Peirce's 1870 “Logic of Relatives”

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Preliminaries
https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

All,

I need to return to my study of Peirce’s 1870 Logic of Relatives,
and I thought it might be more pleasant to do that on my blog than
to hermit away on the wiki where I last left off.

Peirce’s 1870 “Logic of Relatives” • Part 1
===========================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1

Peirce’s text employs lower case letters for logical terms of general reference
and upper case letters for logical terms of individual reference. General terms
fall into types, namely, absolute terms, dyadic relative terms, and higher adic
relative terms, and Peirce employs different typefaces to distinguish these.
The following Tables indicate the typefaces used in the text below for Peirce’s
examples of general terms.

Table 1. Absolute Terms (Monadic Relatives)
https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-absolute-terms-monadic-relatives-2.0.png

Table 2. Simple Relative Terms (Dyadic Relatives)
https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-simple-relative-terms-dyadic-relatives-2.0.png

Table 3. Conjugative Terms (Higher Adic Relatives)
https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-conjugative-terms-higher-adic-relatives-2.0.png

Individual terms are taken to denote individual entities falling under
a general term. Peirce uses upper case Roman letters for individual terms,
for example, the individual horses H, H′, H″ falling under the general term h
for horse.

The path to understanding Peirce’s system and its wider implications
for logic can be smoothed by paraphrasing his notations in a variety
of contemporary mathematical formalisms, while preserving the semantics
as much as possible. Remaining faithful to Peirce’s orthography while
adding parallel sets of stylistic conventions will, however, demand close
attention to typography-in-context. Current style sheets for mathematical
texts specify italics for mathematical variables, with upper case letters
for sets and lower case letters for individuals. So we need to keep an
eye out for the difference between the individual X of the genus x and
the element x of the set X as we pass between the two styles of text.

References
==========

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives,
Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”,
Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.
Reprinted, Collected Papers (CP 3.45–149), Chronological Edition (CE 2, 359–429).
Online:
• https://www.jstor.org/stable/25058006
• https://archive.org/details/jstor-25058006
• https://books.google.com/books?id=fFnWmf5oLaoC

• Peirce, C.S., Collected Papers of Charles Sanders Peirce,
vols. 1–6, Charles Hartshorne and Paul Weiss (eds.),
vols. 7–8, Arthur W. Burks (ed.), Harvard University Press,
Cambridge, MA, 1931–1935, 1958. Cited as (CP volume.paragraph).

• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition,
Peirce Edition Project (eds.), Indiana University Press, Bloomington and
Indianapolis, IN, 1981–. Cited as (CE volume, page).

Resources
=========

• Peirce’s 1870 Logic of Relatives
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 1
https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/

All,

We pick up the text at §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 1
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_1

<QUOTE CSP>

Use of the Letters
==================

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.

(Peirce, CP 3.63)
</QUOTE?

One thing that strikes me about the above passage is a pattern
of argument I can recognize as invoking a closure principle.
This is a figure of reasoning Peirce uses in three other places:
his discussion of continuous predicates, his definition of a
sign relation, and his formulation of the pragmatic maxim itself.

One might also call attention to the following two statements:

<QUOTE CSP>

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.
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 2
https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-selection-2/

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 2
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_2

<QUOTE CSP>

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].

(Peirce, CP 3.65)
</QUOTE>

Peirce’s remarks at CP 3.65 are so replete with remarkable ideas,
some of them so taken for granted in mathematical discourse as usually
to escape explicit mention, others so suggestive of things to come in a
future remote from his time of writing, and yet so smoothly slipped into
the stream of thought that it’s all too easy to overlook their significance,
that all I can do to highlight their impact is to dress them up in different
words, whose main advantage is being more jarring to the mind’s sensibilities.

• This mapping of letters to numbers, or logical terms to mathematical quantities,
is the very core of what quantification theory is all about, definitely more to
the point than the mere “innovation” of using distinctive symbols for the
so-called quantifiers.

• 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.

• 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.

• 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?

• I also observe that Peirce takes the individual objects of a particular universe
of discourse in a generative way, as opposed to a totalizing way, and thus these
contingent individuals afford us with a basis for talking freely about collections,
constructions, properties, qualities, subsets, and higher types built up thereon.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 3
https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/

All,

We move on to the next part of §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 3
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_3

<QUOTE CSP>

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.

(Peirce, CP 3.66)
</QUOTE>

The quantifier mapping from terms to numbers that Peirce signifies by means
of the square bracket notation [t] has one of its principal uses in providing
a basis for the computation of frequencies, probabilities, and all the other
statistical measures constructed from them, and thus in affording 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 abduce on 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 another way of saying it
addresses the question: What may be hoped? We have to face the possibility 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.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 4
https://inquiryintoinquiry.com/2014/01/31/peirces-1870-logic-of-relatives-selection-4/

All,

Here is the next part of §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 4
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_4

<QUOTE CSP>

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.

(Peirce, CP 3.67)
</QUOTE>

A wealth of issues arises here that I hope to take up in depth
at a later point, but for the moment I shall be able to mention
only the barest sample of them in passing.

The two papers precedinge 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.

In setting up a correspondence between “letters” and “numbers”,
Peirce constructs a structure-preserving map from a logical domain
to a numerical domain. That he does this deliberately is evidenced
by the care that he takes with the conditions under which the chosen
aspects of structure are preserved, along with his recognition of the
critical fact that zeroes are preserved by the mapping.

Incidentally, Peirce appears to have an inkling of the problems
that would later be caused by using the plus sign for inclusive
disjunction, but his advice was overridden by the dialects of
applied logic that developed in various communities, retarding
the exchange of information among engineering, mathematical, and
philosophical specialties all throughout the subsequent century.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 5
https://inquiryintoinquiry.com/2014/02/04/peirces-1870-logic-of-relatives-selection-5/
Note. Please follow the link above for better formatting of Peirce's text,
as some of his typographical distinctions are lost in the transcript below.

All,

On to the next part of §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 5
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_5

<QUOTE CSP>

The Signs for Multiplication
============================

I shall adopt for the conception of multiplication the application of
a relation, in such a way that, for example, ℓw shall denote whatever is
lover of a woman. This notation is the same as that used by Mr. De Morgan,
although he appears not to have had multiplication in his mind.

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) = sm +, sw.

(ℓ +, s)w will denote whatever is lover or servant to a woman, and:

(ℓ +, s)w = ℓw +, ℓw.

(sℓ)w will denote whatever stands to a woman in the relation of servant of a lover, and:

(sℓ)w = s(ℓw).

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:

x1 = 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.

(Peirce, CP 3.68)
<QUOTE>

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 those characters.

Sign relations, like any brand of non-trivial triadic 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 we would normally use to
control the complexity, like neglecting one of the domains,
in effect, projecting the triadic sign relation onto one
of its dyadic faces, or focusing on a single ordered triple
(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 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.

Figure 1 shows 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.

Figure 1. Equivalent Terms “v” = “p”
https://inquiryintoinquiry.files.wordpress.com/2014/02/lor-1870-figure-1.jpg

Depending on whether we interpret the terms “v” and “p” as applying to
persons who hold the offices at one particular time or as applying to
all persons who have held the offices over an extended period of history,
their denotations may be either singular of plural, respectively.

Terms referring to many objects are known as having “general denotations”
or “plural referents”. They may be represented in the above style of
picture by drawing an ellipsis of three nodes like “o o o” at the
object ends of sign relational triples.

For a more complicated example, Figure 2 shows how I would picture
Peirce’s example of an equivalence between terms which comes about
by applying the distributive law for relative multiplication over
absolute summation.

Figure 2. Equivalent Terms “s(m +, w)” = “sm +, sw”
https://inquiryintoinquiry.files.wordpress.com/2014/02/lor-1870-figure-2.jpg

Resources
=========

Sign Relations
https://oeis.org/wiki/Sign_relation

Triadic Relations
https://oeis.org/wiki/Triadic_relation

Information = Comprehension × Extension
https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 6
https://inquiryintoinquiry.com/2014/02/05/peirces-1870-logic-of-relatives-selection-6/

Note. Please follow the link above for better formatting of Peirce's text,
as some of his typographical distinctions are lost in the transcript below.

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 6
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_6

The application of a relation is one of the most basic operations
in Peirce’s logic. Because relation applications are so pervasive and
because Peirce treats them on the pattern of algebraic multiplication,
the part of §3 concerned with “The Signs for Multiplication” will occupy
our attention for many days to come.

<QUOTE CSP>

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:

gxy

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),

goh

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.

goh = go(H +, H′+, H″ +, etc.) = goH +, goH′+, goH″ +, 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 goh. 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:

gℓwh,

we have written giver of a woman to a lover of her,
and if we add brackets, thus,

g(ℓw)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:

g₁₂ℓ₁wh = g₁₁ℓ₂hw = g₂₍₋₁₎hℓ₁w,

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,

ℓ₀

denotes the lover of “that” lover or the lover of himself, just as
goh 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:

ℓ₀ = L₀ +, L′₀ +, L″₀ +, etc.

A subjacent sign of infinity may indicate
that the correlate is indeterminate, so that:

ℓ_∞

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:

ℓ₁ = ℓ,

g₁₁ = g₁ = g.

This enables us to retain our former expressions ℓw, goh, etc.

(Peirce, CP 3.69–70)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic Of Relatives” • Discussion 3
https://inquiryintoinquiry.com/2021/12/10/peirces-1870-logic-of-relatives-discussion-3/

| All other sciences without exception depend upon the
| principles of mathematics; and mathematics borrows
| nothing from them but hints.
|
| C.S. Peirce • “Logic of Number”

| A principal intention of this essay is to separate what
| are known as algebras of logic from the subject of logic,
| and to re-align them with mathematics.
|
| G. Spencer Brown • Laws of Form

Re: Peirce’s 1870 “Logic Of Relatives” • Overview
https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/
Re: Laws of Form
https://groups.io/g/lawsofform/topic/peirce_s_1870_logic_of/87355251
::: James Bowery
https://groups.io/g/lawsofform/message/1328
https://groups.io/g/lawsofform/message/1332

Dear James,

I am pleased to see you engaging the material
on Peirce's Logic of Relatives. For my part
I'll need to lay out several more Selections
before the major themes of Peirce's essay
begin to emerge from the supporting but
sometimes distracting details.

In the meantime two clues to the Big Picture can be gleaned from the
paired epigraphs I put up in lights at the top of the post. For what
we have here is a return to the thrilling days of yesteryear when the
mathematics of logic was still mathematics, shortly before Frege (maybe
unwittingly) and Russell (in a way less wittingly) detoured it down the
linguistic U‑turn to nominalism.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Sets as Sums
https://inquiryintoinquiry.com/2014/02/06/peirces-1870-logic-of-relatives-sets-as-sums/

Comment on Peirce’s 1870 “Logic of Relatives” • Sets as Sums
============================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Comment_:_Sets_as_Logical_Sums

All,

Peirce’s way of representing sets as logical sums may seem archaic,
but it’s quite often used in mathematics and remains the tool of
choice in many branches of algebra, combinatorics, computing,
and statistics to this day.

Peirce applied this genre of representation to logic in fairly
novel ways and the degree to which he elaborated its use in the
logic of relative terms is certainly original with him, but this
particular device, going under the handle of “generating functions”,
goes way back, well before anyone thought of sticking a flag in set
theory as a separate territory or of trying to fence off our native
possessions of classes and collections with explicit decrees of axioms.

And back in the days when a “computer” was simply a person who computed,
well before the advent of electronic computers we take for granted today,
mathematicians commonly used generating functions as a rough and ready sort
of addressable memory to organize, store, and keep track of their accounts on
a wide variety of formal objects.

Let’s look at a few simple examples of generating functions,
much as I encountered them during my own first adventures
in the Realm of Combinatorics.

Suppose 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 e have, in choosing a subset
of {a, b, c}, to exclude the element “a” (signified by the “1”), or
else to include it (signified by the “a”), proceeding in a similar
fashion with the other elements in their turn.

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 way
that it strikes exactly one of the two bumpers
on each 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 the product (1 + a)(1 + b)(1 + c), one obtains:

1 + a + b + c + ab + ac + bc + abc.

This informs us that the subsets of choice are:

∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.

And so they are.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 7
https://inquiryintoinquiry.com/2014/02/07/peirces-1870-logic-of-relatives-selection-7/

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 7
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_7

<QUOTE CSP>

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:

[Display] Giver of a Horse to a Lover of a Woman
https://inquiryintoinquiry.files.wordpress.com/2021/12/peirces-1870-lor-e280a2-giver-of-a-horse-to-a-lover-of-a-woman.png

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 “goh” a single
letter for “oh”.

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 [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 [Gothic], where only one
was necessary. But it is convenient to use both.

(Peirce, CP 3.71–72)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax
https://inquiryintoinquiry.com/2014/02/12/peirces-1870-logic-of-relatives-proto-graphical-syntax/

Comment on Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax
======================================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Comment_:_Proto-Graphical_Syntax

All,

It is clear from our last Selection that Peirce is already on
the verge of a graphical syntax for the logic of relative terms.
Indeed, it is likely he had already reached that point in his
own thinking some time before.

For instance, it seems quite impossible for a person with any graphical
sensitivity whatever to scan that last variation on “giver of a horse
to a lover of a woman” without drawing or at least imagining lines
of identity to connect the corresponding marks of reference,
as shown in the following Figure.

Figure 3. Giver of a Horse to a Lover of a Woman
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-glwh.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Discussion 4
https://inquiryintoinquiry.com/2021/12/15/peirces-1870-logic-of-relatives-discussion-4/

Re: Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax
https://inquiryintoinquiry.com/2014/02/12/peirces-1870-logic-of-relatives-proto-graphical-syntax/
Re: FB | Ancient Logic
https://www.facebook.com/groups/204404706587604/posts/1543628845998510
::: Henning Engebretsen
https://www.facebook.com/groups/204404706587604/posts/1543628845998510/?comment_id=1543632945998100

<QUOTE HE:>
What's your point, it's obviously too graphical, but
perhaps you are driving at something else. Explain?
</QUOTE>

Dear Henning,

I wasn't sure if your “too” was intended to mean “also graphical”
or “overly graphical” but what I'm gearing up to do here is a careful
survey of the source from which radiated all our most lucid graphical
systems of logic, from Peirce's own entitative and existential graphs,
to Spencer Brown's calculus of indications, to John Sowa's conceptual
graphs. The first glimmerings of that evolution go back further than
widely appreciated, to Peirce's 1870 “Logic of Relatives” at least,
and I'm hoping in time to make that clear.

Resource
========

Peirce’s 1870 “Logic of Relatives” • Overview
https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 8
https://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-selection-8/

Note. Please follow the link above for the proper formatting of Peirce's text,
as many of his typographical distinctions are lost in the following transcript.

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 8
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_8

<QUOTE CSP>

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,sw

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

goh

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,,sw is not the same as ℓ,sw (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,sw = ℓ,sw,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,₀w = sw

and the other is infinity; for as 1^∞ is indeterminate in ordinary algebra,
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,_∞

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_∞.

I shall for brevity frequently express this by an antique figure one 1.

“Anything” by:

1₀.

I shall often also write a straight 1 for anything.

(Peirce, CP 3.73)
</QUOTE>

Regards.

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.1
https://inquiryintoinquiry.com/2014/02/18/peirces-1870-logic-of-relatives-comment-8-1/

Peirce’s 1870 “Logic of Relatives” • Comment 8.1
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.1

All,

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.

Let us invent a concrete example to illustrate the use of Peirce’s notation.
Imagine a discourse whose universe X will remind us 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 afford ourselves the convenience of abbreviated terms
while preserving Peirce’s conventions about capitalization, we may
use the alternate terms “u” for the universe X and “Jeste” for the
character Clown. This permits the above description of the universe
of discourse to be rewritten 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 as follows.

“B”, “C”, “D”, “E”, “I”, “J”, “O”

Each of these terms denotes in a singular fashion
the corresponding individual in X.

By way of “general terms” in this discussion,
we may begin with the following set.

“b” = “black”

“m” = “man”

“w” = “woman”

The denotation of a general term may be given
by means of an equation between terms.

b = O

m = C +, I +, J +, O

w = B +, D +, E

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.2
https://inquiryintoinquiry.com/2014/02/18/peirces-1870-logic-of-relatives-comment-8-2/

All,

I continue with my commentary on CP 3.73, developing the “Othello” example
as a way of illustrating Peirce’s formalism.

Peirce’s 1870 “Logic of Relatives” • Comment 8.2
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.2

In the development of the story so far, we have a universe
of discourse characterized by the following equations.

1 = B +, C +, D +, E +, I +, J +, O

b = O

m = C +, I +, J +, O

w = B +, D +, E

This much forms a basis for the collection of absolute terms
to be used in the example. Let us now consider how we might
represent an exemplary collection of relative terms.

Consider the genesis of relative terms, for example:

“lover of____”

“betrayer to____of____”

“winner over of____to____from____”

We may regard these fill-in-the-blank forms as being derived by
a kind of rhematic abstraction from the corresponding instances
of absolute terms.

The following examples illustrate the relationships existing among
absolute terms, relative terms, relations, and elementary relations.

• The relative term “lover of____” can be derived from
the absolute term “lover of____Emilia” by removing the
absolute term “Emilia”.

Iago is a lover of Emilia, so the relate-correlate pair I:E
is an element of the dyadic relation associated with the
relative term “lover of____”.

• The relative term “betrayer to____of____” can be derived from
the absolute term “betrayer to Othello of Desdemona” by removing
the absolute terms “Othello” and “Desdemona”.

Iago is a betrayer to Othello of Desdemona, so the elementary relative
term I:O:D is an element of the triadic relation associated with the
relative term “betrayer to____of____”.

• The relative term “winner over of____to____from____” can be derived from
the absolute term “winner over of Othello to Iago from Cassio” by removing
the absolute terms “Othello”, “Iago”, and “Cassio”.

Iago is a winner over of Othello to Iago from Cassio, so the elementary
relative term I:O:I:C is an element of the tetradic relation associated
with the relative term “winner over of____to____from____”.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.3
https://inquiryintoinquiry.com/2014/02/18/peirces-1870-logic-of-relatives-comment-8-3/

Peirce’s 1870 “Logic of Relatives” • Comment 8.3
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.3

All,

I continue with my commentary on CP 3.73, developing the “Othello” example
as a way of illustrating Peirce’s formalism.

It is critically important to distinguish a “relation” from a “relative term”.

• The “relation” is an object of thought which may be
regarded “in extension” as a set of ordered tuples
known as its “elementary relations”.

• The “relative term” is a sign which denotes certain objects,
called its “relates”, as these are determined in relation to
certain other objects, called its “correlates”. Under most
circumstances the relative term may be taken to denote the
corresponding relation.

Returning to the “Othello” example, let us consider the
dyadic relatives “lover of____” and “servant of____”.

The relative term ℓ equivalent to the rhematic expression
“lover of____” is given by the following equation.

ℓ = B:C +, C:B +, D:O +, E:I +, I:E +, O:D

In the interests of simplicity, let’s put aside all distinctions
of rank and fealty, collapsing the motley crews of servant and
subordinate under the heading of a single service, denoted by
the relative term s for “servant of____”. The terms of this
unified service are given by the following equation.

s = C:O +, E:D +, I:O +, J:D +, J:O

The elementary relation I:C under s might be implied by
the plot of the play but since it is so hotly arguable
I will leave it out of the toll.

One thing more we need to watch out for: There are many conventions
in the field regarding the ordering of terms in their applications and
different conventions are more convenient under different circumstances,
so there’s little chance any one of them can be canonized once and for all.
In our current reading we apply relative terms from right to left and our
conception of relative multiplication, or relational composition, needs
to be adjusted accordingly.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.4
https://inquiryintoinquiry.com/2014/02/19/peirces-1870-logic-of-relatives-comment-8-4/

Peirce’s 1870 “Logic of Relatives” • Comment 8.4
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.4

All,

To familiarize ourselves with the forms of calculation
available in Peirce’s notation, let us compute a few of
the simplest products we find in the “Othello” universe.

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 dyadic 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:

ℓ1 = lover of anything
= (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
= anything except J

ℓb = lover of a black
= (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)
× O
= D

ℓm = lover of a man
= (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)
× (C +, I +, J +, O)

= B +, D +, E

ℓw = lover of a woman
= (B:C +, C:B +, D:O +, E:I +, I:E +, O:D)
× (B +, D +, E)
= C +, I +, O

s1 = servant of anything
= (C:O +, E:D +, I:O +, J:D +, J:O)
× (B +, C +, D +, E +, I +, J +, O)
= C +, E +, I +, J

sb = servant of a black
= (C:O +, E:D +, I:O +, J:D +, J:O)
× O

= C +, I +, J

sm = servant of a man
= (C:O +, E:D +, I:O +, J:D +, J:O)
× (C +, I +, J +, O)

= C +, I +, J

sw = 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 sees the relative terms ℓ and s
do not commute, in other words, ℓs is not equal to sℓ.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.4 (revised)
https://inquiryintoinquiry.com/2014/02/19/peirces-1870-logic-of-relatives-comment-8-4/

Thanks, Mary, I've edited the examples. I think the problem is using
“a black” as a substantive term, which has objectionable connotations
even in contexts where using “black” as an adjective is currently okay.
I'll have to check whether it was Shakespeare's or Peirce's usage, but
I really should have caught it in any case.

Regards,

Jon

On 12/21/2021 4:16 PM, Mary Libertin wrote:
> Hi Jon,
>
> I enjoy reading some of your posts. They are quite a challenge.
> The passages you work with in this current post are directly
> connected with Peirce, which is as it should be.
>
> But can you change the racist language “a black” to another term
> or at least use an asterisk, an asterisk that addresses the issues
> with modern readers, basically an acknowledgment that you are aware
> of the current response to such offensive language? It doesn’t change
> the logic at all.
>
> Thanks!
> Dr. Mary Libertin
> Emeritus professor of English
>

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.5
https://inquiryintoinquiry.com/2014/02/20/peirces-1870-logic-of-relatives-comment-8-5/

Peirce’s 1870 “Logic of Relatives” • Comment 8.5
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.5

All,

I'm breaking the email version of this Comment into sections
on account of the abundance of Figures and attachments in it.

[Section 1]

Since multiplication by a dyadic relative term is a logical analogue
of matrix multiplication in linear algebra, all the products computed
above can be represented by “logical matrices”, that is, by arrays of
boolean {0, 1} coordinate values. Absolute terms and dyadic relatives
are represented as 1-dimensional and 2-dimensional arrays, respectively.

The equations defining the absolute terms are given again below,
first as logical sums of individual terms and then as n-tuples
of boolean coordinates.

Display 1. Othello Universe
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-universe-small.png

Since we are going to be regarding these tuples as column arrays,
it is convenient to arrange them into a table of the following form.

Display 2. Othello Column Array
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-column-array-large.png

Here are the dyadic 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.

Display 3. ℓ = B:C +, C:B +, D:O +, E:I +, I:E +, O:D
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-l.png

Display 4. s = C:O +, E:D +, I:O +, J:D +, J:O
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-s.png

To be continued ...

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.5
https://inquiryintoinquiry.com/2014/02/20/peirces-1870-logic-of-relatives-comment-8-5/

Peirce’s 1870 “Logic of Relatives” • Comment 8.5
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.5

All,

Earlier we took the dyadic relatives ℓ and s and
multiplied them on the absolute terms 1, O, m, w.
Those 8 products are now represented as products
of logical matrices. (I'll post just the links
to the PNGs as there are such a large number.)

ℓ1 = lover of anything

https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-l1.png

ℓO = lover of Othello
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-lo.png

ℓm = lover of a man

https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-lm.png

ℓw = lover of a woman

https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-lw.png

s1 = servant of anything

https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-s1.png

sO = servant of Othello
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-so.png

sm = servant of a man

https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-sm.png

sw = servant of a woman

https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-sw.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.5
https://inquiryintoinquiry.com/2014/02/20/peirces-1870-logic-of-relatives-comment-8-5/

Peirce’s 1870 “Logic of Relatives” • Comment 8.5
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.5

All,

Last but not least, here are the matrix representations of
the two products of dyadic relatives we computed earlier.

ℓs = lover of a servant of ———
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-ls.png

sℓ = servant of a lover of ———
https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-logical-matrix-sl.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.6
https://inquiryintoinquiry.com/2014/02/23/peirces-1870-logic-of-relatives-comment-8-6/

Peirce’s 1870 “Logic of Relatives” • Comment 8.6
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_8.6

All,

The foregoing has hopefully filled in enough background that we
can begin to make sense of the more mysterious parts of CP 3.73.

<QUOTE CSP>

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.

(Peirce, CP 3.73)
</QUOTE>

In any system where elements are organized according to types
there tend to be any number of ways in which elements of one
type are naturally associated with elements of another type.
If the association is anything like a logical equivalence,
but with the first type being lower and the second type
being higher in some sense, then one may speak of a
“semantic ascent” from the lower to the higher type.

For example, it is common in mathematics to associate an
element _a_ of a set A with the constant function fₐ : X → A
which has fₐ(x) = a for all x in X, where X is an arbitrary
set which is fixed in the context of discussion. Indeed, the
correspondence is so close that one often uses the same name “a”
to denote both the element a in A and the function a = fₐ : X → A,
relying on context or an explicit type indication to tell them apart.

For another example, we have the “tacit extension” of a k-place relation
L ⊆ X₁ × … × Xₖ to a (k+1)-place relation L′ ⊆ X₁ × … × Xₖ₊₁ which we get
by letting L' = L × Xₖ₊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 extension 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 a noble
= (C:C +, I:I +, J:J +, O:O)
×
(C +, D +, O)
= C +, O

n,m = noble that is a man
= (C:C +, D:D +, O:O)

×
(C +, I +, J +, O)

= C +, O

w,n = woman that is a noble
= (B:B +, D:D +, E:E)
×
(C +, D +, O)
= D

n,m = noble that is a woman
= (C:C +, D:D +, O:O)

×
(B +, D +, E)

= D

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 9
https://inquiryintoinquiry.com/2014/02/23/peirces-1870-logic-of-relatives-selection-9/

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 9
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_9

<QUOTE CSP>

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.

(Peirce, CP 3.74)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.1
https://inquiryintoinquiry.com/2014/02/24/peirces-1870-logic-of-relatives-comment-9-1/

Re: Peirce’s 1870 “Logic of Relatives” • Selection 8
https://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-selection-8/

Re: Peirce’s 1870 “Logic of Relatives” • Selection 9
https://inquiryintoinquiry.com/2014/02/23/peirces-1870-logic-of-relatives-selection-9/

All,

Perspective on Peirce’s use of the comma operator at CP 3.73 and CP 3.74 can
be gained by dropping back a few years and seeing how George Boole explained
his twin conceptions of “selective operations” and “selective symbols”.

<QUOTE Boole>
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 [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 [x^2 = x] of the literal symbols.

(Boole, Laws of Thought, 44–45)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.2
https://inquiryintoinquiry.com/2014/02/26/peirces-1870-logic-of-relatives-comment-9-2/

Peirce’s 1870 “Logic of Relatives” • Comment 9.2
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.2

All,

In setting up his discussion of selective operations and their
corresponding selective symbols, Boole writes the following.

<QUOTE Boole>

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, 43)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.3
https://inquiryintoinquiry.com/2014/02/26/peirces-1870-logic-of-relatives-comment-9-3/

Peirce’s 1870 “Logic of Relatives” • Comment 9.3
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.3

All,

An “idempotent element” x in an algebraic system is one
which obeys the idempotent law, that is, it satisfies the
equation xx = x. Under most circumstances it is usual to
write this as x² = x.

If the algebraic system in question falls under the additional laws
necessary to carry out the required transformations then x² = x is
convertible to x - x² = 0, and this in turn to x(1 - x) = 0.

If the algebraic system satisfies the requirements of a boolean algebra
then the equation x(1 - x) = 0 amounts to saying 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 Boole attaches to “selective
operations”. It will also link up with the statements Peirce makes regarding
his dimension-raising comma operation.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.4
https://inquiryintoinquiry.com/2014/02/27/peirces-1870-logic-of-relatives-comment-9-4/

Peirce’s 1870 “Logic of Relatives” • Comment 9.4
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.4

All,

Boole rationalizes the properties of what we now call “boolean
multiplication”, roughly equivalent to logical conjunction, in
terms of the laws which govern selective operations. Peirce,
in his turn, taking a radical step of analysis which has seldom
been recognized for what it would lead to, does not consider this
multiplication to be a fundamental operation, but derives it as
a by-product of relative multiplication by a comma relative.
In this way Peirce makes logical conjunction a special case
of relative composition.

This opens up a wide field of inquiry, “the operational significance of
logical terms”, 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
we entertained when last stopping there.

Here is the list of absolute terms we had been considering before:

Figure 1. Absolute Terms 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-absolute-terms-1-m-n-w.png

Here is the list of “comma inflexions” or
“diagonal extensions” of those terms:

1, = anything 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 the diagonal extension of *1* is
the same thing as the identity relation _1_.

Earlier we computed the following products, obtained by
applying the diagonal extensions of absolute terms to
the same set of absolute terms.

m,n = man that is a noble = C +, O

n,m = noble that is a man = C +, O

w,n = woman that is a noble = D

n,m = noble that is a woman = D

From that we take our first clue as to why the commutative law holds for
logical conjunction. More in the way of practical insight could be had
by working systematically through the collection of products generated
by the operational means at hand, namely, the products obtained by
appending a comma to each of the terms 1, m, n, w then applying
the resulting relatives to those selfsame terms again.

Before we venture into that territory, however, let us equip our intuitions
with the forms of graphical and matrical representation which served us so
well in our previous adventures.

Regards,

Jon

[Lenard Troncale]

Dear Jon,

I am very impressed by your constant output. I have never answered any of these messages, but I keep them all on my university server, now a listing of 432 of them. I just don’t know where to begin answering them. My own work is on my own Systems Processes Theory for which we have begun a SIG in ISSS. I suppose you know of the Singer’s work and that of Warfield as extended by the Simpson’s (he Joe and his wife, and Kevin Dye) until his (Joe’s) recent untimely death. I just wanted to compliment on your pursuit of these philosophies.

Dr. Len Troncale,
Professor Emeritus and Past Chair
Dept. of Biological Sciences,
Founding Director Emeritus, Inst. for Advanced Systems Studies,
College of Science
Currently Lecturer, Masters in Systems Engineering
IME Dept., College of Engineering,
California State Polytechnic University
3801 W. Temple Ave.
Pomona, California 91768
Eomega Covenant Office2 & Claremont Office:
232 Harrison, Suite B, Claremont, CA, 91711
President, General Systems Research, Development, and Consulting (GSRDC)
29th ISSS President; ISSS Managing Director, 1982-1989, ISSP Managing Director
Personal Cell/Mobile Phone: 909-374-6115
Email: lrtro...@cpp.edu

> On Dec 31, 2021, at 12:40 PM, Jon Awbrey <jaw...@att.net> wrote:
>
> Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.1

> https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Finquiryintoinquiry.com%2F2014%2F02%2F24%2Fpeirces-1870-logic-of-relatives-comment-9-1%2F&amp;data=04%7C01%7Clrtroncale%40cpp.edu%7C4994292054e445878daf08d9cc9dc8a4%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C1%7C637765800329394427%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&amp;sdata=LOdkvVz9%2B5boVsoB4WLUaL3IhxRe6FBWjLfEOAIMVCc%3D&amp;reserved=0

>
> Re: Peirce’s 1870 “Logic of Relatives” • Selection 8

> https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Finquiryintoinquiry.com%2F2014%2F02%2F17%2Fpeirces-1870-logic-of-relatives-selection-8%2F&amp;data=04%7C01%7Clrtroncale%40cpp.edu%7C4994292054e445878daf08d9cc9dc8a4%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C1%7C637765800329394427%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&amp;sdata=PdUdsuj94kt1z2gQ2iRPPuAkwkf3mXKLmEfSzG897A4%3D&amp;reserved=0

>
> Re: Peirce’s 1870 “Logic of Relatives” • Selection 9

> https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Finquiryintoinquiry.com%2F2014%2F02%2F23%2Fpeirces-1870-logic-of-relatives-selection-9%2F&amp;data=04%7C01%7Clrtroncale%40cpp.edu%7C4994292054e445878daf08d9cc9dc8a4%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C1%7C637765800329394427%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&amp;sdata=CPnX1E9VYuNPQWUZHV90hb1KkMzF8T%2FdeJ2lCGUHf6o%3D&amp;reserved=0

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[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.5
https://inquiryintoinquiry.com/2014/03/04/peirces-1870-logic-of-relatives-comment-9-5/

Peirce’s 1870 “Logic of Relatives” • Comment 9.5
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.5

All,

I'm breaking the email version of this Comment into sections
on account of the abundance of Figures and attachments in it.

[Section 1]

Peirce’s comma operation, in its application to an absolute term,
is tantamount to the representation of that term’s denotation as
an idempotent transformation, which is commonly represented as a
diagonal matrix. Hence the alternate name, “diagonal extension”.

An idempotent element x is given by the abstract condition that
xx = x but elements like those are commonly encountered in more
concrete circumstances, acting as operators or transformations
on other sets or spaces, and in that action they will often be
represented as matrices of coefficients.

Let’s see how this looks in the matrix and graph pictures
of absolute and relative terms.

Absolute Terms
==============

Display 1. Absolute Terms 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-absolute-terms-1-m-n-w.png

Previously, we represented absolute terms
as column arrays. The above four terms are
given by the columns of the following table.

Display 2. Othello Columns 1 M N W Large
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-columns-1-m-n-w-large.png

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Discussion 5
http://inquiryintoinquiry.com/2022/01/08/peirces-1870-logic-of-relatives-discussion-5/

Re: Conceptual Graphs
https://lists.cs.uni-kassel.de/hyperkitty/list/c...@lists.iccs-conference.org/thread/XE5S3FXYNYISP5RHFBA4KXXM22IKEWBL/
::: Peiyuan Zhu
https://lists.cs.uni-kassel.de/hyperkitty/list/c...@lists.iccs-conference.org/message/XE5S3FXYNYISP5RHFBA4KXXM22IKEWBL/

<QUOTE PZ:>
I’m studying imprecise probabilities which initially works as
an extension in Boole's Laws of Thoughts. It seems like Boole
was solving a set of algebraic equations for probabilities where
some of the probabilities do not have precise values therefore
need to be bounded. Has anyone studied Boole’s algebraic system
of probabilities? Is Peirce extending Boole's algebraic system
in his Logic of Relatives?
</QUOTE>

Dear Peiyuan,

Issues related to the ones you mention will come up in the Selections and
Commentary I'm posting on Peirce's 1870 Logic of Relatives, the full title
of which, “Description of a Notation for the Logic of Relatives, Resulting
from an Amplification of the Conceptions of Boole's Calculus of Logic”, is
sufficient hint of the author's intent, namely, to extend the correspondence
Boole discovered between the calculus of propositions and the statistics of
simple events to a correspondence between the calculus of relations and the
statistics of complex events, contingency matrices, higher order correlations,
and ultimately the full range of information theory.

But it will take a while to develop all that …

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.5 (cont.)

https://inquiryintoinquiry.com/2014/03/04/peirces-1870-logic-of-relatives-comment-9-5/

Peirce’s 1870 “Logic of Relatives” • Comment 9.5
================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.5

[Part 2]

All,

The types of graphs known as “bigraphs” or “bipartite graphs”
can be used to picture simple relative terms, dyadic relations,
and their corresponding logical matrices. One way to bring
absolute terms and their corresponding sets of individuals
into the bigraph picture is to mark the nodes in some way,
for example, hollow nodes for non-members and filled nodes
for members of the indicated set, as shown below.

Bigraph Nodes
=============

Figure 4.1. 1 = B +, C +, D +, E +, I +, J +, O
https://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-4-1.jpg

Figure 4.2. m = C +, I +, J +, O
https://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-4-2.jpg

Figure 4.3. n = C +, D +, O
https://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-4-3.jpg

Figure 4.4. w = B +, D +, E
https://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-4-4.jpg

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.5 (complete)

https://inquiryintoinquiry.com/2014/03/04/peirces-1870-logic-of-relatives-comment-9-5/

Peirce’s 1870 “Logic of Relatives” • Comment 9.5

https://inquiryintoinquiry.com/2014/03/04/peirces-1870-logic-of-relatives-comment-9-5/

All,

Thinking it would be useful to have the various forms of absolute
and relative terms before us all on one page, I've revamped the
graphics to the point where they have a good chance of fitting.
So let's see ...

⁂ ⁂ ⁂

Peirce’s comma operation, in its application to an absolute term,
is tantamount to the representation of that term’s denotation as
an idempotent transformation, which is commonly represented as a
diagonal matrix. Hence the alternate name, “diagonal extension”.

An idempotent element x is given by the abstract condition that
xx = x but elements like those are commonly encountered in more
concrete circumstances, acting as operators or transformations
on other sets or spaces, and in that action they will often be
represented as matrices of coefficients.

Let’s see how this looks in the matrix and graph pictures of
absolute and relative terms.

Absolute Terms
==============

Display 1. Absolute Terms 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-absolute-terms-1-m-n-w.png

Previously, we represented absolute terms as column arrays.

The above four terms are given by the columns of the following Table.

Column Arrays
=============

Display 2. Column Arrays
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-columns-1-m-n-w-large.png

The types of graphs known as “bigraphs” or “bipartite graphs”
can be used to picture simple relative terms, dyadic relations,
and their corresponding logical matrices. One way to bring
absolute terms and their corresponding sets of individuals
into the bigraph picture is to mark the nodes in some way,

for example, hollow nodes for non‑members and filled nodes

for members of the indicated set, as shown below.

Dichromatic Nodes
=================

Display 3. Dichromatic Nodes 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-dichromatic-nodes-1-m-n-w.png

Diagonal Extensions
===================

Display 4. Diagonal Extensions 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-diagonal-extensions-1-m-n-w.png

Naturally enough, the diagonal extensions are represented
by diagonal matrices.

Diagonal Matrices
=================

Display 5. Diagonal Matrices 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-diagonal-matrices-1-m-n-w.png

Cast into the bigraph picture of dyadic relations,
the diagonal extension of an absolute term takes on
a very distinctive sort of “straight-laced” character,
as shown below.

Idempotent Bigraphs
===================

Display 6. Idempotent Bigraphs 1 M N W
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-idempotent-bigraphs-1-m-n-w.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.6
https://inquiryintoinquiry.com/2014/03/08/peirces-1870-logic-of-relatives-comment-9-6/

Peirce’s 1870 “Logic of Relatives” • Comment 9.6
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.6

All,

By way of fixing the current array of relational concepts in our minds, let us
work through a sample of products from our relational multiplication table that
will serve to illustrate the application of a comma relative to an absolute term,
presented in both matrix and bigraph pictures.

Example 1. Comma Product 1,1 = 1
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-product-11-1.png

Example 2. Comma Product 1,m = m
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-product-1m-m.png

Example 3. Comma Product m,1 = m
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-product-m1-m.png

Example 4. Comma Product m,n
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-product-mn-1.png

Example 5. Comma Product n,m
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-product-nm.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.7
https://inquiryintoinquiry.com/2014/03/08/peirces-1870-logic-of-relatives-comment-9-7/

Peirce’s 1870 “Logic of Relatives” • Comment 9.7
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_9.7

All,

From this point forward we may think of idempotents, selectives,
and zero-one diagonal matrices as being roughly equivalent notions.
The only reason I say “roughly” is that we are comparing ideas at
different levels of abstraction in proposing those connections.

We have covered the way Peirce uses his invention of the comma modifier
to assimilate boolean multiplication, logical conjunction, and what we
may think of as “serial selection” under his more general account of
relative multiplication.

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.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 10
https://inquiryintoinquiry.com/2014/03/09/peirces-1870-logic-of-relatives-selection-10/

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 10
=================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_10

<QUOTE CSP>

The Signs for Multiplication (cont.)

The sum x + x generally denotes no logical term.
But x,_∞ + x,_∞ may be considered as denoting
some two x’s. It is natural to write

[Display 1] x + x = 2.x and x,_∞ + x,_∞ = 2.x,_∞
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-selection-10-display-1.png

where the dot shows that this multiplication is invertible.
We may also use the antique figures so that

[Display 2]
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-selection-10-display-2.png

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,

[Display 3] ℓ2.w and 2.ℓw
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-selection-10-display-3.png

are unequal; but the husbands of two women
are the same as two husbands of women, that is,

[Display 4]
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-selection-10-display-4.png

(Peirce, CP 3.75)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.1
https://inquiryintoinquiry.com/2014/03/10/peirces-1870-logic-of-relatives-comment-10-1/

Re: Peirce’s 1870 “Logic of Relatives” • Selection 10 (CP 3.75)
https://inquiryintoinquiry.com/2014/03/09/peirces-1870-logic-of-relatives-selection-10/

Peirce’s 1870 “Logic of Relatives” • Comment 10.1
=================================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.1

All,

What Peirce is attempting to do at CP 3.75 is absolutely amazing.
I did not run across anything on a par with it again until the
mid 1980s when I began studying the application of mathematical
category theory to computation and logic. Gauging the success
of Peirce’s attempt would take a return to his earlier paper
“Upon the Logic of Mathematics” (1867) to pick up the ideas
about arithmetic he sets out there.

Another branch of the investigation would require us to examine the
syntactic mechanics of “subjacent signs” Peirce uses to establish
linkages among relational domains. The indices employed for this
purpose amount to a category of diacritical and interpretive signs
which includes, among other things, the comma functor we have just
been discussing.

Combining the two branches of this investigation opens a wider
context for the study of relational compositions, distilling the
essence of what it takes to relate relations, possibly complex,
to other relations, possibly simple.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.2
https://inquiryintoinquiry.com/2014/03/14/peirces-1870-logic-of-relatives-comment-10-2/

Peirce’s 1870 “Logic of Relatives” • Comment 10.2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.2

All,

To say a relative term “imparts a relation” is to say 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 these constraints,
determinations, and informations, as imparted by relative terms, are
compounded in the formation of syntax.

Let us go back and look more carefully at just how it happens that Peirce’s
adjacent terms and subjacent indices manage to impart their respective measures
of information about relations. Consider the examples shown in Figures 7 and 8,
where connecting lines of identity have been drawn between the corresponding
occurrences of the subjacent marks of reference: †, ‡, ∥, §, ¶.

Figure 7. Lover of a Servant of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-lsw-2.0.png

Figure 8. Giver of a Horse to a Lover of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-glwh-2.0.png

One way to approach the problem of “information fusion” in Peirce’s syntax
is to soften the distinction between adjacent terms and subjacent signs and
treat the types of constraints they separately signify more on a par with
each other. To that purpose, let us consider a way of thinking about
relational composition that emphasizes the set-theoretic constraints
involved in the construction of a composite relation.

For example, given the relations L ⊆ X × Y and M ⊆ Y × Z,
Table 9 and Figure 10 present two ways of picturing the
constraints involved in constructing the relational
composition L ∘ M ⊆ X × Z.

Table 9. Relational Composition L ◦ M
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-composition-table-l-e297a6-m.png

The way to read Table 9 is to imagine you are playing a game
which involves placing tokens on the squares of a board marked
in just that way. The rules are you have to place a single token
on each marked square in the middle of the board in such a way that
all the indicated constraints are satisfied. That is, 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 the blank squares empty.

Furthermore, the tokens placed in each row and column have to
obey the relational constraints indicated at the heads of the
corresponding row and column. Thus, the two tokens from X have
to denote 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 in the relations L and M, respectively.

The upshot is, when all that has been done, when the L, M, and 1
relations are satisfied, then the row marked L ∘ M will automatically
bear the tokens of a pair of elements in the composite relation L ∘ M.

Figure 10 shows a different way of viewing the same situation.

Figure 10. Relational Composition L ◦ M
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-composition-figure-l-e297a6-m.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.3
https://inquiryintoinquiry.com/2014/03/25/peirces-1870-logic-of-relatives-comment-10-3/

Peirce’s 1870 “Logic of Relatives” • Comment 10.3
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.3

All,

We have been using several styles of picture to illustrate relative terms and the
relations they denote. Let’s now examine the relationships which exist among the
variety of visual schemes. Two examples of relative multiplication we considered
before are diagrammed again in Figures 11 and 12.

Figure 11. Lover of a Servant of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-lsw-2.0.png


Figure 12. Giver of a Horse to a Lover of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-glwh-2.0.png

Figures 11 and 12 employ one of the styles of syntax Peirce used for relative
multiplication, to which I added lines of identity to connect the corresponding
marks of reference. Forms like these show the anatomy of the relative terms
themselves, while the forms in Table 13 and Figure 14 are adapted to show
the structures of the objective relations they denote.

Table 13. Relational Composition L ◦ S
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-composition-table-l-e297a6-s.png

Figure 14. Relational Composition L ◦ S
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-composition-figure-l-e297a6-s.png

There are many ways Peirce might have gotten from his 1870 Notation for
the Logic of Relatives to his more evolved systems of Logical Graphs.
It is interesting to speculate on how the metamorphosis might have
been accomplished by way of transformations acting on these nascent
forms of syntax and taking place not too far from the pale of its
means, that is, as nearly as possible according to the rules and
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 rèlate of some
relation. Thus, the principal rèlate, 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 15 demonstrates this possibility, posing a transitional
case between the style of syntax in Figure 11 and the picture
of composition in Figure 14.

Figure 15. Anything that is a Lover of a Servant of Anything
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-universal-bracket-l-e297a6-s.png

In this composite sketch the diagonal extension 1 of the universe *1* is
invoked up front to anchor an explicit line of identity for the leading
rèlate of the composition, while the terminal argument w is generalized
to the whole universe *1*. Doing this amounts to an act of abstraction
from the particular application to w. This form of universal bracketing
isolates the serial composition of the relations L and S to form the
composite L ◦ S.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.4
https://inquiryintoinquiry.com/2014/03/26/peirces-1870-logic-of-relatives-comment-10-4/

Peirce’s 1870 “Logic of Relatives” • Comment 10.4
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.4

All,

From now on the forms of analysis exemplified in the last set of
Figures and Tables will serve as a convenient bridge between the
logic of relative terms and the mathematics of their extensional
relations.

We may think of Table 13 as illustrating a “spreadsheet model” of
relational composition while Figure 14 may be thought of as making
a start toward a “hypergraph model” of generalized compositions.
I’ll explain the hypergraph model in more detail at a later point.

The transitional form of analysis represented by Figure 15 may be
called the “universal bracketing” of relatives as relations.

Table 13. L ◦ S Table
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-l-e297a6-s-table.png

Figure 14. L ◦ S Figure
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-l-e297a6-s-figure.png

Figure 15. Universal Bracket L ◦ S
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-universal-bracket-l-e297a6-s.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.5
https://inquiryintoinquiry.com/2014/04/06/peirces-1870-logic-of-relatives-comment-10-5/

Peirce’s 1870 “Logic of Relatives” • Comment 10.5
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.5

All,

We have sufficiently covered the application of the comma functor
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 commafications of dyadic relative terms and
the corresponding diagonal extensions to triadic relations.

<QUOTE CSP>

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,sw 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.

(Peirce, CP 3.73)
</QUOTE>

Just to plant our feet on a more solid stage, let us 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:

Display 1
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.5-display-1.png

And next we derive the following results:

Display 2
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.5-display-2.png

Now what are we to make of that?

If we operate in accordance with Peirce’s example of goh as the
“giver of a horse to an owner of that horse” then we may assume the
associative law and the distributive law are in force, allowing us to
derive this equation:

Display 3
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.5-display-3.png

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 case
(J:J:D)(J:D)(D) = Jis non‑null.

It follows that:

Display 4
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.5-display-4.png

And so what Peirce says makes sense in this case.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.6
https://inquiryintoinquiry.com/2014/04/16/peirces-1870-logic-of-relatives-comment-10-6/

Peirce’s 1870 “Logic of Relatives” • Comment 10.6
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.6

All,

As Peirce observes, it is not possible to work with relations
in general without eventually abandoning all the more usual
algebraic principles, in due time the associative law and
even the distributive law, just as we already gave up the
commutative law. It cannot be helped, as we cannot reflect
on a law except from a perspective outside it, in any case,
virtually so.

This could be done from the standpoint of the combinator calculus,
and there are places where Peirce verges on systems of a like character,
but here we are making a deliberate effort to stay within the syntactic
neighborhood of Peirce’s 1870 Logic of Relatives. Not too coincidentally,
it is for the sake of making smoother transitions between narrower and wider
regimes of algebraic law that we have been developing the paradigm of Figures
and Tables indicated above.

In the next several episodes, then, I’ll examine the cases Peirce uses
illustrate the next level of complexity in the multiplication of relative
terms, as shown in the Figures below.

Figure 16. Giver of a Horse to a Lover of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-glwh-2.0.png

Figure 17. Giver of a Horse to an Owner of It
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-goh.png

Figure 18. Lover that is a Servant of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-lsw-1.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.7
https://inquiryintoinquiry.com/2014/04/18/peirces-1870-logic-of-relatives-comment-10-7/

Peirce’s 1870 “Logic of Relatives” • Comment 10.7
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.7

All,

Here is what I get when I analyze Peirce’s
“giver of a horse to a lover of a woman” example
along the same lines as the dyadic compositions.

We may begin with the mark-up shown in Figure 19.

Figure 19. Giver of a Horse to a Lover of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-glwh-2.0.png

If we analyze this in accord with the spreadsheet model of
relational composition then the core of it is a particular
way of composing a triadic “giving” relation G ⊆ T × U × V
with a dyadic “loving” relation L ⊆ U × W so as to obtain
a specialized type of triadic relation (G ◦ L) ⊆ T × V × W.
The applicable constraints on tuples are shown in Table 20.

Table 20. Relational Composition G ◦ L
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-composition-table-g-e297a6-l.png

The hypergraph picture of the abstract composition is given in Figure 21.

Figure 21. Anything that is a Giver of Anything to a Lover of Anything
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-composite-of-triadic-g-on-dyadic-l.png

Regards,

Jon

[Vorachet Jaroensawas]

Dear Jon.

I am so sorry to interrupt this thread but I will get to my point quickly. Regarding the message sent by Len in the past weeks, it notified me about the sad news about Joe. I once joined his Google meet conference and studied some of his security models. Personally, I would like to learn more about the best place I can access some of his works. Similarly, Joe and Jon contributed a lot of progress and well-summarized detail to the young learners with this group and I was one of them. Anyone who can advise, please share the information to vora...@gmail.com.

Thank you so much.

Vorachet

> The SysSciWG wiki is at https://sites.google.com/site/syssciwg/.

>
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[Lenard Troncale]

Dear Vorachet:

It turns out that Kevin Dye has been running the meetings online since Joe’s untimely passing away. They continue Joe’s fixation on extending and computerizing his logical operators. Or at least that is my understanding. I am mailed about the next meetings regularly. But I do not intend to attend in the near future. Kevin’s email address is:
ke...@futureworldscenter.org
Contact him.

Len Troncale

>>> https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Finquiryintoinquiry.com%2F2014%2F02%2F24%2Fpeirces-1870-logic-of-relatives-comment-9-1%2F&amp;data=04%7C01%7Clrtroncale%40cpp.edu%7C9eb7a0c00e574892297608d9e5a09aa1%7C164ba61e39ec4f5d89ffaa1f00a521b4%7C0%7C1%7C637793300421930421%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&amp;sdata=EJfcYo5n33vbE3R8xaxWGX0JXIM21JG5R2sDQaF2YaQ%3D&amp;reserved=0

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>>> Re: Peirce’s 1870 “Logic of Relatives” • Selection 8

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>>>
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[jsi...@soe.ucsc.edu]

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[Jon Awbrey]

Cf: Peirce’s 1870 “Logic Of Relatives” • Comment 3
http://inquiryintoinquiry.com/2022/02/03/peirces-1870-logic-of-relatives-comment-3/
Re: Peirce’s 1870 “Logic of Relatives” • Comment 10.6
https://inquiryintoinquiry.com/2014/04/16/peirces-1870-logic-of-relatives-comment-10-6/
Re: Peirce’s 1870 “Logic of Relatives” • Comment 10.7
https://inquiryintoinquiry.com/2014/04/18/peirces-1870-logic-of-relatives-comment-10-7/

Figure 21. Composite of Triadic G on Dyadic L
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-composite-of-triadic-g-on-dyadic-l.png

All,

In passing to more complex combinations of relative terms
and the objective relations they denote, as we began to do
in Comments 10.6 and 10.7, I made use of words like “composite”
and “composition” along with the usual composition sign “◦” to
describe their structures. That amounts to loose speech on
my part and I may have to reform my Sprach at a later stage
of the Spiel.

At any rate, we need to distinguish the more complex forms of
combination encountered here from the ordinary composition of
dyadic relations symbolized by “◦” whose result must stay within
the class of dyadic relations. We can draw that distinction by
means of an adjective or a substantive term — so long as we see it
we can parse the words later.

Resources
=========

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.8
https://inquiryintoinquiry.com/2014/04/22/peirces-1870-logic-of-relatives-comment-10-8/

Peirce’s 1870 “Logic of Relatives” • Comment 10.8
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.8

All,

Our progress through the 1870 Logic of Relatives brings us
in sight of a critical transition point, one which turns on
the “teridentity” relation.

The markup for Peirce’s “giver of a horse to an owner of it”
is shown again in Figure 22.

Figure 22. Giver of a Horse to an Owner of It
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-goh.png

The hypergraph picture of the abstract composition is given in Figure 23.

Figure 23. Anything that is a Giver of Anything to an Owner of It
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-e280a2goe280a2.png

If we analyze this in accord with the spreadsheet model of
relational composition then the core of it is a particular

way of composing a triadic “giving” relation G ⊆ X × Y × Z
with a dyadic “owning” relation O ⊆ Y × Z in such a way as
to determine a specialized dyadic relation (G ◦ O) ⊆ X × Z.
Table 24 schematizes the associated constraints on tuples.

Table 24. Relational Composition G ◦ O
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-relational-composition-table-g-e297a6-o.png

So we see the notorious teridentity relation, which
I left equivocally denoted by the same symbol as the
identity relation 1, is already implicit in Peirce’s
discussion at this point.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.8
https://inquiryintoinquiry.com/2014/04/22/peirces-1870-logic-of-relatives-comment-10-8/

Attachments to go with the previous post ...

Figure 22. Giver of a Horse to an Owner of It
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-goh.png

Figure 23. Anything that is a Giver of Anything to an Owner of It
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-e280a2goe280a2.png

Table 24. Relational Composition G ◦ O
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-relational-composition-table-g-e297a6-o.png

Regards.

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.9
https://inquiryintoinquiry.com/2014/04/23/peirces-1870-logic-of-relatives-comment-10-9/

Peirce’s 1870 “Logic of Relatives” • Comment 10.9
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.9


<QUOTE AMSB>

Ergo in numero quo numeramus repetitio unitatum facit pluralitatem;
in rerum vero numero non facit pluralitatem unitatum repetitio,
vel si de eodem dicam “gladius unus mucro unus ensis unus”.

Therefore in the case of that number by which we number,
the repetition of ones makes a plurality;
but in the number consisting in things
the repetition of ones does not make a plurality,
as, for example, if I say of one and the same thing,
“one sword, one brand, one blade”.

Boethius (Anicius Manlius Severinus Boethius, c. 480–524 A.D.),
De Trinitate (The Trinity Is One God Not Three Gods),
The Theological Tractates, H.F. Stewart, E.K. Rand, S.J. Tester (trans.),
New Edition, Loeb Classical Library, Harvard/Heinemann, 1973.

</QUOTE>

All,

The use of the concepts of identity and teridentity is not to identify
a thing-in-itself with itself, much less twice or thrice over — there
is no need and 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 requiring unification does
not reside in the object but in the phenomena — in the appearances
which might have been appearances of different objects but are bound
by the indicated identities to be just so many aspects, facets, parts,
roles, or signs of one and the same object.

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.

Figure 23. Anything that is a Giver of Anything to an Owner of It
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-e280a2goe280a2.png

The use of the teridentity concept in the “giver of a horse to an owner of it”
is to say the thing appearing with respect to its quality under an absolute term,
“a horse”, the thing appearing with respect to its existence as the correlate of
a dyadic relative, “a potential possession”, and the thing appearing with respect
to its synthesis as the correlate of a triadic relative, “a gift”, are one and
the same thing.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.10
https://inquiryintoinquiry.com/2014/04/24/peirces-1870-logic-of-relatives-comment-10-10/

Peirce’s 1870 “Logic of Relatives” • Comment 10.10
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.10

All,

The last of Peirce’s three examples involving the composition of
triadic relatives with dyadic relatives is shown again in Figure 25.

Figure 25. Lover that is a Servant of a Woman
https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-lsw-1.png

The hypergraph picture of the abstract composition is given in Figure 26.

Figure 26. Anything that is a Lover that is a Servant of Anything
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-e280a2lse280a2.png

This example illustrates the way Peirce analyzes the logical conjunction,
we might even say the parallel conjunction, of a pair of dyadic relatives
in terms of the comma extension and the same style of composition that we
saw in the last example, that is, according to a pattern of anaphora that
invokes the teridentity relation.

Laying out the above analysis of logical conjunction on the
spreadsheet model of relational composition, the gist of it is
the diagonal extension of a dyadic “loving” relation L ⊆ X × Y
to a triadic “being and loving” relation L ⊆ X × X × Y, which is
then composed with a dyadic “serving” relation S ⊆ X × Y so as to
determine a dyadic relation L,S ⊆ X × Y. Table 27 schematizes the
associated constraints on tuples.

Table 27. Relational Composition L,S
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-relational-composition-table-ls.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.11

Peirce’s 1870 “Logic of Relatives” • Comment 10.11
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.11

All,

Let us return to the point where we left off unpacking the contents of CP 3.73.
Here Peirce remarks that the comma operator can be iterated at will.

<QUOTE CSP:>

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 goh means a man that is a rich individual
and is a black [person] that is that rich individual. But this has
no other meaning than m,b,r or a man that is a black [person] 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.

(Peirce, CP 3.73)
https://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-selection-8/
</QUOTE>

Again, let’s check whether this makes sense on the stage of our small but
dramatic model. Let’s say Desdemona and Othello are rich and, among the
persons of the play, only they. On this premiss we obtain a sample of
absolute terms sufficiently ample to work through Peirce’s example.

Display 1. Absolute Terms
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.11-display-1.png

One application of the comma operator yields the following dyadic relatives.

Display 2. Dyadic Relatives
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.11-display-2.png

Another application of the comma operator generates the following triadic relatives.

Display 3. Triadic Relatives
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.11-display-3.png

Assuming the associativity of multiplication among dyadic relatives,
the product m,b,r may be computed by a brute force method to yield
the following result.

Display 4. m,b,r = O
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.11-display-4.png

This says that a man that is black that is rich is Othello,
which is true on the premisses of our present universe of
discourse.

Following the standard associative combinations of goh, the product
m,,b,r is multiplied out along the following lines, where the trinomials
of the form (X:Y:Z)(Y:Z)(Z) are the only ones producing a non‑null result,
namely, (X:Y:Z)(Y:Z)(Z) = X.

Display 5. m,,b,r = O
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.11-display-5.png

So we have that m,,b,r = m,b,r.

In closing, observe how the teridentity relation has turned up again
in this context, as the second comma‑ing of the universal term itself.

Display 6. Teridentity 1,,
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.11-display-6.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.11

https://inquiryintoinquiry.com/2014/04/25/peirces-1870-logic-of-relatives-comment-10-11/

Forgot to link the much better formatted blog copy ...

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.12
https://inquiryintoinquiry.com/2014/04/26/peirces-1870-logic-of-relatives-comment-10-12/

Peirce’s 1870 “Logic of Relatives” • Comment 10.12
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Commentary_Note_10.12

All,

Potential ambiguities in Peirce’s two versions of the “rich black man”
example can be resolved by providing them with explicit graphical markups,
as shown in Figures 28 and 29.

Figure 28. Man that is Black that is Rich
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-mbr.png

Figure 29. Man that is a Rich Individual and is
a Black Person that is that Rich Individual
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-mbr-1.png

On the other hand, as the forms of relational composition become
more complex, the corresponding algebraic products of elementary
relatives, for example, (x:y:z)(y:z)(z), will not always determine
unique results without the addition of more information about the
intended linkings of terms.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 11
https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/

All,

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 11
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Selection_11

<QUOTE CSP>

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 × 3 is a pair of triplets, and 3 × 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] = [tf]

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.

(Peirce, CP 3.76)
</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.1
https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-comment-11-1/

Peirce’s 1870 “Logic of Relatives” • Comment 11.1
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.1

Dear Reader,

We have reached a suitable place to pause in our reading of Peirce’s text —
actually, it’s more like a place to run as fast as we can along a parallel
track — where I can pay off a few of the expository IOUs I’ve been using
to pave the way to this point.

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
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 dyadic relations, namely, the bigraph representation and
the matrix representation, each of which has its differential advantages
in different types of situations.

But we lack a comparable picture of how to compute the richer variety of
relational compositions involving triadic or higher adicity relations.
As a matter of fact, we run into a non-trivial classification problem
simply to enumerate the different types of compositions arising in
those cases.

Therefore let us inaugurate a systematic study of relational composition,
general enough to articulate the “generative potency” of Peirce’s 1870
Logic of Relatives.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.2
https://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-comment-11-2/

Peirce’s 1870 “Logic of Relatives” • Comment 11.2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.2

All,

Let’s bring together the various things Peirce has said about
the “number of function” up to this point in the paper.

CSP : 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].

(Peirce, CP 3.65)
https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-selection-2/

CSP : 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.

(Peirce, CP 3.66)
https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/


CSP : 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.

(Peirce, CP 3.67)
https://inquiryintoinquiry.com/2014/01/31/peirces-1870-logic-of-relatives-selection-4/

CSP : 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 × 3 is a pair of triplets, and 3 × 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] = [tf] 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.

(Peirce, CP 3.76)

https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.3
https://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-comment-11-3/

Peirce’s 1870 “Logic of Relatives” • Comment 11.3
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.3

All,

Before I can discuss Peirce’s “number of” function in greater detail
I will need to deal with an expositional difficulty I have been
carefully dancing around all this time, but one which will
no longer abide its assigned place under the rug.

Functions have long been understood, from well before Peirce’s time to ours,
as special cases of dyadic relations, so the “number of” function is already
to be numbered among the class of dyadic relatives we’ve been dealing with
all this time. But Peirce’s manner of representing a dyadic relative term
mentions the “rèlate” first and the “correlate” second, a convention going
over into functional terms as making the functional value first and the
functional argument second.

The problem is, almost anyone brought up in our present time frame is
accustomed to thinking of a function as a set of ordered pairs where
the order in each pair lists the functional argument first and the
functional value second.

Syntactic wrinkles of this sort can be ironed out smoothly enough
in a framework of flexible interpretive conventions, but not without
introducing an order of anachronism into Peirce’s text I want to avoid
as much as possible. This will require me to experiment with various
styles of compromise. Among other things, the interpretation of
Peirce’s 1870 “Logic of Relatives” can be facilitated by introducing
a few items of background material on relations in general, as regarded
from a combinatorial point of view.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.4
https://inquiryintoinquiry.com/2014/05/01/peirces-1870-logic-of-relatives-comment-11-4/

Peirce’s 1870 “Logic of Relatives” • Comment 11.4
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.4

All,

The task before us is to clarify the relationships among
relative terms, relations, and the special cases of relations
given by equivalence relations, functions, and so on.

The first obstacle to get past is the order convention Peirce’s
orientation to relative terms causes him to use for functions.
To focus on a concrete example of immediate use in this discussion,
let’s take the “number of” function Peirce denotes by means of
square brackets and re-formulate it as a dyadic relative term v
in the following way.

v(t) := [t] = the number of the term t.

To set the dyadic relative term v within a suitable context of
interpretation, let's suppose that v corresponds to a relation
V ⊆ R × S where R is the set of real numbers and S is a suitable
syntactic domain, here described as a set of terms. The dyadic
relation V is at first sight a function from S to R. There is,
however, a great likelihood we cannot always assign a number to
every term in whatever syntactic domain S we happen to pick, so
we may eventually be forced to treat the dyadic relation V as a
partial function from S to R. All things considered, then, let’s
try the following budget of strategies and compromises.

First, let’s adapt the arrow notation for functions in such a way
as to allow detaching the functional orientation from the order in
which the names of domains are written on the page. Second, let’s
change the notation for partial functions, or pre-functions, to mark
more clearly their distinction from functions. This produces the
following scheme.

q : X → Y means q is functional at X.

q : X ← Y means q is functional at Y.

q : X ⇀ Y means q is pre-functional at X.

q : X ↼ Y means q is pre-functional at Y.

Until it becomes necessary to stipulate otherwise, let’s assume v
is a function in R of S, written v : R ← S, amounting to a functional
alias of the dyadic relation V ⊆ R × S and associated with the dyadic
relative term v whose rèlate lies in the set R of real numbers and
whose correlate lies in the set S of syntactic terms.

Note. Please refer to the following article on Relation Theory
for the definitions of functions and pre‑functions used in the
above discussion.

Relation Theory
https://oeis.org/wiki/Relation_theory

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.5
https://inquiryintoinquiry.com/2014/05/02/peirces-1870-logic-of-relatives-comment-11-5/

Peirce’s 1870 “Logic of Relatives” • Comment 11.5
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.5

All,

Everyone knows the right sort of diagram can be a great aid in rendering
complex matters comprehensible. With that in mind, let’s extract what
we need from the Relation Theory article to illuminate Peirce’s 1870
Logic of Relatives and use it to fashion what icons we can within
the current frame of discussion.

Relation Theory
https://oeis.org/wiki/Relation_theory

For the immediate present, we may begin with dyadic relations and
describe the most frequently encountered species of relations and
functions in terms of their local and numerical incidence properties.

Let P ⊆ X × Y be an arbitrary dyadic relation.
The following properties of P can then be defined.

Display 1
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.5-display-1.png

If P ⊆ 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 alternate lower case name, say “p”, and writing p : X ⇀ Y.

Just by way of formalizing the definition:

Display 2
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.5-display-2.png

To illustrate these properties, let us fashion a generic enough example of
a dyadic relation, E ⊆ X × Y, where X = Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
and where the bigraph picture of E is shown in Figure 30.

Figure 30. Dyadic Relation E
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-30.jpg

If we scan along the X dimension from 0 to 9 we see that the incidence degrees
of the X nodes with the Y domain are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0 in that order.

If we scan along the Y dimension from 0 to 9 we see that the incidence degrees
of the Y nodes with the X domain are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0 in that order.

Thus, E is not total at either X or Y since there are nodes in both X and Y
having incidence degrees less than 1.

Also, E is not tubular at either X or Y since there are nodes in both X and Y
having incidence degrees greater than 1.

Clearly then the relation E cannot qualify as a pre-function,
much less as a function on either of its relational domains.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.6
https://inquiryintoinquiry.com/2014/05/04/peirces-1870-logic-of-relatives-comment-11-6/

Peirce’s 1870 “Logic of Relatives” • Comment 11.6
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.6

All,

Let’s continue working our way through the above definitions,
constructing appropriate examples as we go.

1. Relation E₁ ⊆ X × Y exemplifies the quality of “totality at X”.
•• https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-31.jpg

2. Relation E₂ ⊆ X × Y exemplifies the quality of “totality at Y”.
•• https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-32.jpg

3. Relation E₃ ⊆ X × Y exemplifies the quality of “tubularity at X”.
•• https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-33.jpg

4. Relation E₄ ⊆ X × Y exemplifies the quality of “tubularity at Y”.
•• https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-34.jpg

So E₃ is a pre-function e₃ : X ⇀ Y and E₄ is a pre-function e₄ : X ↼ Y.

Resources
=========

Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus

Relational Concepts
https://oeis.org/wiki/Logic_Syllabus#Relational_concepts

Relation Theory
https://oeis.org/wiki/Relation_theory

Relative Term, Rhema, Rheme
https://oeis.org/wiki/Relative_term

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.7
https://inquiryintoinquiry.com/2014/05/05/peirces-1870-logic-of-relatives-comment-11-7/

Peirce’s 1870 “Logic of Relatives” • Comment 11.7
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.7

All,

We come now to the special cases of dyadic relations
known as functions. It will serve a dual purpose in
the present exposition to take the class of functions
as a source of object examples for clarifying the more
abstruse concepts of Relation Theory.

To begin, let us recall the definition of a
“local flag” L_{a @ j} of a k-adic relation L.

Display 1
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.7-display-1-1.png

For a dyadic relation L ⊆ X × Y the notation for local flags
can be simplified in two ways.

First, the local flags L_{u @ 1} and L_{v @ 2} are often more
conveniently notated as L_{u @ X} and L_{v @ Y}, respectively.

Second, the notation may be streamlined even further by making
the following
definitions.

Display 2
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.7-display-2.png

In light of these conventions, the local flags of a dyadic relation
L ⊆ X × Y may be comprehended under the following descriptions.

Display 3
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.7-display-3.png

The following definitions are also useful.

Display 4
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.7-display-4.png

A sufficient illustration is supplied by the earlier example E.

Figure 35. Dyadic Relation E
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-30.jpg

Figure 36 shows the local flag E_{3 @ X} of E.

Figure 36. Local Flag E_{3 @ X}
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-36.jpg

Figure 37 shows the local flag E_{2 @ Y} of E.

Figure 37. Local Flag E_{2 @ Y}
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-37.jpg

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.8
https://inquiryintoinquiry.com/2014/05/06/peirces-1870-logic-of-relatives-comment-11-8/

Peirce’s 1870 “Logic of Relatives” • Comment 11.8
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.8

All,

Let’s take a closer look at the “numerical incidence properties” of relations,
concentrating on the assorted regularity conditions defined in the article on
Relation Theory ( https://oeis.org/wiki/Relation_theory ).

For example, L has the property of being “c-regular at j” if and only if
the cardinality of the local flag L_{x @ j} is equal to 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 like fashion, one may define the numerical incidence properties
“(< c)-regular at j”, “(> c)-regular at j”, and so on. For ease of
reference, a number of such definitions are recorded below.

Display 1. Definitions
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.8-definitions.png

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
that domain, in short, c-regular at j.

For example, let G = {r, s, t} and H = {1, 2, 3, 4, 5, 6, 7, 8, 9}
and consider the dyadic relation F ⊆ G × H bigraphed below.

Figure 38. Dyadic Relation F
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-38.jpg

We observe that F is 3-regular at G and 1-regular at H.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.9
https://inquiryintoinquiry.com/2014/05/07/peirces-1870-logic-of-relatives-comment-11-9/

Peirce’s 1870 “Logic of Relatives” • Comment 11.9
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.9

All,

Among the variety of regularities affecting dyadic relations
we pay special attention to the c-regularity conditions where
c is equal to 1.

Let P ⊆ X × Y be an arbitrary dyadic relation.

The following properties can be defined.

Definitions 1
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.9-definitions-1.png

We previously examined dyadic relations exemplifying each of these
regularity conditions. Then we introduced a few bits of terminology
and special-purpose notations for working with tubular relations.

Definitions 2
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.9-definitions-2.png

We arrive by way of this winding stair at the special cases of
dyadic relations P ⊆ X × Y variously described as “1-regular”,
“total and tubular”, or “total prefunctions” on specified domains,
X or Y or both, and which are more often celebrated as “functions”
on those domains.

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 a relation P ⊆ X × Y is “total and tubular” at X
is to say P is 1-regular at X. Thus, we may formalize
the following definitions.

Definitions 3
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.9-definitions-3.png

For example, let X = Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and let
F ⊆ X × Y be the dyadic relation depicted in the bigraph below.

Figure 39. Dyadic Relation F
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-39.jpg

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.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.10
https://inquiryintoinquiry.com/2014/05/07/peirces-1870-logic-of-relatives-comment-11-10/

Peirce’s 1870 “Logic of Relatives” • Comment 11.10
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.10

All,

A dyadic relation F ⊆ X × Y which qualifies as
a function f : X → Y may then enjoy a number of
further distinctions.

Definitions
https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.10-definitions.png

For example, the function f : X → Y shown below is neither
total nor tubular at its codomain Y so it can enjoy none of
the properties of being surjective, injective, or bijective.

Figure 40. Function f : X → Y
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-40.jpg

An easy way to extract a surjective function from any function
is to reset its codomain to its range. For example, the range
of the function f above is Y' = {0, 2, 5, 6, 7, 8, 9}. 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 a mapping onto Y'.

Figure 41. Function g : X → Y'
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-41.jpg

The function h : Y' → Y is injective.

Figure 42. Function h : Y' → Y
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-42.jpg

The function m : X → Y is bijective.

Figure 43. Function m : X → Y
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-43.jpg

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.11
https://inquiryintoinquiry.com/2014/05/09/peirces-1870-logic-of-relatives-comment-11-11/

Peirce’s 1870 “Logic of Relatives” • Comment 11.11
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.11

All,

The preceding exercises were intended to beef-up our “functional literacy” skills
to the point where we can read our functional alphabets backwards and forwards and
recognize the local functionalities immanent in relative terms no matter where they
reside among the domains of relations. These skills will serve us in good stead as
we work to build a catwalk from Peirce’s platform of 1870 to contemporary scenes on
the logic of relatives, and back again.

By way of extending a few very tentative planks,
let us experiment with the following definitions.

• A relative term “p” and the corresponding relation P ⊆ X × Y
are both called “functional on rèlates” if and only if
P is a function at X. We write this in symbols as P : X → Y.

• A relative term “p” and the corresponding relation P ⊆ X × Y
are both called “functional on correlates” if and only if
P is a function at Y. We write this in symbols as 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 × Y, P : X → Y, and P : X ← Y,
as the case may be, when and if it serves to clarify matters.

From this current, perhaps transient, perspective, it appears
our next task is to examine how the known properties of relations
are modified when aspects of functionality are 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 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.

Figure 44. Anything that is a P of a Q of Anything
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-universal-bracket-p-e297a6-q.png

From this we extract the “hypergraph picture” of relational composition.

Figure 45. Relational Composition P ◦ Q
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-p-e297a6-q-figure.png

All the information contained in these Figures can be
expressed in the form of a constraint satisfaction table,
or “spreadsheet picture” of relational composition.

Table 46. Relational Composition P ◦ Q
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-relational-composition-table-p-e297a6-q.png

The following plan of study then presents itself: to see what easy mileage
we can get in our exploration of functions by adopting the above templates
as the primers of a paradigm.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.12
https://inquiryintoinquiry.com/2014/05/12/peirces-1870-logic-of-relatives-comment-11-12/

Peirce’s 1870 “Logic of Relatives” • Comment 11.12
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.12

All,

Since functions are special cases of dyadic relations
and since the space of dyadic relations is closed under
relational composition — that is, the composition of two
dyadic relations is again a dyadic relation — we know the
relational composition of two functions has to be a dyadic
relation.

If the relational composition of two functions is necessarily
a function, too, then we would be justified in speaking of
“functional composition” and also in saying the space of
functions is closed under this functional composition.

Just for novelty’s sake, let’s try to prove this
for relations which are functional on correlates.

The task is this — We are given a pair of dyadic relations:

• P ⊆ X × Y and Q ⊆ Y × Z

The dyadic relations P and Q are assumed to be functional
on correlates, a premiss we express as follows.

• P : X ← Y and Q : Y ← Z

We are charged with deciding whether the relational
composition P ◦ Q ⊆ X × Z is also functional on
correlates, in symbols, whether P ◦ Q : X ← Z.

It always helps to begin by recalling the pertinent definitions.

For a dyadic relation L ⊆ X × Y, we have the following equivalence.

• L is a function L : X ← Y ⇔ L is 1-regular at Y.

As for the definition of relational composition, it is enough
to consider the coefficient of the composite relation on an
arbitrary ordered pair, i:j. For that we have the following
formula, where the summation indicated is logical disjunction.

• (P ◦ Q)_{ij} = ∑ₖ P_{ik} Q_{kj}

So let’s begin.

• P : X ← Y, or the fact that P is 1-regular at Y, means
there is exactly one ordered pair i:k in P for each k in Y.

• Q : Y ← Z, or the fact that Q is 1-regular at Z, means
there is exactly one ordered pair k:j in Q for each j in Z.

• As a result, there is exactly one ordered pair i:j in P ◦ Q
for each j in Z, which means P ◦ Q is 1-regular at Z, and so
we have the function P ◦ Q : X ← Z.

And we are done.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.13
https://inquiryintoinquiry.com/2014/05/14/peirces-1870-logic-of-relatives-comment-11-13/

Peirce’s 1870 “Logic of Relatives” • Comment 11.13
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.13

All,

As we make our way toward the foothills of Peirce’s 1870 Logic of Relatives
there are several pieces of equipment we must not leave the plains without,
namely, the utilities variously known as arrows, morphisms, homomorphisms,
structure-preserving maps, among other names, depending on the altitude
of abstraction we happen to be traversing at the moment in question.
As a moderate to middling but not too beaten track, let’s examine
a few ways of defining morphisms that will serve us in the present
discussion.

Suppose we are given three functions J, K, L
satisfying the following conditions.

• J : X ← Y

• K : X ← X × X

• L : Y ← Y × Y

• J(L(u, v)) = K(Ju, Jv)

Our sagittarian leitmotif can be rubricized in the following slogan.

• “The J-image of the L-product is the K-product of the J-images.”

Figure 47 presents us with a picture of the situation in question.

Figure 47. Structure Preserving Transformation J : K ← L
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-structure-preserving-transformation-jkl.png

Table 48 gives the constraint matrix version of the same thing.

Table 48. Structure Preserving Transformation J : K ← L
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-j-makes-k-from-l-table.png

One way to read the Table is in terms of the informational redundancies
it summarizes. For example, one way to read it says that satisfying the
constraint in the L row along with all the constraints in the J columns
automatically satisfies the constraint in the K row. Quite by design,
that is one way to understand the equation J(L(u, v)) = K(Ju, Jv).

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.14
https://inquiryintoinquiry.com/2014/05/15/peirces-1870-logic-of-relatives-comment-11-14/

Peirce’s 1870 “Logic of Relatives” • Comment 11.14
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.14

All,

Let’s now look at a more familiar example of a morphism J,
say, one of the mappings of reals into reals commonly known
as logarithm functions, where you get to pick your favorite base.

Here we have K(r, s) = r + s and L(u, v) = u ⋅ v and the formula
J(L(u, v)) = K(Ju, Jv) becomes J(u ⋅ v) = J(u) + J(v), where ordinary
multiplication and addition are indicated by a dot (⋅) and a plus sign (+),
respectively.

Figure 49 shows how the multiplication, addition, and logarithm operations fit together.

Figure 49. Logarithm Arrow J : {+} ← {⋅}
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-logarithm-arrow-j-makes-from-e28b85.png

Thus, where the “image operation” J is the logarithm map,
the “source operation” is the numerical product, and the
“target operation” is the numerical sum, we have the
following rule of thumb.

• “The image of the product is the sum of the images.”

• J(u ⋅ v) = J(u) + J(v)

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.15
https://inquiryintoinquiry.com/2014/05/15/peirces-1870-logic-of-relatives-comment-11-15/

Peirce’s 1870 “Logic of Relatives” • Comment 11.15
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.15

All,

I’m going to elaborate a little further on the subject
of arrows, morphisms, or structure-preserving mappings,
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” preserved by a structure-preserving map is just the
structure we all know and love as a triadic relation. Very typically,
it will be the type of triadic relation that defines the type of binary
operation that obeys the rules of a mathematical structure known as
a “group”, that is, a structure satisfying the axioms for closure,
associativity, identities, and inverses.

For example, in the case of the logarithm map J we have the following data.

• J : Reals ← Reals (properly restricted)

• K : Reals ← Reals × Reals where K(r, s) = r + s

• L : Reals ← Reals × Reals 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 brackets as a name for the triadic relation that defines the corresponding
group, we have the following set-up.

• J : [+] ← [⋅]

• [+] ⊆ Reals × Reals × Reals

• [⋅] ⊆ Reals × Reals × Reals

It often happens that both group operations are indicated
by the same sign, usually one from the set { ⋅ , ∗ , + } 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 one of the following
two variants.

• “The image of the sum is the sum of the images.”

• “The image of the product is the sum of the images.”

Figure 50 presents a generic picture for groups G and H.

Figure 50. Group Homomorphism J : G ← H
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-group-homomorphism-j-makes-g-from-h.png

In a setting where both groups are written with a plus sign, perhaps
even constituting the 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 analogous to the distributive multiplication of a factor J over
a sum (u + v). That is why morphisms are regarded as generalizations
of linear functions and are frequently referred to in those terms.

Regards,

Jon

[Jon Awbrey]

Peirce’s 1870 “Logic of Relatives” • Comment 11.16
https://inquiryintoinquiry.com/2014/05/21/peirces-1870-logic-of-relatives-comment-11-16/

Peirce’s 1870 “Logic of Relatives” • Comment 11.16
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.16

All,

We now have enough material on morphisms to go back and cast a
more studied eye on what Peirce is doing with that “number of”
function, whose application to a logical term t is indicated by
writing the term in square brackets, as [t]. It is convenient
to have a prefix notation for the function mapping a term t to
a number [t] but Peirce previously reserved the letter “n” for
logical “not”, so let’s use v(t) as a variant for [t].

My plan will be nothing less plodding than to work through the
statements Peirce made in defining and explaining the “number of”
function up to our present place in the paper, namely, the budget
of points collected in Comment 11.2.

https://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-comment-11-2/

NOF 1
=====

<QUOTE CSP:>

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].

(Peirce, CP 3.65)
</QUOTE>

The role of the “number of” function may be formalized by assigning it
a name and a type, in the present discussion v : S → R, where S is a
suitable set of signs, a “syntactic domain”, containing all the logical
terms whose numbers we need to evaluate in a given context, and where R
is the set of real numbers.

Transcribing Peirce’s example:

Let m = man
and t = tooth of___.

Then v(t) = [t] = [tm]/[m]

To spell it out in words, the number of the relative term “tooth of___”
in a universe of perfect human dentition is equal to the number of teeth
of humans divided by the number of humans, that is, 32.

The dyadic relative term t determines a dyadic relation T ⊆ X × Y,
where X and Y contain all the teeth and all the people, respectively,
under discussion.

A rough indication of the bigraph for T might be drawn as follows,
showing just the first few items in the toothy part of X and the
peoply part of Y.

Figure 51. Dyadic Relation T
https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-511.jpg

Notice that the “number of” function v : S → R needs the
data represented by the entire bigraph for T in order to
compute the value [t].

Finally, one observes this component of T is a function in
the direction T : X → Y, since we are counting only teeth
which occupy exactly one mouth of a tooth-bearing creature.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.17
https://inquiryintoinquiry.com/2014/05/27/peirces-1870-logic-of-relatives-comment-11-17/

Peirce’s 1870 “Logic of Relatives” • Comment 11.17
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.17

All,

I think the reader is beginning to get an inkling of the crucial importance
of the “number of” function in Peirce’s way of looking at logic. It is one
plank in the bridge from logic to the theories of probability, statistics,
and information, in which setting logic forms but a limiting case at one
scenic turnout on the expanding vista. It is one of the ways Peirce forges
a link between the “eternal”, logical, or rational realm and the “secular”,
empirical, or real domain.

With that note of encouragement and exhortation,
let us return to the details of the text.

NOF 2
=====

<QUOTE CSP:>

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.

(Peirce, CP 3.66)
https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/

</QUOTE?

Peirce is here remarking on the principle that the measure v on
logical terms preserves or respects the prevailing relations of
implication, inclusion, or subsumption which impose an ordering on
those terms. In these passages, 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 he will introduce distinctive symbols for logical orders.
The links among terms, sets, and numbers can be pursued in all directions
and Peirce has already indicated in an earlier paper how he would construct
the integers from sets, that is, from the aggregate denotations of terms.
I will try to get back to that another time.

We have a statement of the following form.

• If f < m then the number of Frenchmen is less than the number of men.

This goes into symbolic form as follows.

• f < m ⇒ [f] < [m].

In this setting the “<” on the left is a logical ordering on
syntactic terms while the “<” on the right is an arithmetic
ordering on real numbers.

The question that arises in this case is whether a map between
two ordered sets is “order-preserving”. In order to formulate
the question in more general terms, we may begin with the
following set-up.

• Let X₁ be a set with the ordering <₁.

• Let X₂ be a set with the ordering <₂.

An order relation is typically defined by a set of axioms that
determines its properties. Since we have frequent occasion to
view the same set in the light of several different order relations,
we often resort to explicit specifications like (X, <₁), (X, <₂),
and so on to indicate a set with a given ordering.

A map F : (X₁, <₁) → (X₂, <₂) is order-preserving if and only if
a statement of a particular form holds for all x and y in (X₁, <₁),
namely, the following.

• x <₁ y ⇒ F(x) <₂ F(y).

The “number of” map v : (S, <₁) → (R, <₂) has just this character,
as exemplified in the case at hand.

• f < m ⇒ [f] < [m]

• f < m ⇒ v(f) < v(m)

The “<” on the left is read as proper inclusion, in other words,
“subset of but not equal to”, while the “<” on the right is read
as the usual less than relation.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.18

Peirce’s 1870 “Logic of Relatives” • Comment 11.18
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.18

All,

An “order-preserving map” is a special case of a structure-preserving map and
the idea of preserving structure, as used in mathematics, means preserving some
but not necessarily all the structure of the source domain in the transition to
the target domain. In that vein, we may speak of “structure preservation in
measure”, the suggestion being that a property able to be qualified in manner is
potentially able to be quantified in degree, admitting answers to questions like,
“How structure-preserving is it?”

Let’s see how this applies to Peirce’s “number of” function v : S → R.
Let “—<” denote the implication relation on logical terms, let “≤” denote
the less than or equal to relation on real numbers, and let x, y be any
pair of absolute terms in the syntactic domain S. Then we observe the
following relationships.

• x —< y ⇒ v(x) ≤ v(y)

Equivalently:

• x —< y ⇒ [x] ≤ [y]

Nowhere near the number of logical distinctions on the left sides
of the implication arrows are typically preserved as one passes to
the linear orderings of real numbers on their right sides but that
is not required in order to call the map v : S → R order-preserving,
or what is known as an “order morphism”.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.18
https://inquiryintoinquiry.com/2014/05/28/peirces-1870-logic-of-relatives-comment-11-18/

[forgot the main link]

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.19
https://inquiryintoinquiry.com/2014/05/29/peirces-1870-logic-of-relatives-comment-11-19/

Peirce’s 1870 “Logic of Relatives” • Comment 11.19
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.19

All,

Up to this point in the 1870 Logic of Relatives, Peirce has
introduced the “number of” function on logical terms, v : S → R
such that v : s ↦ [s], and discussed the extent to which its use
as a measure satisfies the relevant measure-theoretic principles,
beginning with the following two.

• The “number of” map exhibits a certain type of “uniformity property”,
where the value of the measure on a uniformly qualified population is
in fact actualized by each member of the population.

• The “number of” map satisfies an “order morphism principle”, where
the partial order of logical terms under implication or inclusion is
reflected to a moderate degree by the linear order of their measures.

In Selection 4 Peirce takes up the action of the “number of” function on
two types of more or less additive operations we normally consider in logic.

Selection 4 (CP 3.67)
https://inquiryintoinquiry.com/2014/01/31/peirces-1870-logic-of-relatives-selection-4/

NOF 3.1
=======

CSP: “It is plain that both the regular non-invertible addition and the
invertible addition satisfy the absolute conditions.” (CP 3.67).

Peirce uses the sign “+,” to indicate what he calls the
“regular non-invertible addition”, corresponding to the
inclusive disjunction of logical terms or the union of
their extensions as sets.

Peirce uses the sign “+” to indicate what he calls the
“invertible addition”, corresponding to the exclusive
disjunction of logical terms or the symmetric difference
of their extensions as sets.

NOF 3.2
=======

CSP: “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 the above remark would require us to pick up the
precise technical sense in which Peirce is using the word “collection” and
that would 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 is 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 true in the strictest sense,
we could remove the question marks from the following dubious equations.

• v(x +, y) =?= v(x) + v(y)

Equivalently:

• [x +, y] =?= [x] + [y]

Of course, things are not quite that simple when it comes to
inclusive disjunctions and set‑theoretic unions, so it is usual
to introduce the concept of a “sub‑additive measure” to describe
the principle that does hold here, namely, the following.

• v(x +, y) ≤ v(x) + v(y)

Equivalently:

• [x +, y] ≤ [x] + [y]

That is why Peirce trims his discussion of the point with the following hedge.

NOF 3.3
=======

CSP: “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 thing on behalf of the additive identity element, as follows.

NOF 3.4
=======

CSP: “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.” (CP 3.67).

With respect to the nullity 0 in S and the number 0 in R,
we have the following equation.

• v(0) = [0] = 0.

In sum, therefore, it can be said:

A measure only serves which also preserves
a due respect for the function of a vacuum
in nature.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.20
https://inquiryintoinquiry.com/2014/06/01/peirces-1870-logic-of-relatives-comment-11-20/

Peirce’s 1870 “Logic of Relatives” • Comment 11.20
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.20

All,

We come to the last of Peirce’s statements about the “number of” function,
first quoted in Selection 11 and again with the whole set in Comment 11.2.

https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/
https://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-comment-11-2/

NOF 4.1
=======

<QUOTE CSP:>

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 × 3 is a pair of triplets, and 3 × 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].

(Peirce, CP 3.76)
</QUOTE>

Peirce here observes what may be called a “contingent morphism”.
On a condition he gives, the mapping of logical terms to their
corresponding numbers preserves the multiplication of relative
terms after the fashion of the following formula.

• v(xy) = v(x) v(y).

Equivalently:

• [xy] = [x][y].

The condition for this to hold is expressed by Peirce in the following manner.

“There are just as many x’s per y as there are, per things, things of the universe.”

Peirce’s phrasing on this point is admittedly hard to parse but I think
if we stick with his story to the end we can see what he is driving at.

NOF 4.2
=======

<QUOTE CSP:>

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] = [tf]

holds arithmetically. (Peirce, CP 3.76).
</QUOTE>

Now that is something we can sink our teeth into and trace the
bigraph representation of the situation. It will help to recall
our first examination of the “tooth of” relation and to adjust the
picture we sketched of it on that occasion.

Transcribing Peirce’s example:

Let m = man
and t = tooth of___.

Then v(t) = [t] = [tm]/[m].

That is to say, the number of the relative term “tooth of”

is equal to the number of teeth of humans divided by the

number of humans. In a universe of perfect human dentition
this gives a quotient of 32.

The dyadic relative term t determines a dyadic relation T ⊆ X × Y,

where X contains all the teeth and Y contains all the people under
discussion.

To make the case as simple as possible and still cover the point,
suppose there are just four people in our universe of discourse
and just two of them are French. The bigraph product below
shows the pertinent facts of the case.

Figure 52. Bigraph Product T ◦ F
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-bigraph-product-t-e297a6-f.png

In this picture the order of relational composition flows down the page.
For convenience in composing relations, the absolute term f = Frenchman
is inflected by the comma functor to form the dyadic relative term
f, = Frenchman that is___, which in turn determines the idempotent
representation of Frenchmen as a subset of mankind, F ⊆ Y × Y.

Now let’s see if we can use this picture
to make sense of the following statement.

NOF 4.2
=======

<QUOTE CSP:>

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] = [tf]

holds arithmetically. (Peirce, CP 3.76).
</QUOTE>

In statistical terms, Peirce is saying this: If the population of
Frenchmen is a *fair sample* of the general population with regard
to the factor of dentition, then the morphic equation,

• [tf] = [t][f],

whose transpose gives the equation,

• [t] = [tf]/[f],

is every bit as true as the defining equation in this circumstance, namely,

• [t] = [tm]/[m].

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.21
https://inquiryintoinquiry.com/2014/06/02/peirces-1870-logic-of-relatives-comment-11-21/

Peirce’s 1870 “Logic of Relatives” • Comment 11.21
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.21

All,

One more example and one more general observation and we’ll be
caught up with our homework on Peirce’s “number of” function.

NOF 4.3
=======

<QUOTE CSP:>

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.

(Peirce, CP 3.76)
https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/

</QUOTE>

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
one of the following statements.

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 the following paraphrase of what
Peirce intends.

• Men are just as likely to be black as things in general are likely to be black.

Stated in terms of conditional probability, we have the following equation.

• P(b|m) = P(b).

From the definition of conditional probability:

• P(b|m) = P(bm)/P(m).

Equivalently:

• P(bm) = P(b|m)P(m).

Taking everything together, we have the following result.

• P(bm) = P(b|m)P(m) = P(b)P(m).

That, of course, is the definition of independent events,
as applied to the event of being Black and the event of
being a Man. We may take that as the most likely reading
of Peirce’s statement about frequencies:

• [m,b] = [m,][b].

The terms of that equation can be normalized to produce
the corresponding statement about probabilities.

• P(mb) = P(m)P(b).

Let’s see if that reading checks out.

Let N be the number of things in general. Expressed in Peirce’s
notation we have the equation [*1*] = N. On the assumption that
m and b are associated with independent events, we obtain the
following sequence of equations.

• [m,b] = P(mb)N = P(m)P(b)N = P(m)[b] = [m,][b].

As a result, 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”.
That seems to make sense.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.22
https://inquiryintoinquiry.com/2014/06/04/peirces-1870-logic-of-relatives-comment-11-22/

Peirce’s 1870 “Logic of Relatives” • Comment 11.22
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.22

All,

Let’s look at that last example from a different angle.

NOF 4.3
=======

<QUOTE CSP:>
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.

(Peirce, CP 3.76)
https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/
</QUOTE>

Viewed in various lights the formula [m,b] = [m,][b] presents itself
as an aimed arrow, fair sampling, or stochastic independence condition.

Peirce’s example assumes a universe of things in general encompassing the
denotations of the absolute terms m = man and b = black. That allows us
to illustrate the case in relief, by returning to our earlier staging of
Othello and examining the premiss that “men are just as apt to be black
as things in general” within the frame of that that empirical if fictional
universe of discourse.

We have the following data.

• b = O
• m = C +, I +, J +, O
• *1* = B +, C +, D +, E +, I +, J +, O

• b, = O:O
• m, = C:C +, I:I +, J:J +, O:O
• *1*, = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

The “fair sampling condition” amounts to saying men are just as likely
to be black as things in general are likely to be black. In other words,
men are a fair sample of things in general with respect to the predicate
of being black.

On that condition the following equation holds.

• [m,b] = [m,][b].

Assuming [b] is not zero, the next equation follows.

• [m,] = [m,b]/[b].

As before, it is convenient to represent the absolute term b = black
by means of the corresponding idempotent term b, = black that is___.

Let is next consider the bigraph for the following relational product.

• m,b = man that is black.

We may represent that in the following equivalent form.

• m,b, = man that is black that is___.

Figure 53. Bigraph Product M,B,
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-othello-product-mb.png

The facts of the matter in the Othello case
are such that the following formula holds.

• m,b = b.

And that in turn is equivalent to each of the following statements.

• m ∧ b = b
• b ⇒ m
• b —< m

Those last implications puncture any notion of statistical independence
for b and m in the universe of discourse at hand but it will repay us to
explore the details of the case a little further. Putting all the general
formulas and particular facts together, we arrive at the following summation
of the situation in the Othello case.

If the fair sampling condition were true,
it would have the following consequence.

• [m,] = [m,b]/[b] = [b]/[b] = 1.

On the contrary, we have the following fact.

• [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 terms, we would have:

• P(mb) = 4/49.

In point of fact, however, we have:

• P(mb) = P(b) = 1/7.

Another way to see it is to observe that:

• P(b|m) = 1/4 while P(b) 1/7.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.23
https://inquiryintoinquiry.com/2014/06/05/peirces-1870-logic-of-relatives-comment-11-23/

Peirce’s 1870 “Logic of Relatives” • Comment 11.23
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.23

All,

Peirce’s description of logical conjunction and conditional
probability via the logic of relatives and the mathematics of
relations is critical to understanding the relationship between
logic and measurement, in effect, the qualitative and quantitative
aspects of inquiry. To ground that connection firmly in mind, I will
try to sum up as succinctly as possible, in more current notation, the
lesson we ought to take away from Peirce’s last “number of” example, since
I know the account I have given so far may appear to have wandered widely.

NOF 4.3
=======

<QUOTE CSP:>
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.

(Peirce, CP 3.76)
https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/
</QUOTE>

Viewed in different lights the formula [m,b] = [m,][b] presents itself
as an aimed arrow, fair sampling, or statistical independence condition.
The concept of independence was illustrated in the previous installment
by means of a case where independence fails. The details of that case
are summarized below.

Figure 54. Bigraph Product M,B,
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-othello-product-mb.png

The condition that “men are just as apt to be black as things in general”
is expressed in terms of conditional probabilities as P(b|m) = P(b), which
means that the probability of the event b given the event m is equal to the
unconditional probability of the event b.

In the Othello example it is enough to observe that P(b|m) = 1/4 while
P(b) = 1/7 in order to recognize the bias or dependency of the sampling map.

The reduction of a conditional probability to an absolute probability,
as P(A|Z) = P(A), is one of the ways we come to recognize the condition
of independence, P(AZ) = P(A)P(Z), via the definition of conditional
probability, P(A|Z) = P(AZ)/P(Z).

By way of recalling the derivation, the definition of conditional probability
plus the independence condition yields the following sequence of equations.

• P(A|Z) = P(AZ)/P(Z) = P(A)P(Z)/P(Z) = P(A).

As Hamlet discovered, there’s a lot to be learned from turning a crank.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.24
https://inquiryintoinquiry.com/2014/06/08/peirces-1870-logic-of-relatives-comment-11-24/

Peirce’s 1870 “Logic of Relatives” • Comment 11.24
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_11.24

Note. Please see the blog post linked above for the proper formatting,
as some of what's discussed below depends on the typography in various
editions and transcriptions of Peirce's text.

All,

We come to the last of Peirce's observations
about the “number of” function from CP 3.76.

NOF 4.4
=======

<QUOTE CSP:>

It is to be observed that

• [\mathit{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.

(Peirce, CP 3.76 and CE 2, 376)
https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-selection-11/
</QUOTE>

There are problems with the printing of the text at this point.
To recall the conventions we are using in this transcription,
\mathit{1} is the italic 1 denoting the dyadic identity relation
\mathfrak{1} is the “antique figure one” which Peirce defines
as 1_∞ = “something”.

Collected Papers CP 3 gives [\mathit{1}] = \mathfrak{1}, which does not make sense.
Chronological Edition CE 2 gives the 1's in different styles of italics but reading
the equation as [\mathit{1}] = 1 makes better sense if the latter “1” is the numeral
denoting the natural number 1 and not the absolute term “1” denoting the universe of
discourse. The quantity [\mathit{1}] is defined as the average number of things
related by the identity relation \mathit{1} to one individual, and so it makes sense
that [\mathit{1}] = 1 in N, where N is the set of non-negative integers {0, 1, 2, …}.

With respect to the relative term \mathit{1} in the syntactic domain S
and the number 1 in the non-negative integers N we have the following.

• v(\mathit{1}) = [\mathit{1}] = 1.

At long last, then, the “number of” mapping v : S → R
has another one of the properties required of an arrow
from logical terms in S to real numbers in R.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 12
https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/

All,

On to the next part of §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 12
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Selection_12

<QUOTE CSP:>

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^(ℓw) will
denote whatever is a servant of everything that is lover of a woman.
So that

• (s^ℓ)^w = s^(ℓw).

https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-s5el5ew-s5elw.png

(Peirce, CP 3.77)

</QUOTE>

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.1
https://inquiryintoinquiry.com/2014/06/10/peirces-1870-logic-of-relatives-comment-12-1/

Peirce’s 1870 “Logic of Relatives” • Comment 12.1
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.1

All,

To get a better sense of why Peirce’s formulas in Selection 12
mean what they do, and to prepare the ground for understanding
more complex relational expressions, it will help to assemble
the following materials and definitions.

• X is a set singled out in a particular discussion
as the “universe of discourse”.

• W ⊆ X is the monadic relation, or set, whose elements
fall under the absolute term w = woman. The elements
of W are referred to as the “denotation” or “extension”
of the term w.

• L ⊆ X × X is the dyadic relation associated with the
relative term ℓ = lover of___.

• S ⊆ X × X is the dyadic relation associated with the
relative term s = servant of___.

• \mathsf{W} = Mat(W) = Mat(w) is the 1-dimensional matrix
representation of the set W and the term w.

• \mathsf{L} = Mat(L) = Mat(ℓ) is the 2-dimensional matrix
representation of the relation L and the relative term ℓ.

• \mathsf{S} = Mat(S) = Mat(s) is the 2-dimensional matrix
representation of the relation S and the relative term s.

The “local flags” of the relation L are defined as follows.

• u ∗ L
= L_u@1
= {(u, x) ∈ L}
= ordered pairs in L with u in the 1st place.

• L ∗ v
= L_v@2
= {(x, v) ∈ L}
= ordered pairs in L with v in the 2nd place.

The “applications” of the relation L are defined as follows.

• u ∙ L
= proj₂(u ∗ L)
= {x ∈ X : (u, x) ∈ L}
= loved by u.

• L ∙ v
= proj₁(L ∗ v)
= {x ∈ X : (x, v) ∈ L}
= lover of v.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce's 1870 “Logic of Relatives” • Comment 12.2 [part 1 of 2]
https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-comment-12-2/

Peirce's 1870 “Logic of Relatives” • Comment 12.2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.2

All,

Let us make a few preliminary observations about the operation of
“logical involution” which Peirce introduces in the following words.

<QUOTE CSP:>

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.

(Peirce, CP 3.77)
https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/
</QUOTE>

In ordinary arithmetic the involution x^y, or the exponentiation of
x to the power y, is the repeated application of the multiplier x
for as many times as there are ones making up the exponent y.

In analogous fashion, the logical involution ℓ^w is the repeated application
of the term ℓ for as many times as there are individuals under the term w.
On Peirce’s interpretive rules, the repeated applications of the base term ℓ
are distributed across the individuals of the exponent term w. In particular,
the base term ℓ is not applied successively in the manner that would give
something on the order of “a lover of a lover of ... a lover of a woman”.

By way of example, suppose a universe of discourse numbers among its
elements just three women, W′, W″, W‴. In Peirce's notation the fact
may be written as follows.

• w = W′ +, W″ +, W‴

In that case the following equation would hold.

• ℓ^w = ℓ^(W′ +, W″ +, W‴) = (ℓW′),(ℓW″),(ℓW‴)

The equation says a lover of every woman in the aggregate W′ +, W″ +, W‴
is a lover of W′ that is a lover of W″ that is a lover of W‴. In other
words, a lover of every woman in the universe at hand is a lover of W′
and a lover of W″ and a lover of W‴.

The denotation of the term ℓ^w is a subset of X which may be obtained by
the following procedure. For each flag of the form L ∗ x with x in W
collect the subset proj₁(L ∗ x) of elements which appear as the first
components of the pairs in L ∗ x and then take the intersection of all
those subsets. Putting it all together, we have the following equation.

Denotation Equation ℓ^w
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-denotation-equation-l5ew.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce's 1870 “Logic of Relatives” • Comment 12.2 (part 2 of 3)

https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-comment-12-2/

Peirce's 1870 “Logic of Relatives” • Comment 12.2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.2

All,

It is instructive to examine the matrix representation of ℓ^w at this point,
not the least because it effectively dispels the mystery of the name “involution”.
First, we make the following observation. To say j is a lover of every woman is
to say j loves k if k is a woman. This can be rendered in symbols as follows.

• j loves k ⇐ k is a woman.

Reading the formula ℓ^w as “j loves k if k is a woman” highlights the operation of
converse implication inherent in it, and this in turn reveals the analogy between
implication and involution that accounts for the aptness of the latter name.

The operations defined by the formulas x^y = z and (x ⇐ y) = z for
x, y, z in the boolean domain B = {0, 1} are tabulated as follows.

Table. Involution ≅ Implication
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-involution-e28985-implication.png

It is clear the two operations are isomorphic, being effectively
the same operation of type B × B → B. All that remains is to see
how operations like these on values in B induce the corresponding
operations on sets and terms.

To be continued ...

Regards,

Jon

[Jon Awbrey]

Cf: Peirce's 1870 “Logic of Relatives” • Comment 12.2 (part 3 of 3)

https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-comment-12-2/

Peirce's 1870 “Logic of Relatives” • Comment 12.2
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.2

All,

Finally, let's see how operations on boolean values

induce the corresponding operations on sets and terms.

The term ℓ^w determines a selection of individuals from the universe
of discourse X which may be computed via the corresponding operation
on coefficient matrices. If the terms ℓ and w are represented by the
matrices L = Mat(ℓ) and W = Mat(w), respectively, then the operation on
terms which produces the term ℓ^w must be represented by a corresponding
operation on matrices, L^W = Mat(ℓ)^Mat(w), which gives the matrix Mat(ℓ^w).
In short, the involution operation on matrices must be defined in such a way
that the following equation holds.

Equation 1. Matrix Involution L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-involution-l5ew.png

The fact that ℓ^w denotes individuals in a subset of X tells us its
matrix representation L^W is a 1‑dimensional array of coefficients
in B indexed by the elements of X. The value of the matrix L^W at
the index u in X is written (L^W)_u and computed as follows.

Equation 2. Matrix Computation L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-computation-l5ew.png

Regards,

Jon

[Jon Awbrey]

Peirce’s 1870 “Logic of Relatives” • Comment 12.3 (part 1 of 2)
https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-comment-12-3/

Peirce’s 1870 “Logic of Relatives” • Comment 12.3
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.3

All,

We now have two ways of computing a logical involution raising
a dyadic relative term to the power of a monadic absolute term,
for example, ℓ^w for “lover of every woman”.

The first method applies set-theoretic operations to the extensions
of absolute and relative terms, expressing the denotation of the
term ℓ^w as the intersection of a set of relational applications.

Equation 1. Denotation Equation L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-denotation-equation-l5ew-e288a9le28899xe28888w.png

The second method operates in the matrix representation,
expressing the value of the matrix L^W = Math(ℓ^w) at an
argument u as a product of coefficient powers.

Equation 2. Matrix Computation L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-computation-l5ew.png

Abstract formulas like these are more easily grasped
with the aid of a concrete example and a picture of
the relations involved. Next time we'll take up
one such example.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.3 (part 2 of 2)

https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-comment-12-3/

Peirce’s 1870 “Logic of Relatives” • Comment 12.3
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.3

All,

Last time we looked at two ways of computing a logical involution which
raises a dyadic relative term to the power of a monadic absolute term,

for example, ℓ^w for “lover of every woman”.

The first method computes the denotation of the term ℓ^w
by intersecting the family of sets produced by applying ℓ
to the elements of w, as given by the following equation.

Equation 1. Denotation Equation for L^W

https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-denotation-equation-l5ew-e288a9le28899xe28888w.png

The second method operates in the matrix representation,

raising the matrix for ℓ to the power of the matrix for w,
as given by the following equation.

Equation 2. Matrix Computation for L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-computation-l5ew.png

To gain a more intuitive grasp of what these formulas mean,
let's cook up a concrete example and draw a picture of the
relations involved.

Involution Example
==================

Consider a universe of discourse X subject to the following data.

• X = {a, b, c, d, e, f, g, h, i}
• W = {d, f}
• L = {b:a, b:c, c:b, c:d, e:d, e:e, e:f, g:f, g:h, h:g, h:i}

Figure 55 shows the placement of W within X
and the placement of L within X × X.

Figure 55. Bigraph for the Involution L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-bigraph-involution-l5ew.png

To highlight the role of W more clearly, the Figure represents
the absolute term “w” by means of the relative term “w,” which
conveys the same information.

Computing the denotation of ℓ^w by way of the class intersection
formula, we can show our work as follows.

Equation 3. Class Intersection Formula for L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-class-intersection-l5ew.png

With Figure 55 in mind we can visualize
the computation of (L^W)_u as follows.

1. Pick a specific u in the bottom row of the Figure.

2. Pan across the elements v in the middle row of the Figure.

3. If u links to v then L_uv = 1, otherwise L_uv = 0.

4. If v in the middle row links to v in the top row then W_v = 1,
otherwise W_v = 0.

5. Compute the value of (L_uv)^W_v, which equals the value of the
converse implication L_uv ⇐ W_v, for each v in the middle row.

6. If any of the values (L_uv)^W_v is 0 then the product
∏_v (L_uv)^W_v is 0, otherwise it is 1.

As a general observation, we know the value of (L_uv)^W_v goes to 0
just as soon as we find a v in X such that L_uv = 0 and W_v = 1,
in other words, such that (u, v) is not L but v is in W.
If there is no such v then (L_uv)^W_v = 1.

Running through the program for each u in X, the only case producing
a non-zero result is ((L_uv)^W_v)_e = 1. That portion of the work
can be summarized as follows.

Equation 4. Matrix Coefficient Formula for L^W
https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-coefficient-l5ew.png

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.4
https://inquiryintoinquiry.com/2014/06/14/peirces-1870-logic-of-relatives-comment-12-4/

Peirce’s 1870 “Logic of Relatives” • Comment 12.4
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.4

All,

Peirce next considers a pair of compound involutions, stating
an equation between them analogous to a law of exponents from
ordinary arithmetic, namely, (a^b)^c = a^{bc}.

Law of Exponents (a^b)^c = a^(bc)
https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-a5eb5ec-a5ebc-1.png

<QUOTE CSP:>

Then (s^ℓ)^w will denote whatever stands to every woman
in the relation of servant of every lover of hers; and
s^(ℓw) will denote whatever is a servant of everything
that is lover of a woman. So that (s^ℓ)^w = s^(ℓw).

https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-s5el5ew-s5elw.png

(Peirce, CP 3.77)
https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/
</QUOTE>

Articulating the compound relative term s^(ℓw)
in set-theoretic terms is fairly immediate.

Denotation Equation s^(ℓw)
https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-denotation-equation-s5elw.png

On the other hand, translating the compound relative term (s^ℓ)^w
into its set-theoretic equivalent is less immediate, the hang-up
being we have yet to define the case of logical involution raising
one dyadic relative term to the power of another. As a result, it
looks easier to proceed through the matrix representation, drawing
once again on the inspection of a concrete example.

Involution Example 2
====================

Consider a universe of discourse X subject to the following data.

• X = {a, b, c, d, e, f, g, h, i}

• L = {b:a, b:c, c:b, c:d, e:d, e:e, e:f, g:f, g:h, h:g, h:i}

• S = {b:a, b:c, d:c, d:d, d:e, f:e, f:f, f:g, h:g, h:i}

Figure 56. Bigraph for the Involution S^L
https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-bigraph-involution-s5el.png

There is a “servant of every lover of” link between u and v
if and only if u∙S ⊇ L∙v. But the vacuous inclusions, that is,
the cases where L∙v = ∅, have the effect of adding non‑intuitive
links to the mix.

The computational requirements are evidently met by the following formula.

Matrix Computation for S^L
https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-matrix-computation-s5el.png

In other words, (S^L)_xy = 0 if and only if
there exists a p in X such that S_xp = 0 and
L_py = 1.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.5
https://inquiryintoinquiry.com/2014/06/15/peirces-1870-logic-of-relatives-comment-12-5/

Peirce’s 1870 “Logic of Relatives” • Comment 12.5
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Commentary_Note_12.5

All,

The equation (s^ℓ)^w = s^(ℓw) can be verified by
establishing the corresponding equation in matrices.

• (S^L)^W = S^(LW)

If A and B are two 1-dimensional matrices over the
same index set X then A = B if and only if A_x = B_x
for every x in X. Thus, a routine way to check the
validity of (S^L)^W = S^(LW) is to check whether
the following equation holds for arbitrary x in X.

• ((S^L)^W)_x = (S^(LW))_x

Taking both ends toward the middle, we proceed as follows.

Matrix Equation ((S^L)^W)_x = (S^(LW))_x
https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-s5el5ew_x-s5elw_x.png

The products commute, so the equation holds. In essence, the matrix
identity turns on the fact that the law of exponents (a^b)^c = a^(bc)
in ordinary arithmetic holds when the values a, b, c are restricted to
the boolean domain B = {0, 1}. Interpreted as a logical statement, the
law of exponents (a^b)^c = a^(bc) amounts to a theorem of propositional
calculus otherwise expressed in the following ways.

• ((a ⇐ b) ⇐ c) = (a ⇐ b ∧ c)

• (c ⇒ (b ⇒ a)) = (c ∧ b ⇒ a)

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Intermezzo
https://inquiryintoinquiry.com/2014/06/18/peirces-1870-logic-of-relatives-intermezzo/

Peirce’s 1870 “Logic of Relatives”

https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

Update • 10 April 2022

All,

This brings me to the end of the notes on Peirce’s 1870
Logic of Relatives I began posting to the web in various
online discussion groups a dozen (now a score) years ago.
There are apart from that only scattered notes and bits of
discussion with other people I archived on the InterSciWiki
talk page of the OEIS Wiki article linked above.

I rushed through my last few comments a little too hastily, giving
no more than sketches of proofs for Peirce’s logical formulas, and
I won’t be reasonably well convinced of them until I examine a few
more concrete examples and develop one or two independent lines of
proof. So I have that much unfinished business to do before moving
on to the rest of Peirce’s paper.

But I’ll take a few days to catch my breath, rummage through those
old notes of mine to see if they hide any hints worth salvaging,
and then start fresh, raveling out the rest of Peirce’s clues
to the maze of logical relatives.

Regards,

Jon

[Jon Awbrey]

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 13
https://inquiryintoinquiry.com/2022/04/13/peirces-1870-logic-of-relatives-selection-13/

All,

I continue with my Selections from and Comments on Peirce's
1870 Logic of Relatives, one of those works which convinced
me so long ago I would need to learn a lot more mathematics
before I'd have any hope of understanding what he was about.
What I've put on the Web to date is linked in this Overview:

Peirce’s 1870 “Logic of Relatives” • Overview
https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 13
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#Selection_13

<QUOTE CSP:>
The Sign of Involution (cont.)

A servant of every man and woman will be denoted by s^(m +, w)
and (s^m),(s^w) will denote a servant of every man that is a
servant of every woman. So that

s(m+,w) = (s^m),(s^w)
https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-s5emw-s5ems5ew.png