why you don't actually see the concept of
 +
teridentity being exemplified here.
 +
 +
If it's only a matter of having lost the context of the
 +
present discussion over the break, then you may find the
 +
previous notes archived at the distal ends of the ur-links
 +
that I append below (except for the first nine discussion
 +
notes that got lost in a disk crash at the Arisbe Dev site).
 +
 +
HC: Also, I think we can have a system of notation in which
 +
    teridentity is needed.  Those points seem reasonably clear.
 +
 +
The advantage of a concept is the integration of a species of manifold.
 +
The necessity of a concept is the incapacity to integrate it otherwise.
 +
 +
Of course, no one should be too impressed with a concept that
 +
is only the artifact of a particular system of representation.
 +
So before we accord a concept the status of addressing reality,
 +
and declare it a term of some tenured office in our intellects,
 +
we would want to see some evidence that it helps us to manage
 +
a reality that we cannot see a way to manage any other way.
 +
 +
Granted.
 +
 +
Now how in general do we go about an investiture of this sort?
 +
That is the big question that would serve us well to consider
 +
in the process of the more limited investigation of identity.
 +
Indeed, I do not see how it is possible to answer the small
 +
question if no understanding is reached on the big question.
 +
 +
HC: What remains relatively unclear is why we should need a system of notation
 +
    in which teridentity appears or is needed as against one in which it seems
 +
    not to be needed -- since assertion of identity can be made for any number
 +
    of terms in the standard predicate calculus.
 +
 +
This sort of statement totally non-plusses me.
 +
It seems like a complete non-sequitur or even
 +
a contradiction in terms to me.
 +
 +
The question is about the minimal adequate resource base for
 +
defining, deriving, or generating all of the concepts that we
 +
need for a given but very general type of application that we
 +
conventionally but equivocally refer to as "logic".  You seem
 +
to be saying something like this:  We don't need 3-identity
 +
because we have 4-identity, 5-identity, 6-identity, ..., in
 +
the "standard predicate calculus".  The question is not what
 +
concepts are generated in all the generations that follow the
 +
establishment of the conceptual resource base (axiom system),
 +
but what is the minimal set of concepts that we can use to
 +
generate the needed collection of concepts.  And there the
 +
answer is, in a way that is subject to the usual sorts of
 +
mathematical proof, that 3-identity is the minimum while
 +
2-identity is not big enough to do the job we want to do.
 +
 +
Logic Of Relatives 01-41, LOR Discussion Notes 10-17.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 18
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
JA = Jon Awbrey
 +
 +
JA: but now I have to add to my do-list the problems of comparing the
 +
    whole variorum of letters and drafts of letters to Lady Welby.
 +
    I only have the CP 8 and Wiener versions here, so I will
 +
    depend on you for ample excerpts from the Lieb volume.
 +
 +
BM: I made such a kind of comparison some time ago.  I selected
 +
    the following 3 cases on the criterium of alternate "grounds".
 +
    Hoping it could save some labor.  The first rank expressions
 +
    come from the MS 339 written in Oct. 1904 and I label them
 +
    with an (a).  I think that it is interesting to note that
 +
    they were written four years before the letters to Welby
 +
    and just one or two years after the Syllabus which is the
 +
    usual reference for the classification in 3 trichotomies
 +
    and 10 classes.  The second (b) is our initial table (from
 +
    a draft to Lady Welby, Dec. 1908, CP 8.344) and the third
 +
    (c) comes from a letter sent in Dec. 1908 (CP 8.345-8.376).
 +
    A tabular presentation would be better but I can't do it.
 +
    Comparing (c) against (a) and (b) is informative, I think.
 +
 +
Is this anywhere that it can be linked to from Arisbe?
 +
I've seen many pretty pictures of these things over the
 +
years, but may have to follow my own gnosis for a while.
 +
 +
Pages I have bookmarked just recently,
 +
but not really had the chance to study:
 +
 +
http://www.digitalpeirce.org/hoffmann/p-sighof.htm
 +
http://www.csd.uwo.ca/~merkle/thesis/Introduction.html
 +
http://members.door.net/arisbe/menu/library/aboutcsp/merkle/hci-abstract.htm
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 19
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
JA = Jon Awbrey
 +
 +
I now have three partially answered messages on the table,
 +
so I will just grab this fragment off the top of the deck.
 +
 +
BM: Peirce gives the following definition (CP 8.343):
 +
 +
BM, quoting CSP:
 +
 +
    | It is likewise requisite to distinguish
 +
    | the 'Immediate Interpretant', i.e. the
 +
    | Interpretant represented or signified in
 +
    | the Sign, from the 'Dynamic Interpretant',
 +
    | or effect actually produced on the mind
 +
    | by the Sign; and both of these from
 +
    | the 'Normal Interpretant', or effect
 +
    | that would be produced on the mind by
 +
    | the Sign after sufficient development
 +
    | of thought.
 +
    |
 +
    | C.S. Peirce, 'Collected Papers', CP 8.343.
 +
 +
JA: Well, you've really tossed me in the middle of the briar patch now!
 +
    I must continue with my reading from the 1870 LOR, ...
 +
 +
BM: Yes indeed!  I am irritated by having not the necessary
 +
    turn of mind to fully grasp it.  But it seems to be a
 +
    prerequisite in order to understand the very meaning
 +
    of the above table.  It could be the same for:
 +
 +
BM, quoting CSP:
 +
 +
    | I define a 'Sign' as anything which on the one hand
 +
    | is so determined by an Object and on the other hand
 +
    | so determines an idea in a person's mind, that this
 +
    | latter determination, which I term the 'Interpretant'
 +
    | of the sign, is thereby mediately determined by that
 +
    | Object.
 +
 +
BM: The so-called "latter determination" would make the 'Interpretant'
 +
    a tri-relative term into a teridentity involving Sign and Object.
 +
    Isn't it?
 +
 +
BM: I thought previously that the Peirce's phrasing was just applying the
 +
    principle of transitivity.  From O determines S and S determines I,
 +
    it follows:  O determines I.  But this is not the same as teridentity.
 +
    Do you think so or otherwise?
 +
 +
My answers are "No" and "Otherwise".
 +
 +
Continuing to discourse about definite universes thereof,
 +
the 3-identity term over the universe 1 = {A, B, C, D, ...} --
 +
I only said it was definite, I didn't say it wasn't vague! --
 +
designates, roughly speaking, the 3-adic relation that may
 +
be hinted at by way of the following series:
 +
 +
1,,  =  A:A:A +, B:B:B +, C:C:C +, D:D:D +, ...
 +
 +
I did a study on Peirce's notion of "determination".
 +
As I understand it so far, we need to keep in mind
 +
that it is more fundamental than causation, can be
 +
a form of "partial determination", and is roughly
 +
formal, mathematical, or "information-theoretic",
 +
not of necessity invoking any temporal order.
 +
 +
For example, when we say "The points A and B determine the line AB",
 +
this invokes the concept of a 3-adic relation of determination that
 +
does not identify A, B, AB, is not transitive, as transitivity has
 +
to do with the composition of 2-adic relations and would amount to
 +
the consideration of a degenerate 3-adic relation in this context.
 +
 +
Now, it is possible to have a sign relation q whose sum enlists
 +
an elementary sign relation O:S:I where O = S = I.  For example,
 +
it makes perfect sense to me to say that the whole universe may
 +
be a sign of itself to itself, so the conception is admissable.
 +
But this amounts to a very special case, by no means general.
 +
More generally, we are contemplating sums like the following:
 +
 +
q  =  O1:S1:I1 +, O2:S2:I2 +, O3:S3:I3 +, ...
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 20
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
JR = Joe Ransdell
 +
 +
HC: Though I certainly hesitate to think that we are separated
 +
    from the world by a veil of signs, it seems clear, too, on
 +
    Peircean grounds, that no sign can ever capture its object
 +
    completely.
 +
 +
JR: Any case of self-representation is a case of sign-object identity,
 +
    in some sense of "identity".  I have argued in various places that
 +
    this is the key to the doctrine of immediate perception as it occurs
 +
    in Peirce's theory.
 +
 +
To put the phrase back on the lathe:
 +
 +
| We are not separated from the world by a veil of signs --
 +
| we are the veil of signs.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 21
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
AS = Armando Sercovich
 +
 +
AS: We are not separated from the world by a veil of signs nor we are a veil of signs.
 +
    Simply we are signs.
 +
 +
AS, quoting CSP:
 +
 +
    | The *man-sign* acquires information, and comes to mean more than he did before.
 +
    | But so do words.  Does not electricity mean more now than it did in the days
 +
    | of Franklin?  Man makes the word, and the word means nothing which the man
 +
    | has not made it mean, and that only to some man.  But since man can think
 +
    | only by means of words or other external symbols, these might turn round
 +
    | and say:  "You mean nothing which we have not taught you, and then only
 +
    | so far as you address some word as the interpretant of your thought".
 +
    | In fact, therefore, men and words reciprocally educate each other;
 +
    | each increase of a man's information involves, and is involved by,
 +
    | a corresponding increase of a word's information.
 +
    |
 +
    | Without fatiguing the reader by stretching this parallelism too far, it is
 +
    | sufficient to say that there is no element whatever of man's consciousness
 +
    | which has not something corresponding to it in the word;  and the reason is
 +
    | obvious.  It is that the word or sign which man uses *is* the man itself.
 +
    | For, as the fact that every thought is a sign, taken in conjunction with
 +
    | the fact that life is a train of thought, proves that man is a sign;  so,
 +
    | that every thought is an *external* sign proves that man is an external
 +
    | sign.  That is to say, the man and the external sign are identical, in
 +
    | the same sense in which the words 'homo' and 'man' are identical.  Thus
 +
    | my language is the sum total of myself;  for the man is the thought ...
 +
    |
 +
    |'Charles S. Peirce:  Selected Writings (Values in a Universe of Chance)',
 +
    | edited with an introduction and notes by Philip P. Wiener, Dover,
 +
    | New York, NY, 1966. Originally published under the subtitle
 +
    | in parentheses above, Doubleday & Company, 1958.
 +
 +
I read you loud and clear.
 +
Every manifold must have
 +
its catalytic converter.
 +
 +
<Innumerate Continuation:>
 +
 +
TUC = The Usual CISPEC
 +
 +
TUC Alert:
 +
 +
| E.P.A. Says Catalytic Converter Is
 +
| Growing Cause of Global Warming
 +
| By Matthew L. Wald
 +
| Copyright 1998 The New York Times
 +
| May 29, 1998
 +
| -----------------------------------------------------------------------
 +
| WASHINGTON -- The catalytic converter, an invention that has sharply
 +
| reduced smog from cars, has now become a significant and growing cause
 +
| of global warming, according to the Environmental Protection Agency
 +
 +
Much as I would like to speculate ad libitum on these exciting new prospects for the
 +
application of Peirce's chemico-algebraic theory of logic to the theorem-o-dynamics
 +
of auto-semeiosis, I must get back to "business as usual" (BAU) ...
 +
 +
And now a word from our sponsor ...
 +
 +
http://www2.naias.com/
 +
 +
Reporting from Motown ---
 +
 +
Jon Awbrey
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 22
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
 +
HC: You quote the following passage from a prior posting of mine:
 +
 +
HC: What remains relatively unclear is why we should need a system of notation
 +
    in which teridentity appears or is needed as against one in which it seems
 +
    not to be needed -- since assertion of identity can be made for any number
 +
    of terms in the standard predicate calculus.
 +
 +
HC: You comment as follows:
 +
 +
JA: This sort of statement totally non-plusses me.
 +
    It seems like a complete non-sequitur or even
 +
    a contradiction in terms to me.
 +
 +
JA: The question is about the minimal adequate resource base for
 +
    defining, deriving, or generating all of the concepts that we
 +
    need for a given but very general type of application that we
 +
    conventionally but equivocally refer to as "logic".  You seem
 +
    to be saying something like this:  We don't need 3-identity
 +
    because we have 4-identity, 5-identity, 6-identity, ..., in
 +
    the "standard predicate calculus".  The question is not what
 +
    concepts are generated in all the generations that follow the
 +
    establishment of the conceptual resource base (axiom system),
 +
    but what is the minimal set of concepts that we can use to
 +
    generate the needed collection of concepts.  And there the
 +
    answer is, in a way that is subject to the usual sorts of
 +
    mathematical proof, that 3-identity is the minimum while
 +
    2-identity is not big enough to do the job we want to do.
 +
 +
HC: I have fallen a bit behind on this thread while attending to some other
 +
    matters, but in this reply, you do seem to me to be coming around to an
 +
    understanding of the issues involved, as I see them.  You put the matter
 +
    this way, "We don't need 3-identity because we have 4-identity, 5-identity,
 +
    6-identity, ..., in the 'standard predicate calculus'".  Actually, as I think
 +
    you must know, there is no such thing as "4-identity", "5-identity", etc., in
 +
    the standard predicate calculus.  It is more that such concepts are not needed,
 +
    just as teridentity is not needed, since the general apparatus of the predicate
 +
    calculus allows us to express identity among any number of terms without special
 +
    provision beyond "=".
 +
 +
No, that is not the case.  Standard predicate calculus allows the expression
 +
of predicates I_k, for k = 2, 3, 4, ..., such that I_k (x_1, ..., x_k) holds
 +
if and only if all x_j, for j = 1 to k, are identical.  So predicate calculus
 +
contains a k-identity predicate for all such k.  So whether "they're in there"
 +
is not an issue.  The question is whether these or any other predicates can be
 +
constructed or defined in terms of 2-adic relations alone.  And the answer is
 +
no, they cannot.  The vector of the misconception counterwise appears to be
 +
as various a virus as the common cold, and every bit as resistant to cure.
 +
I have taken the trouble to enumerate some of the more prevalent strains,
 +
but most of them appear to go back to the 'Principia Mathematica', and
 +
the variety of nominalism called "syntacticism" -- Ges-und-heit! --
 +
that was spread by it, however unwittedly by some of its carriers.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 23
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In trying to answer the rest of your last note,
 +
it seems that we cannot go any further without
 +
achieving some concrete clarity as to what is
 +
denominated by "standard predicate calculus",
 +
that is, "first order logic", or whatever.
 +
 +
There is a "canonical" presentation of the subject, as I remember it, anyway,
 +
in the following sample of materials from Chang & Keisler's 'Model Theory'.
 +
(There's a newer edition of the book, but this part of the subject hasn't
 +
really changed all that much in ages.)
 +
 +
Model Theory 01-39
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 24
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
 +
HC: I might object that "teridentity" seems to come
 +
    to a matter of "a=b & b=c", so that a specific
 +
    predicate of teridentity seems unnecessary.
 +
 +
I am presently concerned with expositing and interpreting
 +
the logical system that Peirce laid out in the LOR of 1870.
 +
It is my considered opinion after thirty years of study that
 +
there are untapped resources remaining in this work that have
 +
yet to make it through the filters of that ilk of syntacticism
 +
that was all the rage in the late great 1900's.  I find there
 +
to be an appreciably different point of view on logic that is
 +
embodied in Peirce's work, and until we have made the minimal
 +
effort to read what he wrote it is just plain futile to keep
 +
on pretending that we have already assimilated it, or that
 +
we are qualified to evaluate its cogency.
 +
 +
The symbol "&" that you employ above denotes a mathematical object that
 +
qualifies as a 3-adic relation.  Independently of my own views, there
 +
is an abundance of statements in evidence that mathematical thinkers
 +
from Peirce to Goedel consider the appreciation of facts like this
 +
to mark the boundary between realism and nominalism in regard to
 +
mathematical objects.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 25
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
JA = Jon Awbrey
 +
 +
HC: I might object that "teridentity" seems to come
 +
    to a matter of "a=b & b=c", so that a specific
 +
    predicate of teridentity seems unnecessary.
 +
 +
JA: I am presently concerned with expositing and interpreting
 +
    the logical system that Peirce laid out in the LOR of 1870.
 +
    It is my considered opinion after thirty years of study that
 +
    there are untapped resources remaining in this work that have
 +
    yet to make it through the filters of that ilk of syntacticism
 +
    that was all the rage in the late great 1900's.  I find there
 +
    to be an appreciably different point of view on logic that is
 +
    embodied in Peirce's work, and until we have made the minimal
 +
    effort to read what he wrote it is just plain futile to keep
 +
    on pretending that we have already assimilated it, or that
 +
    we are qualified to evaluate its cogency.
 +
 +
JA: The symbol "&" that you employ above denotes a mathematical object that
 +
    qualifies as a 3-adic relation.  Independently of my own views, there
 +
    is an abundance of statements in evidence that mathematical thinkers
 +
    from Peirce to Goedel consider the appreciation of facts like this
 +
    to mark the boundary between realism and nominalism in regard to
 +
    mathematical objects.
 +
 +
HC: I would agree, I think, that "&" may be thought of
 +
    as a function mapping pairs of statements onto the
 +
    conjunction of that pair.
 +
 +
Yes, indeed, in the immortal words of my very first college algebra book:
 +
"A binary operation is a ternary relation".  As it happens, the symbol "&"
 +
is equivocal in its interpretation -- computerese today steals a Freudian
 +
line and dubs it "polymorphous" -- it can be regarded in various contexts
 +
as a 3-adic relation on syntactic elements called "sentences", on logical
 +
elements called "propositions", or on truth values collated in the boolean
 +
domain B = {false, true} = {0, 1}.  But the mappings and relations between
 +
all of these interpretive choices are moderately well understood.  Still,
 +
no matter how many ways you enumerate for looking at a B-bird, the "&" is
 +
always 3-adic.  And that is sufficient to meet your objection, so I think
 +
I will just leave it there until next time.
 +
 +
On a related note, that I must postpone until later:
 +
We seem to congrue that there is a skewness between
 +
the way that most mathematicians use logic and some
 +
philosophers talk about logic, but I may not be the
 +
one to set it adjoint, much as I am inclined to try.
 +
At the moment I have this long-post-poned exponency
 +
to carry out.  I will simply recommend for your due
 +
consideration Peirce's 1870 Logic Of Relatives, and
 +
leave it at that.  There's a cornucopiousness to it
 +
that's yet to be dreamt of in the philosophy of the
 +
1900's.  I am doing what I can to infotain you with
 +
the Gardens of Mathematical Recreations that I find
 +
within Peirce's work, and that's in direct response
 +
to many, okay, a couple of requests.  Perhaps I can
 +
not hope to attain the degree of horticultural arts
 +
that Gardners before me have exhibited in this work,
 +
but then again, who could?  Everybody's a critic --
 +
but the better ones read first, and criticize later.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 26
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
 +
HC: But on the other hand, it is not customary to think of "&" as
 +
    a relation among statements or sentences -- as, for instance,
 +
    logical implication is considered a logical relation between
 +
    statements or sentences.
 +
 +
Actually, it is the custom in many quarters to treat all of the
 +
boolean operations, logical connectives, propositional relations,
 +
or whatever you want to call them, as "equal citizens", having each
 +
their "functional" (f : B^k -> B) and their "relational" (L c B^(k+1))
 +
interpretations and applications.  From this vantage, the interpretive
 +
distinction that is commonly regarded as that between "assertion" and
 +
mere "contemplation" is tantamount to a "pragmatic" difference between
 +
computing the values of a function on a given domain of arguments and
 +
computing the inverse of a function vis-a-vis a prospective true value.
 +
This is the logical analogue of the way that our mathematical models
 +
of reality have long been working, unsuspected and undisturbed by
 +
most philosophers of science, I might add.  If only the logical
 +
side of the ledger were to be developed rather more fully than
 +
it is at present, we might wake one of these days to find our
 +
logical accounts of reality, finally, at long last, after an
 +
overweaningly longish adolescence, beginning to come of age.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 27
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
 +
HC: For, if I make an assertion A&B, then I am not asserting
 +
    that the statement A stands in a relation to a statement B.
 +
    Instead, I am asserting the conjunction A&B (which logically
 +
    implies both the conjuncts in view of the definition of "&").
 +
 +
Please try to remember where we came in.  This whole play of
 +
animadversions about 3-adicity and 3-identity is set against
 +
the backdrop of a single point, over the issue as to whether
 +
3-adic relations are wholly dispensable or somehow essential
 +
to logic, mathematics, and indeed to argument, communication,
 +
and reasoning in general.  Some folks clamor "Off with their
 +
unnecessary heads!" -- other people, who are forced by their
 +
occupations to pay close attention to the ongoing complexity
 +
of the processes at stake, know that, far from finding 3-ads
 +
in this or that isolated corner of the realm, one can hardly
 +
do anything at all in the ways of logging or mathing without
 +
running smack dab into veritable hosts of them.
 +
 +
I have just shown that "a=b & b=c" involves a 3-adic relation.
 +
Some people would consider this particular 3-adic relation to
 +
be more complex than the 3-identity relation, but that may be
 +
a question of taste.  At any rate, the 3-adic aspect persists.
 +
 +
HC: If "&" counts as a triadic relation, simply because it serves
 +
    to conjoin two statements into a third, then it would seem that
 +
    any binary relation 'R' will count as triadic, simply because
 +
    it places two things into a relation, which is a "third" thing.
 +
    By the same kind of reasoning a triadic relation, as ordinarily
 +
    understood would be really 4-adic.
 +
 +
The rest of your comments are just confused,
 +
and do not use the terms as they are defined.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 28
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
JA = Jon Awbrey
 +
JR = Joseph Ransdell
 +
 +
JA: Notice that Peirce follows the mathematician's usual practice,
 +
    then and now, of making the status of being an "individual" or
 +
    a "universal" relative to a discourse in progress.  I have come
 +
    to appreciate more and more of late how radically different this
 +
    "patchwork" or "piecewise" approach to things is from the way of
 +
    some philosophers who seem to be content with nothing less than
 +
    many worlds domination, which means that they are never content
 +
    and rarely get started toward the solution of any real problem.
 +
    Just my observation, I hope you understand.
 +
 +
JR: Yes, I take this as underscoring and explicating the import of
 +
    making logic prior to rather than dependent upon metaphysics.
 +
 +
I think that Peirce, and of course many math folks, would take math
 +
as prior, on a par, or even identical with logic.  Myself I've been
 +
of many minds about this over the years.  The succinctest picture
 +
that I get from Peirce is always this one:
 +
 +
| [Riddle of the Sphynx]
 +
|
 +
| Normative science rests largely on phenomenology and on mathematics;
 +
| Metaphysics on phenomenology and on normative science.
 +
|
 +
| C.S. Peirce, CP 1.186 (1903)
 +
|
 +
|
 +
|                          o Metaphysics
 +
|                        /|
 +
|                        / |
 +
|                      /  |
 +
|    Normative Science o  |
 +
|                    / \  |
 +
|                    /  \ |
 +
|                  /    \|
 +
|      Mathematics o      o Phenomenology
 +
|
 +
|
 +
| ROTS.  http://stderr.org/pipermail/inquiry/2004-March/001262.html
 +
 +
Logic being a normative science must depend on math and phenomenology.
 +
 +
Of course, it all depends on what a person means by "logic" ...
 +
 +
JA: I also observe that Peirce takes the individual objects of
 +
    a particular universe of discourse in a "generative" way,
 +
    not a "totalizing" way, and thus they afford us with the
 +
    basis for talking freely about collections, constructions,
 +
    properties, qualities, subsets, and "higher types", as
 +
    the phrase is mint.
 +
 +
JR: Would this be essentially the same as regarding quantification as
 +
    distributive rather than collective, i.e. we take the individuals
 +
    of a class one-by-one as selectable rather than as somehow given
 +
    all at once, collectively?
 +
 +
Gosh, that's a harder question.  Your suggestion reminds me
 +
of the way that some intuitionist and even some finitist
 +
mathematicians talk when they reflect on math practice.
 +
I have leanings that way, but when I have tried to
 +
give up the classical logic axioms, I have found
 +
them too built in to my way of thinking to quit.
 +
Still, a healthy circumspection about about our
 +
often-wrongly vaunted capacties to conceive of
 +
totalities is a habitual part of current math.
 +
Again, I think individuals are made not born,
 +
that is, to some degree factitious and mere
 +
compromises of this or that conveniency.
 +
This is one of the reasons that I have
 +
been trying to work out the details
 +
of a functional approach to logic,
 +
propostional, quantificational,
 +
and relational.
 +
 +
Cf: INTRO 30.  http://stderr.org/pipermail/inquiry/2004-November/001765.html
 +
In: INTRO.  http://stderr.org/pipermail/inquiry/2004-November/thread.html#1720
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 29
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
JA = Jon Awbrey
 +
GR = Gary Richmond
 +
 +
Re: LOR.COM 11.24.  http://stderr.org/pipermail/inquiry/2004-November/001836.html
 +
In: LOR.COM.        http://stderr.org/pipermail/inquiry/2004-November/thread.html#1755
 +
 +
JA: The manner in which these arrows and qualified arrows help us
 +
    to construct a suspension bridge that unifies logic, semiotics,
 +
    statistics, stochastics, and information theory will be one of
 +
    the main themes that I aim to elaborate throughout the rest of
 +
    this inquiry.
 +
 +
GR: Pretty ambitious, Jon.  I'm sure you're up to it.
 +
 +
GR: I'd like to anticipate 3 versions:  The mathematical (cactus diagrams, etc.),
 +
    the poetic, and the commonsensical -- ordinary language for those who are
 +
    NEITHER logicians NOR poets.
 +
 +
GR: Are you up to THAT?
 +
 +
Riddle A Body:  "Time Enough, And Space, Excalibrate Co-Arthurs Should Apply"
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 30
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
JA = Jon Awbrey
 +
GR = Gary Richmond
 +
 +
Re: LOR.DIS 29.  http://stderr.org/pipermail/inquiry/2004-November/001838.html
 +
In: LOR.DIS.    http://stderr.org/pipermail/inquiry/2004-November/thread.html#1768
 +
 +
JA: Riddle A Body:  "Time Enough, And Space, Excalibrate Co-Arthurs Should Apply"
 +
 +
GR: Well said, and truly!
 +
 +
Body A Riddle:  TEASE CASA = Fun House.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 31
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Many illusions of selective reading -- like the myth that Peirce did not
 +
discover quantification over indices until 1885 -- can be dispelled by
 +
looking into his 1870 "Logic of Relatives".  I started a web study of
 +
this in 2002, reworked again in 2003 and 2004, the current version
 +
of which can be found here:
 +
 +
LOR.      http://stderr.org/pipermail/inquiry/2004-November/thread.html#1750
 +
LOR-COM.  http://stderr.org/pipermail/inquiry/2004-November/thread.html#1755
 +
LOR-DIS.  http://stderr.org/pipermail/inquiry/2004-November/thread.html#1768
 +
 +
I've only gotten as far as the bare infrastructure of Peirce's 1870 LOR,
 +
but an interesting feature of the study is that, if one draws the pictures
 +
that seem almost demanded by his way of linking up indices over expressions,
 +
then one can see a prototype of his much later logical graphs developing in
 +
the text.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Discussion Note 32
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM: Several discussions could take place there,
 +
    as to the reasons for the number of divisions,
 +
    the reasons of the titles themselves.  Another
 +
    one is my translation from "normal interpretant"
 +
    into "final interpretant" (which one is called
 +
    elsewhere "Eventual Interpretant" or "Destinate
 +
    Interpretant" by CSP).  I let all this aside
 +
    to focus on the following remark:
 +
 +
BM: 6 divisions correspond to individual correlates:
 +
 +
    (S, O_i, O_d, I_i, I_d, I_n),
 +
 +
    3 divisions correspond to dyads:
 +
 +
    (S : O_d, S : I_d, S : I_n),
 +
 +
    and the tenth to a triad:
 +
 +
    (S : O_d : I_n).
 +
 +
    This remark would itself deserve
 +
    a lot of explanations but one
 +
    more time I let this aside.
 +
 +
BM: Then we have the following very clear statement from Peirce:
 +
 +
  | It follows from the Definition of a Sign
 +
  | that since the Dynamoid Object determines
 +
  | the Immediate Object,
 +
  | which determines the Sign,
 +
  | which determines the Destinate Interpretant
 +
  | which determines the Effective Interpretant
 +
  | which determines the Explicit Interpretant
 +
  |
 +
  | the six trichotomies, instead of determining 729 classes of signs,
 +
  | as they would if they were independent, only yield 28 classes; and
 +
  | if, as I strongly opine (not to say almost prove) there are four other
 +
  | trichotomies of signs of the same order of importance, instead of making
 +
  | 59049 classes, these will only come to 66.
 +
  |
 +
  | CSP, "Letter to Lady Welby", 14 Dec 1908, LW, p. 84.
 +
 +
BM: The separation made by CSP between 6 divisions and four others
 +
    seems to rely upon the suggested difference between individual
 +
    correlates and relations.  We get the idea that the 10 divisions
 +
    are ordered on the whole and will end into 66 classes (by means of
 +
    three ordered modal values on each division:  maybe, canbe, wouldbe).
 +
    Finally we have too the ordering for the divisions relative to the
 +
    correlates that I write in my notation:
 +
 +
    Od -> Oi -> S -> If -> Id -> Ii.
 +
 +
BM: This order of "determinations" has bothered many people
 +
    but if we think of it as operative in semiosis, it seems
 +
    to be correct (at least to my eyes).  Thus the question is:
 +
    where, how, and why the "four other trichotomies" fit in this
 +
    schema to obtain a linear ordering on the whole 10 divisions?
 +
    May be the question can be rephrased as:  how intensional
 +
    relationships fit into an extensional one?  Possibly the
 +
    question could be asked the other way.  R. Marty responds
 +
    that in a certain sense the four trichotomies give nothing
 +
    more than the previous six ones but I strongly doubt of this.
 +
 +
BM: I put the problem in graphical form in an attached file
 +
    because my message editor will probably make some mistakes.
 +
    I make a distinction between arrow types drawing because I am
 +
    not sure that the sequence of correlates determinations is of
 +
    the same nature than correlates determination inside relations.
 +
 +
BM: It looks as if the problem amounts to some kind of projection
 +
    of relations on the horizontal axis made of correlates.
 +
 +
BM: If we consider some kind of equivalence (and this seems necessary to
 +
    obtain a linear ordering), by means of Agent -> Patient reductions on
 +
    relations, then erasing transitive determinations leads to:
 +
 +
    Od -> Oi -> S -> S-Od -> If -> S-If -> S-Od-If -> Id -> S-Id -> Ii
 +
 +
BM: While it is interesting to compare the subsequence
 +
    S-Od -> If -> S-If -> S-Od-If with the pragmatic maxim,
 +
    I have no clear idea of the (in-) validity of such a result.
 +
    But I am convinced that the clarity has to come from the
 +
    Logic Of Relatives.
 +
 +
BM: I will be very grateful if you can make something with all that stuff.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM: I also found this passage which may be of some interest
 +
    (CP 4.540, Prolegomena to an Apology of Pragmatism):
 +
 +
| But though an Interpretant is not necessarily a Conclusion, yet a
 +
| Conclusion is necessarily an Interpretant. So that if an Interpretant is
 +
| not subject to the rules of Conclusions there is nothing monstrous in my
 +
| thinking it is subject to some generalization of such rules. For any
 +
| evolution of thought, whether it leads to a Conclusion or not, there is a
 +
| certain normal course, which is to be determined by considerations not in
 +
| the least psychological, and which I wish to expound in my next
 +
| article;†1 and while I entirely agree, in opposition to distinguished
 +
| logicians, that normality can be no criterion for what I call
 +
| rationalistic reasoning, such as alone is admissible in science, yet it
 +
| is precisely the criterion of instinctive or common-sense reasoning,
 +
| which, within its own field, is much more trustworthy than rationalistic
 +
| reasoning. In my opinion, it is self-control which makes any other than
 +
| the normal course of thought possible, just as nothing else makes any
 +
| other than the normal course of action possible; and just as it is
 +
| precisely that that gives room for an ought-to-be of conduct, I mean
 +
| Morality, so it equally gives room for an ought-to-be of thought, which
 +
| is Right Reason; and where there is no self-control, nothing but the
 +
| normal is possible. If your reflections have led you to a different
 +
| conclusion from mine, I can still hope that when you come to read my next
 +
| article, in which I shall endeavor to show what the forms of thought are,
 +
| in general and in some detail, you may yet find that I have not missed
 +
| the truth.
 +
 +
JA: Just as I have always feared, this classification mania
 +
    appears to be communicable! But now I must definitely
 +
    review the Welby correspondence, as all this stuff was
 +
    a blur to my sensibilities the last 10 times I read it.
 +
 +
BM: I think that I understand your reticence. I wonder if:
 +
 +
    a.  the fact that the letters to Lady Welby have been published as such,
 +
        has not lead to approach the matter in a certain way.
 +
 +
    b.  other sources, eventually unpublished, would give another lighting on
 +
        the subject, namely a logical one. I think of MS 339 for example that
 +
        seems to be part of the Logic Notebook. I have had access to some pages
 +
        of it, but not to the whole MS.
 +
 +
BM: A last remark. I don't think that classification is a mania for CSP but I
 +
    know that you know that! It is an instrument of thought and I think that
 +
    it is in this case much more a plan for experimenting than the exposition
 +
    of a conclusion. Experimenting what ? There is a strange statement in a
 +
    letter to W. James where CSP says that what is in question in his "second
 +
    way of dividing signs" is the logical theory of numbers. I give this from
 +
    memory. I have not the quote at hand now but I will search for it if needed.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
BM = Bernard Morand
 +
JA = Jon Awbrey
 +
 +
JA: ... but now I have to add to my do-list the problems of comparing
 +
    the whole variorum of letters and drafts of letters to Lady Welby.
 +
    I only have the CP 8 and Wiener versions here, so I will depend
 +
    on you for ample excerpts from the Lieb volume.
 +
 +
BM: I made such a kind of comparison some time ago. I selected the following
 +
    3 cases on the criterium of alternate "grounds". Hoping it could save
 +
    some labor. The first rank expressions come from the MS 339 written in
 +
    Oct. 1904 and I label them with an (a). I think that it is interesting to
 +
    note that they were written four years before the letters to Welby and
 +
    just one or two years after the Syllabus which is the usual reference for
 +
    the classification in 3 trichotomies and 10 classes. The second (b) is
 +
    our initial table (from a draft to Lady Welby, Dec. 1908, CP 8.344) and
 +
    the third (c) comes from a letter sent in Dec. 1908 (CP 8.345-8.376). A
 +
    tabular presentation would be better but I can't do it. Comparing (c)
 +
    against (a) and (b) is informative, I think.
 +
 +
Division 1
 +
 +
(a) According to the matter of the Sign
 +
 +
(b) According to the Mode of Apprehension of the Sign itself
 +
 +
(c) Signs in respect to their Modes of possible Presentation
 +
 +
Division 2
 +
 +
(a) According to the Immediate Object
 +
 +
(b) According to the Mode of Presentation of the Immediate Object
 +
 +
(c) Objects, as they may be presented
 +
 +
Division 3
 +
 +
(a) According to the Matter of the Dynamic Object
 +
 +
(b) According to the Mode of Being of the Dynamical Object
 +
 +
(c) In respect to the Nature of the Dynamical Objects of Signs
 +
 +
Division 4
 +
 +
(a) According to the mode of representing object by the Dynamic Object
 +
 +
(b) According to the Relation of the Sign to its Dynamical Object
 +
 +
(c) The fourth Trichotomy
 +
 +
Division 5
 +
 +
(a) According to the Immédiate Interpretant
 +
 +
(b) According to the Mode of Presentation of the Immediate Interpretant
 +
 +
(c) As to the nature of the Immediate (or Felt ?) Interpretant
 +
 +
Division 6
 +
 +
(a) According to the Matter of Dynamic Interpretant
 +
 +
(b) According to the Mode of Being of the Dynamical Interpretant
 +
 +
(c) As to the Nature of the Dynamical Interpretant
 +
 +
Division 7
 +
 +
(a) According to the Mode of Affecting Dynamic Interpretant
 +
 +
(b) According to the relation of the Sign to the Dynamical Interpretant
 +
 +
(c) As to the Manner of Appeal to the Dynamic Interpretant
 +
 +
Division 8
 +
 +
(a) According to the Matter of Representative Interpretant
 +
 +
(b) According to the Nature of the Normal Interpretant
 +
 +
(c) According to the Purpose of the Eventual Interpretant
 +
 +
Division 9
 +
 +
(a) According to the Mode of being represented by Representative Interpretant
 +
 +
(b) According to the the relation of the Sign to the Normal Interpretant
 +
 +
(c) As to the Nature of the Influence of the Sign
 +
 +
Division 10
 +
 +
(a) According to the Mode of being represented to represent object by Sign, Truly
 +
 +
(b) According to the Triadic Relation of the Sign to its Dynamical Object and to
 +
    its Normal Interpretant
 +
 +
(c) As to the Nature of the Assurance of the Utterance
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
JA: It may appear that one has side-stepped the issue of empiricism
 +
    that way, but then all that stuff about the synthetic a priori
 +
    raises its head, and we have Peirce's insight that mathematics
 +
    is observational and even experimental, and so I must trail off
 +
    into uncoordinated elliptical thoughts ...
 +
 +
HC: In contrast with this it strikes me that not all meanings of "analytic"
 +
    and "synthetic" have much, if anything, to do with the "analytic and the
 +
    synthetic", say, as in Quine's criticism of the "dualism" of empiricism.
 +
    Surely no one thinks that a plausible analysis must be analytic or that
 +
    synthetic materials tell us much about epistemology.  So, it is not
 +
    clear that anything connected with analyticity or a priori knowledge
 +
    will plausibly or immediately arise from a discussion of analytical
 +
    geometry.  Prevalent mathematical assumptions or postulates, yes --
 +
    but who says these are a prior?  Can't non-Euclidean geometry also
 +
    be treated in the style of analytic geometry?
 +
 +
HC: I can imagine the a discussion might be forced in
 +
    that direction, but the connections don't strike me
 +
    as at all obvious or pressing.  Perhaps Jon would just
 +
    like to bring up the notion of the synthetic apriori?
 +
    But why?
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
HC = Howard Callaway
 +
 +
HC: But I see you as closer to my theme or challenge, when you say
 +
    "The question is about the minimal adequate resource base for
 +
    defining, deriving, or generating all of the concepts that we
 +
    need for a given but very general type of application that we
 +
    conventinally but equivocally refer to as 'logic'".
 +
 +
HC: I think it is accepted on all sides of the discussion that there
 +
    is some sort of "equivalence" between the standard predicate logic
 +
    and Peirce's graphs.
 +
 +
There you would be mistaken, except perhaps for the fact that
 +
"some sort of equivalence" is vague to the depths of vacuity.
 +
It most particularly does not mean "all sorts of equivalence"
 +
or even "all important sorts of equivalence".  It is usually
 +
interpreted to mean an extremely abstract type of syntactic
 +
equivalence, and that is undoubtedly one important type of
 +
equivalence that it is worth examining whether two formal
 +
systems have or not.  But it precisely here that we find
 +
another symptom of syntacticism, namely, the deprecation
 +
of all other important qualities of formal systems, most
 +
pointedly their "analystic, "semantic", and "pragmatic"
 +
qualities, which make all the difference in how well the
 +
system actually serves its users in a real world practice.
 +
You can almost hear the whining and poohing coming from the
 +
syntactic day camp, but those are the hard facts of the case.
 +
 +
HC: But we find this difference in relation to the vocabulary used to express
 +
    identity.  From the point of view of starting with the predicate calculus,
 +
    we don't need "teridentity".  So, this seems to suggest there is something
 +
    of interesting contrast in Peirce's logic, which brings in this concept.
 +
    The obvious question may be expressed by asking why we need teridentity
 +
    in Peirce's system and how Peirce's system may recommend itself in contrast
 +
    to the standard way with related concepts.  This does seem to call for
 +
    a comparative evaluation of distinctive systems.  That is not an easy task,
 +
    as I think we all understand. But I do think that if it is a goal to have
 +
    Peirce's system better appreciated, then that kind of question must be
 +
    addressed.  If "=" is sufficient in the standard predicate calculus,
 +
    to say whatever we may need to say about the identity of terms, then
 +
    what is the advantage of an alternative system which insists on always
 +
    expressing identity of triples?
 +
 +
HC: The questions may look quite different, depending on where we start.
 +
    But in any case, I thought I saw some better appreciation of the
 +
    questions in your comments above.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
It's been that way for about as long as anybody can remember, and
 +
it will remain so, in spite of the spate of history rewriting and
 +
image re-engineering that has become the new rage in self-styled
 +
"analytic" circles.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Work 7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The brands of objection that you continue to make, with no evidence
 +
of reflection on the many explanations that I and others have taken
 +
the time to write out for you, lead me to believe that you are just
 +
not interested in making that effort.  That's okay, life is short,
 +
the arts are long and many, there is always something else to do.
 +
 +
HC: For, if I make an assertion A&B, then I am not asserting
 +
    that the statement A stands in a relation to a statement B.
 +
    Instead, I am asserting the conjunction A&B (which logically
 +
    implies both the conjuncts in view of the definition of "&").
 +
    If "&" counts as a triadic relation, simply because it serves
 +
    to conjoin two statements into a third, then it would seem that
 +
    any binary relation 'R' will count as triadic, simply because
 +
    it places two things into a relation, which is a "third" thing.
 +
    By the same kind of reasoning a triadic relation, as ordinarily
 +
    understood would be really 4-adic.
 +
 +
HC: Now, I think this is the kind of argument you are making, ...
 +
 +
No, it's the kind of argument that you are making.
 +
I am not making that kind of argument, and Peirce
 +
did not make that kind of argument.  Peirce used
 +
his terms subject to definitions that would have
 +
been understandable, and remain understandable,
 +
to those of his readers who understand these
 +
elementary definitions, either though their
 +
prior acquaintance with standard concepts
 +
or through their basic capacity to read
 +
a well-formed, if novel definition.
 +
 +
Peirce made certain observations about the structure of logical concepts
 +
and the structure of their referents.  Those observations are accurate
 +
and important.  He expressed those observations in a form that is clear
 +
to anybody who knows the meanings of the technical terms that he used,
 +
and he is not responsible for the interpretations of those who don't.
 +
 +
HC: ... and it seems to both trivialize the claimed argument
 +
    for teridentity, by trivializing the conception of what
 +
    is to count as a triadic, as contrasted with a binary
 +
    relation, and it also seems to introduce a confusion
 +
    about what is is count as a binary, vs. a triadic
 +
    relation.
 +
 +
Yes, the argument that you are making trivializes
 +
just about everything in sight, but that is the
 +
common and well-known property of any argument
 +
that fails to base itself on a grasp of the
 +
first elements of the subject matter.
 +
 +
HC: If this is mathematical realism, then so much the worse for
 +
    mathematical realism.  I am content to think that we do not
 +
    have a free hand in making up mathematical truth.
 +
 +
No, it's not mathematical realism.  It is your reasoning,
 +
and it exhibits all of the symptoms of syntacticism that
 +
I have already diagnosed.  It's a whole other culture
 +
from what is pandemic in the practice of mathematics,
 +
and it never fails to surprise me that people who
 +
would never call themselves "relativists" in any
 +
other matter of culture suddenly turn into just
 +
that in matters of simple mathematical fact.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Logic Of Relatives -- Old Series
 +
 +
00.  http://suo.ieee.org/ontology/thrd20.html#04416
 +
01.  http://suo.ieee.org/ontology/msg04416.html
 +
02.  http://suo.ieee.org/ontology/msg04417.html
 +
03.  http://suo.ieee.org/ontology/msg04418.html
 +
04.  http://suo.ieee.org/ontology/msg04419.html
 +
05.  http://suo.ieee.org/ontology/msg04421.html
 +
06.  http://suo.ieee.org/ontology/msg04422.html
 +
07.  http://suo.ieee.org/ontology/msg04423.html
 +
08.  http://suo.ieee.org/ontology/msg04424.html
 +
09.  http://suo.ieee.org/ontology/msg04425.html
 +
10.  http://suo.ieee.org/ontology/msg04426.html
 +
11.  http://suo.ieee.org/ontology/msg04427.html
 +
12.  http://suo.ieee.org/ontology/msg04431.html
 +
13.  http://suo.ieee.org/ontology/msg04432.html
 +
14.  http://suo.ieee.org/ontology/msg04435.html
 +
15.  http://suo.ieee.org/ontology/msg04436.html
 +
16.  http://suo.ieee.org/ontology/msg04437.html
 +
17.  http://suo.ieee.org/ontology/msg04438.html
 +
18.  http://suo.ieee.org/ontology/msg04439.html
 +
19.  http://suo.ieee.org/ontology/msg04440.html
 +
20.  http://suo.ieee.org/ontology/msg04441.html
 +
21.  http://suo.ieee.org/ontology/msg04442.html
 +
22.  http://suo.ieee.org/ontology/msg04443.html
 +
23.  http://suo.ieee.org/ontology/msg04444.html
 +
24.  http://suo.ieee.org/ontology/msg04445.html
 +
25.  http://suo.ieee.org/ontology/msg04446.html
 +
26.  http://suo.ieee.org/ontology/msg04447.html
 +
27.  http://suo.ieee.org/ontology/msg04448.html
 +
28.  http://suo.ieee.org/ontology/msg04449.html
 +
29.  http://suo.ieee.org/ontology/msg04450.html
 +
30.  http://suo.ieee.org/ontology/msg04451.html
 +
31.  http://suo.ieee.org/ontology/msg04452.html
 +
32.  http://suo.ieee.org/ontology/msg04453.html
 +
33.  http://suo.ieee.org/ontology/msg04454.html
 +
34.  http://suo.ieee.org/ontology/msg04456.html
 +
35.  http://suo.ieee.org/ontology/msg04457.html
 +
36.  http://suo.ieee.org/ontology/msg04458.html
 +
37.  http://suo.ieee.org/ontology/msg04459.html
 +
38.  http://suo.ieee.org/ontology/msg04462.html
 +
39.  http://suo.ieee.org/ontology/msg04464.html
 +
40.  http://suo.ieee.org/ontology/msg04473.html
 +
41.  http://suo.ieee.org/ontology/msg04478.html
 +
42.  http://suo.ieee.org/ontology/msg04484.html
 +
43.  http://suo.ieee.org/ontology/msg04487.html
 +
44.  http://suo.ieee.org/ontology/msg04488.html
 +
45.  http://suo.ieee.org/ontology/msg04492.html
 +
46.  http://suo.ieee.org/ontology/msg04497.html
 +
47.  http://suo.ieee.org/ontology/msg04498.html
 +
48.  http://suo.ieee.org/ontology/msg04499.html
 +
49.  http://suo.ieee.org/ontology/msg04500.html
 +
50.  http://suo.ieee.org/ontology/msg04501.html
 +
51.  http://suo.ieee.org/ontology/msg04502.html
 +
52.  http://suo.ieee.org/ontology/msg04503.html
 +
53.  http://suo.ieee.org/ontology/msg04504.html
 +
54.  http://suo.ieee.org/ontology/msg04506.html
 +
55.  http://suo.ieee.org/ontology/msg04508.html
 +
56.  http://suo.ieee.org/ontology/msg04509.html
 +
57.  http://suo.ieee.org/ontology/msg04510.html
 +
58.  http://suo.ieee.org/ontology/msg04511.html
 +
59.  http://suo.ieee.org/ontology/msg04512.html
 +
60.  http://suo.ieee.org/ontology/msg04513.html
 +
61.  http://suo.ieee.org/ontology/msg04516.html
 +
62.  http://suo.ieee.org/ontology/msg04517.html
 +
63.  http://suo.ieee.org/ontology/msg04518.html
 +
64.  http://suo.ieee.org/ontology/msg04521.html
 +
65.  http://suo.ieee.org/ontology/msg04539.html
 +
66.  http://suo.ieee.org/ontology/msg04541.html
 +
67.  http://suo.ieee.org/ontology/msg04542.html
 +
68.  http://suo.ieee.org/ontology/msg04543.html
 +
 +
LOR.  Logic of Relatives -- Discussion Notes
 +
 +
00.  http://suo.ieee.org/ontology/thrd19.html#04460
 +
10.  http://suo.ieee.org/ontology/msg04460.html
 +
11.  http://suo.ieee.org/ontology/msg04461.html
 +
12.  http://suo.ieee.org/ontology/msg04471.html
 +
13.  http://suo.ieee.org/ontology/msg04472.html
 +
14.  http://suo.ieee.org/ontology/msg04475.html
 +
15.  http://suo.ieee.org/ontology/msg04476.html
 +
16.  http://suo.ieee.org/ontology/msg04477.html
 +
17.  http://suo.ieee.org/ontology/msg04479.html
 +
18.  http://suo.ieee.org/ontology/msg04480.html
 +
19.  http://suo.ieee.org/ontology/msg04481.html
 +
20.  http://suo.ieee.org/ontology/msg04482.html
 +
21.  http://suo.ieee.org/ontology/msg04483.html
 +
22.  http://suo.ieee.org/ontology/msg04485.html
 +
23.  http://suo.ieee.org/ontology/msg04486.html
 +
24.  http://suo.ieee.org/ontology/msg04493.html
 +
25.  http://suo.ieee.org/ontology/msg04494.html
 +
26.  http://suo.ieee.org/ontology/msg04495.html
 +
27.  http://suo.ieee.org/ontology/msg04496.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Logic Of Relatives -- 2003
 +
 +
LOR.  http://stderr.org/pipermail/inquiry/2003-March/thread.html#186
 +
LOR.  http://stderr.org/pipermail/inquiry/2003-April/thread.html#245
 +
 +
01.  http://stderr.org/pipermail/inquiry/2003-March/000186.html
 +
02.  http://stderr.org/pipermail/inquiry/2003-March/000187.html
 +
03.  http://stderr.org/pipermail/inquiry/2003-March/000188.html
 +
04.  http://stderr.org/pipermail/inquiry/2003-March/000189.html
 +
05.  http://stderr.org/pipermail/inquiry/2003-March/000190.html
 +
06.  http://stderr.org/pipermail/inquiry/2003-March/000191.html
 +
07.  http://stderr.org/pipermail/inquiry/2003-March/000194.html
 +
08.  http://stderr.org/pipermail/inquiry/2003-March/000195.html
 +
09.  http://stderr.org/pipermail/inquiry/2003-April/000245.html
 +
10.  http://stderr.org/pipermail/inquiry/2003-April/000246.html
 +
11.  http://stderr.org/pipermail/inquiry/2003-April/000247.html
 +
12.  http://stderr.org/pipermail/inquiry/2003-April/000248.html
 +
13.  http://stderr.org/pipermail/inquiry/2003-April/000249.html
 +
14.  http://stderr.org/pipermail/inquiry/2003-April/000250.html
 +
15.  http://stderr.org/pipermail/inquiry/2003-April/000251.html
 +
16.  http://stderr.org/pipermail/inquiry/2003-April/000252.html
 +
17.  http://stderr.org/pipermail/inquiry/2003-April/000253.html
 +
18.  http://stderr.org/pipermail/inquiry/2003-April/000254.html
 +
19.  http://stderr.org/pipermail/inquiry/2003-April/000255.html
 +
20.  http://stderr.org/pipermail/inquiry/2003-April/000256.html
 +
21.  http://stderr.org/pipermail/inquiry/2003-April/000257.html
 +
22.  http://stderr.org/pipermail/inquiry/2003-April/000258.html
 +
23.  http://stderr.org/pipermail/inquiry/2003-April/000259.html
 +
24.  http://stderr.org/pipermail/inquiry/2003-April/000260.html
 +
25.  http://stderr.org/pipermail/inquiry/2003-April/000261.html
 +
26.  http://stderr.org/pipermail/inquiry/2003-April/000262.html
 +
27.  http://stderr.org/pipermail/inquiry/2003-April/000263.html
 +
28.  http://stderr.org/pipermail/inquiry/2003-April/000264.html
 +
29.  http://stderr.org/pipermail/inquiry/2003-April/000265.html
 +
30.  http://stderr.org/pipermail/inquiry/2003-April/000267.html
 +
31.  http://stderr.org/pipermail/inquiry/2003-April/000268.html
 +
32.  http://stderr.org/pipermail/inquiry/2003-April/000269.html
 +
33.  http://stderr.org/pipermail/inquiry/2003-April/000270.html
 +
34.  http://stderr.org/pipermail/inquiry/2003-April/000271.html
 +
35.  http://stderr.org/pipermail/inquiry/2003-April/000273.html
 +
36.  http://stderr.org/pipermail/inquiry/2003-April/000274.html
 +
37.  http://stderr.org/pipermail/inquiry/2003-April/000275.html
 +
38.  http://stderr.org/pipermail/inquiry/2003-April/000276.html
 +
39.  http://stderr.org/pipermail/inquiry/2003-April/000277.html
 +
40.  http://stderr.org/pipermail/inquiry/2003-April/000278.html
 +
41.  http://stderr.org/pipermail/inquiry/2003-April/000279.html
 +
42.  http://stderr.org/pipermail/inquiry/2003-April/000280.html
 +
43.  http://stderr.org/pipermail/inquiry/2003-April/000281.html
 +
44.  http://stderr.org/pipermail/inquiry/2003-April/000282.html
 +
45.  http://stderr.org/pipermail/inquiry/2003-April/000283.html
 +
46.  http://stderr.org/pipermail/inquiry/2003-April/000284.html
 +
47.  http://stderr.org/pipermail/inquiry/2003-April/000285.html
 +
48.  http://stderr.org/pipermail/inquiry/2003-April/000286.html
 +
49.  http://stderr.org/pipermail/inquiry/2003-April/000287.html
 +
50.  http://stderr.org/pipermail/inquiry/2003-April/000288.html
 +
51.  http://stderr.org/pipermail/inquiry/2003-April/000289.html
 +
52.  http://stderr.org/pipermail/inquiry/2003-April/000290.html
 +
53.  http://stderr.org/pipermail/inquiry/2003-April/000291.html
 +
54.  http://stderr.org/pipermail/inquiry/2003-April/000294.html
 +
55.  http://stderr.org/pipermail/inquiry/2003-April/000295.html
 +
56.  http://stderr.org/pipermail/inquiry/2003-April/000296.html
 +
57.  http://stderr.org/pipermail/inquiry/2003-April/000297.html
 +
58.  http://stderr.org/pipermail/inquiry/2003-April/000298.html
 +
59.  http://stderr.org/pipermail/inquiry/2003-April/000299.html
 +
60.  http://stderr.org/pipermail/inquiry/2003-April/000300.html
 +
61.  http://stderr.org/pipermail/inquiry/2003-April/000301.html
 +
62.  http://stderr.org/pipermail/inquiry/2003-April/000302.html
 +
63.  http://stderr.org/pipermail/inquiry/2003-April/000303.html
 +
64.  http://stderr.org/pipermail/inquiry/2003-April/000305.html
 +
65.  http://stderr.org/pipermail/inquiry/2003-April/000306.html
 +
66.  http://stderr.org/pipermail/inquiry/2003-April/000307.html
 +
67.  http://stderr.org/pipermail/inquiry/2003-April/000308.html
 +
68.  http://stderr.org/pipermail/inquiry/2003-April/000309.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
LOR.  Logic Of Relatives -- 2004
 +
 +
00.  http://stderr.org/pipermail/inquiry/2004-November/thread.html#1750
 +
01.  http://stderr.org/pipermail/inquiry/2004-November/001750.html
 +
02.  http://stderr.org/pipermail/inquiry/2004-November/001751.html
 +
03.  http://stderr.org/pipermail/inquiry/2004-November/001752.html
 +
04.  http://stderr.org/pipermail/inquiry/2004-November/001753.html
 +
05.  http://stderr.org/pipermail/inquiry/2004-November/001754.html
 +
06.  http://stderr.org/pipermail/inquiry/2004-November/001760.html
 +
07.  http://stderr.org/pipermail/inquiry/2004-November/001769.html
 +
08.  http://stderr.org/pipermail/inquiry/2004-November/001774.html
 +
09.  http://stderr.org/pipermail/inquiry/2004-November/001783.html
 +
10.  http://stderr.org/pipermail/inquiry/2004-November/001794.html
 +
11.  http://stderr.org/pipermail/inquiry/2004-November/001812.html
 +
12.  http://stderr.org/pipermail/inquiry/2004-November/001842.html
 +
13.
 +
 +
LOR.  Logic Of Relatives -- Commentary
 +
 +
00.    http://stderr.org/pipermail/inquiry/2004-November/thread.html#1755
 +
01.    http://stderr.org/pipermail/inquiry/2004-November/001755.html
 +
02.    http://stderr.org/pipermail/inquiry/2004-November/001756.html
 +
03.    http://stderr.org/pipermail/inquiry/2004-November/001757.html
 +
04.    http://stderr.org/pipermail/inquiry/2004-November/001758.html
 +
05.    http://stderr.org/pipermail/inquiry/2004-November/001759.html
 +
06.    http://stderr.org/pipermail/inquiry/2004-November/001761.html
 +
07.    http://stderr.org/pipermail/inquiry/2004-November/001770.html
 +
08.1.  http://stderr.org/pipermail/inquiry/2004-November/001775.html
 +
08.2.  http://stderr.org/pipermail/inquiry/2004-November/001776.html
 +
08.3.  http://stderr.org/pipermail/inquiry/2004-November/001777.html
 +
08.4.  http://stderr.org/pipermail/inquiry/2004-November/001778.html
 +
08.5.  http://stderr.org/pipermail/inquiry/2004-November/001781.html
 +
08.6.  http://stderr.org/pipermail/inquiry/2004-November/001782.html
 +
09.1.  http://stderr.org/pipermail/inquiry/2004-November/001787.html
 +
09.2.  http://stderr.org/pipermail/inquiry/2004-November/001788.html
 +
09.3.  http://stderr.org/pipermail/inquiry/2004-November/001789.html
 +
09.4.  http://stderr.org/pipermail/inquiry/2004-November/001790.html
 +
09.5.  http://stderr.org/pipermail/inquiry/2004-November/001791.html
 +
09.6.  http://stderr.org/pipermail/inquiry/2004-November/001792.html
 +
09.7.  http://stderr.org/pipermail/inquiry/2004-November/001793.html
 +
10.01.  http://stderr.org/pipermail/inquiry/2004-November/001795.html
 +
10.02.  http://stderr.org/pipermail/inquiry/2004-November/001796.html
 +
10.03.  http://stderr.org/pipermail/inquiry/2004-November/001797.html
 +
10.04.  http://stderr.org/pipermail/inquiry/2004-November/001798.html
 +
10.05.  http://stderr.org/pipermail/inquiry/2004-November/001799.html
 +
10.06.  http://stderr.org/pipermail/inquiry/2004-November/001800.html
 +
10.07.  http://stderr.org/pipermail/inquiry/2004-November/001801.html
 +
10.08.  http://stderr.org/pipermail/inquiry/2004-November/001802.html
 +
10.09.  http://stderr.org/pipermail/inquiry/2004-November/001803.html
 +
10.10.  http://stderr.org/pipermail/inquiry/2004-November/001804.html
 +
10.11.  http://stderr.org/pipermail/inquiry/2004-November/001805.html
 +
11.01.  http://stderr.org/pipermail/inquiry/2004-November/001813.html
 +
11.02.  http://stderr.org/pipermail/inquiry/2004-November/001814.html
 +
11.03.  http://stderr.org/pipermail/inquiry/2004-November/001815.html
 +
11.04.  http://stderr.org/pipermail/inquiry/2004-November/001816.html
 +
11.05.  http://stderr.org/pipermail/inquiry/2004-November/001817.html
 +
11.06.  http://stderr.org/pipermail/inquiry/2004-November/001818.html
 +
11.07.  http://stderr.org/pipermail/inquiry/2004-November/001819.html
 +
11.08.  http://stderr.org/pipermail/inquiry/2004-November/001820.html
 +
11.09.  http://stderr.org/pipermail/inquiry/2004-November/001821.html
 +
11.10.  http://stderr.org/pipermail/inquiry/2004-November/001822.html
 +
11.11.  http://stderr.org/pipermail/inquiry/2004-November/001823.html
 +
11.12.  http://stderr.org/pipermail/inquiry/2004-November/001824.html
 +
11.13.  http://stderr.org/pipermail/inquiry/2004-November/001825.html
 +
11.14.  http://stderr.org/pipermail/inquiry/2004-November/001826.html
 +
11.15.  http://stderr.org/pipermail/inquiry/2004-November/001827.html
 +
11.16.  http://stderr.org/pipermail/inquiry/2004-November/001828.html
 +
11.17.  http://stderr.org/pipermail/inquiry/2004-November/001829.html
 +
11.18.  http://stderr.org/pipermail/inquiry/2004-November/001830.html
 +
11.19.  http://stderr.org/pipermail/inquiry/2004-November/001831.html
 +
11.20.  http://stderr.org/pipermail/inquiry/2004-November/001832.html
 +
11.21.  http://stderr.org/pipermail/inquiry/2004-November/001833.html
 +
11.22.  http://stderr.org/pipermail/inquiry/2004-November/001834.html
 +
11.23.  http://stderr.org/pipermail/inquiry/2004-November/001835.html
 +
11.24.  http://stderr.org/pipermail/inquiry/2004-November/001836.html
 +
12.    http://stderr.org/pipermail/inquiry/2004-November/001843.html
 +
13.
 +
 +
LOR-DIS.  Logic Of Relatives -- Discussion
 +
 +
00.  http://suo.ieee.org/ontology/thrd20.html#04460
 +
00.  http://stderr.org/pipermail/inquiry/2004-November/thread.html#1768
 +
00.  http://stderr.org/pipermail/inquiry/2005-January/thread.html#2301
 +
 +
10.  http://suo.ieee.org/ontology/msg04460.html
 +
11.  http://suo.ieee.org/ontology/msg04461.html
 +
12.  http://suo.ieee.org/ontology/msg04471.html
 +
13.  http://suo.ieee.org/ontology/msg04472.html
 +
14.  http://suo.ieee.org/ontology/msg04475.html
 +
15.  http://suo.ieee.org/ontology/msg04476.html
 +
16.  http://suo.ieee.org/ontology/msg04477.html
 +
17.  http://suo.ieee.org/ontology/msg04479.html
 +
18.  http://suo.ieee.org/ontology/msg04480.html
 +
19.  http://suo.ieee.org/ontology/msg04481.html
 +
20.  http://suo.ieee.org/ontology/msg04482.html
 +
21.  http://suo.ieee.org/ontology/msg04483.html
 +
22.  http://suo.ieee.org/ontology/msg04485.html
 +
23.  http://suo.ieee.org/ontology/msg04486.html
 +
24.  http://suo.ieee.org/ontology/msg04493.html
 +
25.  http://suo.ieee.org/ontology/msg04494.html
 +
26.  http://suo.ieee.org/ontology/msg04495.html
 +
27.  http://suo.ieee.org/ontology/msg04496.html
 +
28.  http://stderr.org/pipermail/inquiry/2004-November/001768.html
 +
29.  http://stderr.org/pipermail/inquiry/2004-November/001838.html
 +
30.  http://stderr.org/pipermail/inquiry/2004-November/001840.html
 +
31.  http://stderr.org/pipermail/inquiry/2005-January/002301.html
 +
32.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
</pre>
Jon Awbrey
12,089

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/51102"