Peirce's 1885 “Algebra of Logic” • Discussion 2
• https://inquiryintoinquiry.com/2024/04/03/peirces-1885-algebra-of-logic-discussion-2/

Re: FB | Daniel Everett

One thing I've been trying to understand for a very long time is the changes in Peirce's writing about math and logic from 1865 to 1885. If there's anything I've learned from reading Peirce in the often dim light of intellectual history it is to be wary of progressivist assumptions — but unlike many of his other fans I apply that caution also within the body of his own work. Long story short, from 1865 to 1885 I see progress on several fronts but also bits of backsliding from his more prescient early insights. So it's a puzzle … and it will take more study to ravel out the reasons why.

Resources for reconciling Peirce's two accounts —
1. The 1870 account of logical involution
2. The 1885 account of universal quantification

Peirce's 1870 “Logic of Relatives” • Selection 12 • The Sign of Involution
• https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/
Comments —
(1) https://inquiryintoinquiry.com/2014/06/10/peirces-1870-logic-of-relatives-comment-12-1/
(2) https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-comment-12-2/
(3) https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-comment-12-3/
(4) https://inquiryintoinquiry.com/2014/06/14/peirces-1870-logic-of-relatives-comment-12-4/
(5) https://inquiryintoinquiry.com/2014/06/15/peirces-1870-logic-of-relatives-comment-12-5/

Peirce's 1885 “Algebra of Logic” • Selections
(1) https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/
(2) https://inquiryintoinquiry.com/2024/03/26/peirces-1885-algebra-of-logic-selection-2/
(3) https://inquiryintoinquiry.com/2024/03/30/peirces-1885-algebra-of-logic-selection-3/
(4) https://inquiryintoinquiry.com/2024/04/01/peirces-1885-algebra-of-logic-selection-4/

Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• https://www.jstor.org/stable/2369451

#Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
#Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

Peirce's 1885 “Algebra of Logic” • Discussion 1
• https://inquiryintoinquiry.com/2024/04/02/peirces-1885-algebra-of-logic-discussion-1/

Re: FB | Daniel Everett

DE:
❝One of the most important papers in the history of logic. “On the Algebra of Logic” was the first to introduce the term “quantifier”.

❝Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• https://www.jstor.org/stable/2369451 ❞

As far as quantification by any other word goes, Peirce had already introduced a more advanced and “functional” concept of quantification in his 1870 “Logic of Relatives”. The subsequent passage to Fregean styles of first order logic would turn out to be a retrograde movement toward syntacticism (a species of nominalism), as seen in the general run of what fol‑lowed in the fol‑lowing years.

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

Especially ☞ “The Sign of Involution”
• https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/

The connection between logical involution and universal quantification which Peirce put to use in his 1870 Logic of Relatives will turn up again a century later with the application of category theory to computer science and both of those in turn to logic. Just one more time Peirce was that far ahead of it.

See ☞ Lambek and Scott (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press.
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Lambek

#Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
#Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

Peirce's 1885 “Algebra of Logic” • Selection 1.1
• https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞

❝Any character or proposition either concerns one subject, two subjects, or a plurality of subjects. For example, one particle has mass, two particles attract one another, a particle revolves about the line joining two others. A fact concerning two subjects is a dual character or relation; but a relation which is a mere combination of two independent facts concerning the two subjects may be called “degenerate”, just as two lines are called a degenerate conic. In like manner a plural character or conjoint relation is to be called degenerate if it is a mere compound of dual characters.

❝A sign is in a conjoint relation to the thing denoted and to the mind. If this triple relation is not of a degenerate species, the sign is related to its object only in consequence of a mental association, and depends upon a habit. Such signs are always abstract and general, because habits are general rules to which the organism has become subjected. They are, for the most part, conventional or arbitrary. They include all general words, the main body of speech, and any mode of conveying a judgment. For the sake of brevity I will call them “tokens”.❞ [Note. Peirce more frequently calls these “symbols”.]

#Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
#MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
#AlgebraOfLogic #PredicateCalculus #Quantification #Semiotics
#RelationComposition #RelationConstruction #RelationReduction

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Please follow the above link for the full set of resources.
A few basic articles are linked below.

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

Relation Composition
• https://oeis.org/wiki/Relation_composition

Relation Construction
• https://oeis.org/wiki/Relation_construction

Relation Reduction
• https://oeis.org/wiki/Relation_reduction

Relative Term
• https://oeis.org/wiki/Relative_term

Sign Relation
• https://oeis.org/wiki/Sign_relation

Triadic Relation
• https://oeis.org/wiki/Triadic_relation

Six Ways of Looking at a Triadic Relation ⌬ 1
• https://inquiryintoinquiry.com/2015/02/04/six-ways-of-looking-at-a-triadic-relation-%E2%8C%AC-1/

Mathematical Demonstration and the Doctrine of Individuals
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-1-a/
• https://inquiryintoinquiry.com/2023/05/16/mathematical-demonstration-and-the-doctrine-of-individuals-2-a/

Peirce's 1870 “Logic of Relatives” —
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

#Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
#MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
#PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
#RelationComposition #RelationConstruction #RelationReduction

Survey of Relation Theory
• https://inquiryintoinquiry.com/2023/04/01/survey-of-relation-theory-6/

In this Survey of blog and wiki posts on Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Please follow the above link for the full set of resources.
A few basic articles are linked below.

Elements —
• Relation Theory ( https://oeis.org/wiki/Relation_theory )

Relational Concepts —
• Relation Construction ( https://oeis.org/wiki/Relation_construction )
• Relation Composition ( https://oeis.org/wiki/Relation_composition )
• Relation Reduction ( https://oeis.org/wiki/Relation_reduction )
• Relative Term ( https://oeis.org/wiki/Relative_term )
• Sign Relation ( https://oeis.org/wiki/Sign_relation )
• Triadic Relation ( https://oeis.org/wiki/Triadic_relation )
• Logic of Relatives ( https://oeis.org/wiki/Logic_of_relatives )
• Hypostatic Abstraction ( https://oeis.org/wiki/Hypostatic_abstraction )
• Continuous Predicate ( https://oeis.org/wiki/Continuous_predicate )

Illustrations —

Six Ways of Looking at a Triadic Relation ⌬ 1
• https://inquiryintoinquiry.com/2015/02/04/six-ways-of-looking-at-a-triadic-relation-%E2%8C%AC-1/

Peirce's 1870 “Logic of Relatives” —

Overview
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Preliminaries
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

#Peirce #Logic #LogicOfRelatives #RelationTheory #RelativeTerm
#MonadicRelation #DyadicRelation #TriadicRelation #SignRelation
#PredicateCalculus #ContinuousPredicate #HypostaticAbstraction
#RelationComposition #RelationConstruction #RelationReduction

Peirce's 1870 “Logic of Relatives” • Selection 3.2
• https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/

❝§3. Application of the Algebraic Signs to Logic❞

❝The Signs of Inclusion, Equality, Etc.❞

❝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 \(\mathrm{f} < \mathrm{m}\) is to say that Frenchmen are part of men. Indeed, if \(\mathrm{f} < \mathrm{m},\) then the number of Frenchmen is less than the number of men, and if \(\mathrm{v} = \mathrm{p},\) 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)

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Selection 2.1
• https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-selection-2/

❝§3. Application of the Algebraic Signs to Logic❞

❝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 \((\mathrm{men}),\) 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 #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Selection 1.2
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/

❝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.❞

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:

❝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.❞

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Selection 1.1
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/

We pick up Peirce's text at the following point.

❝§3. Application of the Algebraic Signs to Logic❞

❝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.❞

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 5
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

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 \(\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}\) falling under the general term \(\mathrm{h}\) 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 \(\mathrm{X}\) of the genus \(\mathrm{x}\) and the element \(x\) of the set \(X\) as we pass between the two styles of text.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 4
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

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

The Table displays the single-letter abbreviations and their verbal equivalents for the “conjugative terms” (or “higher adic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of higher adic relative terms, rendered here as LaTeX “mathfrak”, Fraktur, or Gothic.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 3
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

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

The Table displays the single-letter abbreviations and their verbal equivalents for the “simple relative terms” (or “dyadic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of dyadic relative terms, rendered here as LaTeX “mathit” or Italics.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 2
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

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

The Table displays the single-letter abbreviations and their verbal equivalents for the “absolute logical terms” (or “monadic relative terms”) used in Peirce's examples of logical formulas throughout the rest of the paper. Peirce used a distinctive typeface for the absolute term abbreviations, rendered here as LaTeX “mathrm” or Roman.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 1
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

In the beginning was the three-pointed star,
One smile of light across the empty face;
One bough of bone across the rooting air,
The substance forked that marrowed the first sun;
And, burning ciphers on the round of space,
Heaven and hell mixed as they spun.

— #DylanThomas • #InTheBeginning

Peirce’s text uses 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, which Peirce distinguishes through the use of different typefaces. The following Tables show the typefaces used in the present transcript for Peirce's examples of general terms. (I'll post just the image links for now, then the full images and texts in the next three posts.)

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

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

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

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

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

My long ago encounter with Peirce’s 1870 paper, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, was one of the events precipitating my return from the hazier heights of philosophy to the solid plains of mathematics below. Over the years I copied out various drafts of my study notes to the web, consisting of selections from Peirce’s paper along with my running commentary. A few years back I serialized what progress I had made so far to this blog and this Overview consists of links to those installments.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

@bblfish @josd @semwebpro @hochstenbach

One thing I found out early on is how critical it is to get #AlphaGraphs (#BooleanFunctions, #PropositionalCalculus, #ZerothOrderLogic) down tight. If you do that it changes how you view #FOL (#PredicateCalculus, #QuantificationalLogic). That tends to rub people who view FOL as #GOL (#GodsOwnLogic) the wrong way so you have watch out for that if you go down this road.

Here's a primer on \(\alpha\) #LogicalGraphs as I see them —
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no