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