Inquiry Into Inquiry • Discussion 9
• http://inquiryintoinquiry.com/2023/08/19/inquiry-into-inquiry-discussion-9/

Re: Milo Gardner
• https://www.academia.edu/community/VBqzR5?c=Q4jJVy

MG: ❝Do you agree that Peirce was limited to bivalent logic?❞

Taking classical logic as a basis for reasoning is no more limiting than taking Dedekind cuts as a basis for constructing the real number line. For Peirce's relational approach to logic as semiotics the number of dimensions in a relation is more important than the number of values in each dimension. That is where 3 makes a difference over 2.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Inquiry Into Inquiry • Discussion 8
• http://inquiryintoinquiry.com/2023/08/18/inquiry-into-inquiry-discussion-8/

Re: Milo Gardner
• https://www.academia.edu/community/Lbxjg5?c=yqXVog

MG: ❝Peirce sensed that bivalent syntax was superceded by trivalent syntax,
but never resolved that nagging question.❞

My Comment —

The main thing is not a question of syntax but a question of the mathematical models we use to cope with object realities and real objectives (pragmata). Signs, syntax, and systems of representation can make a big difference in how well they represent the object domain and how well they serve the purpose at hand but they remain accessory to those objects and purposes.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Inquiry Into Inquiry • Discussion 7
• http://inquiryintoinquiry.com/2023/08/17/inquiry-into-inquiry-discussion-7/

Dan Everett has prompted a number of discussions on Facebook recently which touch on core issues in Peirce's thought — but threads ravel on and fray so quickly in that medium one rarely get a chance to fill out the warp. Not exactly at random, here's a loose thread I think may be worth the candle.

Re: Facebook • Daniel Everett
• https://www.facebook.com/permalink.php?story_fbid=pfbid0be89MXhhCm8rxahRn4PXif6HHSCmkdiUFfMZ3qS1mNqSzRzUWfqej5a8cyz8TcyJl&id=100093271525294

My Comment —

Compositionality started out as a well-defined concept, arising from the composition of mathematical functions, abstracted to the composition of arrows and functors in category theory, and generalized to the composition of binary, two-place, or dyadic relations. In terms of linguistic complexity it's associated with properly context-free languages. That all keeps compositionality on the dyadic side of the border in Peirce's universe. More lately the term has been volatilized to encompass almost any sort of information fusion, which is all well and good so long as folks make it clear what they are talking about, for which use the term “information fusion” would probably be sufficiently vague.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Inquiry Into Inquiry • Discussion 7
• http://inquiryintoinquiry.com/2023/08/17/inquiry-into-inquiry-discussion-7/

Dan Everett has prompted a number of discussions on Facebook recently which touch on core issues in Peirce's thought — but threads ravel on and fray so quickly in that medium one rarely get a chance to fill out the warp. Not exactly at random, here's a loose thread I think may be worth the candle.

Re: Facebook • Daniel Everett
• https://www.facebook.com/permalink.php?story_fbid=pfbid0be89MXhhCm8rxahRn4PXif6HHSCmkdiUFfMZ3qS1mNqSzRzUWfqej5a8cyz8TcyJl&id=100093271525294

My Comment —

Compositionality started out as a well-defined concept, arising from the composition of mathematical functions, abstracted to the composition of arrows and functors in category theory, and generalized to the composition of binary, two-place, or dyadic relations. In terms of linguistic complexity it's associated with properly context-free languages. That all keeps compositionality on the dyadic side of the border in Peirce's universe. More lately the term has been volatilized to encompass almost any sort of information fusion, which is all well and good so long as folks make it clear what they are talking about, for which use the term “information fusion” would probably be sufficiently vague.

#Peirce #Logic #Inquiry #Inference #Information #InformationFusion #Semiotics
#CategoryTheory #Compositionality #RelationTheory #TriadicRelationIrreducibility

Survey of Precursors Of Category Theory • 4
• http://inquiryintoinquiry.com/2023/08/01/survey-of-precursors-of-category-theory-4/

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background —

Precursors Of Category Theory
• https://oeis.org/wiki/Precursors_Of_Category_Theory

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Blog Series —

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/

Precursors Of Category Theory
1. https://inquiryintoinquiry.com/2013/12/20/precursors-of-category-theory-1/
2. https://inquiryintoinquiry.com/2013/12/30/precursors-of-category-theory-2/
3. https://inquiryintoinquiry.com/2014/01/03/precursors-of-category-theory-3/

Precursors Of Category Theory • Discussion
1. https://inquiryintoinquiry.com/2020/09/13/precursors-of-category-theory-discussion-1/
2. https://inquiryintoinquiry.com/2020/09/21/precursors-of-category-theory-discussion-2/
3. https://inquiryintoinquiry.com/2020/09/25/precursors-of-category-theory-discussion-3/

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
• https://inquiryintoinquiry.com/2012/03/21/c-s-peirce-a-guess-at-the-riddle/

Peirce's Categories
1. https://inquiryintoinquiry.com/2015/10/30/peirces-categories-1/
2. https://inquiryintoinquiry.com/2015/10/31/peirces-categories-2/
3. https://inquiryintoinquiry.com/2015/11/04/peirces-categories-3/
•••
19. https://inquiryintoinquiry.com/2020/05/13/peirces-categories-19/
20. https://inquiryintoinquiry.com/2020/05/14/peirces-categories-20/
21. https://inquiryintoinquiry.com/2020/06/25/peirces-categories-21/

#Aristotle #Peirce #Kant #Carnap #Hilbert #Ackermann #SaundersMacLane
#Abstraction #Analogy #CategoryTheory #Diagrams #FoundationsOfMathematics
#FunctionalLogic #RelationTheory #ContinuousPredicate #HypostaticAbstraction
#CategoryTheory #PeircesCategories #PropositionsAsTypes #TypeTheory #Universals

Survey of Semiotics, Semiosis, Sign Relations • 4
• https://inquiryintoinquiry.com/2023/04/05/survey-of-semiotics-semiosis-sign-relations-4/

C.S. Peirce defines “logic” as “formal semiotic”, using “formal” to distinguish the role of logic as a normative science, over and above the more descriptive study of signs in the wild and their antics at large. Understanding logic as Peirce understood it therefore requires a companionstudy of semiotics, semiosis, and sign relations.

This is a Survey of blog and wiki resources on the theory of signs, also known as “semeiotic” or “semiotics”, and the actions referred to as “semiosis” which transform signs among themselves in relation to their objects, all as based on C.S. Peirce's concept of triadic sign relations.

Please follow the above link for the full set of resources.
Articles and blog series on the core ideas are linked below.

Elements —

• Semeiotic ( https://oeis.org/wiki/Semeiotic )

• Sign Relations ( https://oeis.org/wiki/Sign_relation )

Sources —

C.S. Peirce • On the Definition of Logic
• https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Semiotic
• https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

C.S. Peirce • Objective Logic
• https://inquiryintoinquiry.com/2012/03/09/c-s-peirce-objective-logic/

Blog Series —

• Semeiotic ( https://inquiryintoinquiry.com/2008/07/30/semeiotic/ )

#Peirce #Logic #LogicAsSemiotics #Semeiotic #Semiotics #Semiosis
#RelationTheory #InterpretiveFrameworks #ObjectiveFrameworks
#SystemsOfInterpretation #SignRelations #TriadicRelations

Survey of Relation Theory
• https://inquiryintoinquiry.com/2023/07/26/survey-of-relation-theory-7/

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 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 )
• 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/

Information‑Theoretic Perspective (Escape from Nominalism)

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

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
#RelationComposition #RelationConstruction #RelationReduction

Logic Syllabus • 4
• https://inquiryintoinquiry.com/logic-syllabus/

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

Continuous Predicate • https://oeis.org/wiki/Continuous_predicate
Hypostatic Abstraction • https://oeis.org/wiki/Hypostatic_abstraction
Logic of Relatives • https://oeis.org/wiki/Logic_of_relatives
Logical Matrix • https://oeis.org/wiki/Logical_matrix
Relation • https://oeis.org/wiki/Relation
Relation Composition • https://oeis.org/wiki/Relation_composition
Relation Construction • https://oeis.org/wiki/Relation_construction
Relation Reduction • https://oeis.org/wiki/Relation_reduction
Relation Theory • https://oeis.org/wiki/Relation_theory
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 #LogicSyllabus #ContinuousPredicate #HypostaticAbstraction #LogicOfRelatives
#LogicalMatrix #Relation #RelationComposition #RelationConstruction #RelationReduction
#RelationTheory #RelativeTerm #SignRelation #TriadicRelation

Systems of Interpretation • 1
• https://inquiryintoinquiry.com/2023/05/05/systems-of-interpretation-1-2/

Questions have arisen about the different styles of diagrams and figures used to represent triadic sign relations in Peircean semiotics. What do they mean? Which style is best? Among the most popular pictures some use geometric triangles while others use the three‑pronged graphs Peirce used in his logical graphs to represent triadic relations.

Diagrams and figures, like any signs, can serve to communicate the intended interpretants and thus to coordinate the conduct of interpreters toward the intended objects — but only in communities of interpretation where the conventions of interpretation are understood. Conventions of interpretation are by comparison far more difficult to communicate.

That brings us to the first question we have to ask about the possibility of communication in this area, namely, what conventions of interpretation are needed to make sense of these diagrams, figures, and graphs?

#Peirce #Logic #LogicalGraphs #RelationTheory #Semiotics #Semiosis
#DiagrammaticReasoning #InterpretiveFrameworks #ObjectiveFrameworks
#SystemsOfInterpretation #SignRelations #TriadicRelations #Visualization

@NicoleCRust @knutson_brain

Here's the beginning of a prospective blog series I started … this topic interacts strongly with a host of others I've been struggling to articulate over the years …

Inquiry Into Inquiry • Discussion 6
• https://inquiryintoinquiry.com/2023/04/30/inquiry-into-inquiry-discussion-6/

Re: Nicole Rust
• https://mathstodon.xyz/@NicoleCRust@neuromatch.social/110197230713039748

❝Computations or Processes —
How do you think about the building blocks of the brain?❞

I keep coming back to this thread about levels, along with others on the related issue of paradigms, as those have long been major questions for me. I am trying to clarify my current understanding for a blog post. It will start out a bit like this —

A certain amount of “level” language is natural in the sciences but “level” metaphors come with hidden assumptions about higher and lower places in hierarchies which don't always fit the case at hand. In complex cases what look at first like parallel strata may in time be better comprehended as intersecting domains or mutually recursive and entangled orders of being. When that happens we can guard against misleading imagery by speaking of domains or realms instead of levels.

To be continued …

#Peirce #Logic #LogicalGraphs #DifferentialLogic #CactusLanguage
#Inquiry #InquiryDrivenSystem #InquiryIntoInquiry #NeuralNetwork
#Semiotics #RelationTheory #SignRelation #TriadicRelation #Model
#ObjectiveReality #MathematicalStructure #SymbolicRepresentation

Inquiry Into Inquiry • Discussion 6
• https://inquiryintoinquiry.com/2023/04/30/inquiry-into-inquiry-discussion-6/

Re: Nicole Rust
• https://mathstodon.xyz/@NicoleCRust@neuromatch.social/110197230713039748

❝Computations or Processes — How do you think about the building blocks of the brain?❞

I keep coming back to this thread about levels, along with others on the related issue of paradigms, as those have long been major questions for me. I am trying to clarify my current understanding for a blog post. It will start out a bit like this —

A certain amount of “level” language is natural in the sciences but “level” metaphors come with hidden assumptions about higher and lower places in hierarchies which don't always fit the case at hand. In complex cases what look at first like parallel strata may in time be better comprehended as intersecting domains or mutually recursive and entangled orders of being. When that happens we can guard against misleading imagery by speaking of domains or realms instead of levels.

To be continued …

#Peirce #Logic #LogicalGraphs #DifferentialLogic #CactusLanguage
#Inquiry #InquiryDrivenSystem #InquiryIntoInquiry #NeuralNetwork
#Semiotics #RelationTheory #SignRelation #TriadicRelation #Model
#ObjectiveReality #MathematicalStructure #SymbolicRepresentation

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

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2023/03/28/survey-of-animated-logical-graphs-5/

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph-theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Please follow the above link for the full set of resources. A couple of beginning pieces are linked below.

Logical Graphs • Introduction
• https://inquiryintoinquiry.com/2008/07/29/logical-graphs-introduction/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2008/09/19/logical-graphs-formal-development/

I've been thinking about ways to connect the species of logical graphs I've been developing out of Peirce's entitative and existential graphs with the styles of logical graphs envisioned in the RDF Surfaces group.

One thing arising out of those reflections was I began to tease apart two layers of structure, the one involved in conceiving and computing logical formulas and the other employed in displaying the end results.

At any rate, I'll explore that theme further as we go.

For now, the Survey page linked above will provide an overview of work already done.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #PropositionalCalculus #LogicAsSemiotics
#RelationTheory #SignRelations #Semiotics #W3C #RDF #RDFSurfaces

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2023/03/28/survey-of-animated-logical-graphs-5/

This is a Survey of blog and wiki posts on Logical Graphs, encompassing
several families of graph-theoretic structures originally developed by
Charles S. Peirce as graphical formal languages or visual styles of
syntax amenable to interpretation for logical applications.

Please follow the link above for the full set of resources.
Here I'll just link to a couple of beginning pieces.

Logical Graphs • Introduction
• https://inquiryintoinquiry.com/2008/07/29/logical-graphs-introduction/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2008/09/19/logical-graphs-formal-development/

I've been thinking about ways to connect the species of logical graphs
I've been developing out of Peirce's entitative and existential graphs
with the styles of logical graphs envisioned in the RDF Surfaces group.

One thing arising out of those reflections was I began to tease apart
two layers of structure, the one involved in conceiving and computing
logical formulas and the other employed in displaying the end results.

At any rate, I'll be exploring that theme as we go.

For now, the Survey page linked above will provide an overview of work already done.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #PropositionalCalculus #LogicAsSemiotics
#RelationTheory #SignRelations #Semiotics #W3C #RDF #RDFSurfaces

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2023/04/05/survey-of-semiotics-semiosis-sign-relations-4/

C.S. Peirce defines “logic” as “formal semiotic”, using “formal” to distinguish the role of logic as a normative science, over and above the more descriptive study of signs in the wild and their antics at large. Understanding logic as Peirce understood it therefore requires a companion study of semiotics, semiosis, and sign relations.

This is a Survey of resources on the theory of signs, variously known as “semeiotic” or “semiotics”, and the actions referred to as “semiosis” which transform signs among themselves in relation to their objects, all as based on C.S. Peirce's concept of triadic “sign relations”.

Please follow the above link for the full set of resources. I'll link just a sample of basics below.

Elements —
• Semeiotic ( https://oeis.org/wiki/Semeiotic )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )

Sources —

C.S. Peirce • On the Definition of Logic
• https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Semiotic
• https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

C.S. Peirce • Objective Logic
• https://inquiryintoinquiry.com/2012/03/09/c-s-peirce-objective-logic/

Blog Series —

• Semeiotic ( https://inquiryintoinquiry.com/2008/07/30/semeiotic/ )

#Peirce #Logic #RelationTheory #Semiotics
#Semiosis #SignRelations #TriadicRelations

Peirce's 1870 “Logic of Relatives” • Selection 3.3
• https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/

Comment on ❝The Signs of Inclusion, Equality, Etc.❞

Peirce's use of a square bracket \([t]\) to indicate a mapping from terms to numbers provides a basis for the computation of frequencies, probabilities, and other statistical measures constructed from them, thus 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.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

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❞ (cont.)

❝The Signs of Inclusion, Equality, Etc.❞ (cont.)

❝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.❞

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.

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

Peirce's 1870 “Logic of Relatives” • Selection 3

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

❝The Signs of Inclusion, Equality, Etc.❞

❝I shall follow Boole in taking the sign of equality to signify identity. Thus, if \(\mathrm{v}\) denotes the Vice-President of the United States, and \(\mathrm{p}\) the President of the Senate of the United States,

\[\mathrm{v} = \mathrm{p}\]

❝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

\[\mathrm{f} < \mathrm{m}\]

❝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 \(\leqq,\) “as small as”) will mean “is”. Thus,

\[\mathrm{f} ~-\!\!\!< \mathrm{m}\]

❝means “every Frenchman is a man”, without saying whether there are any other men or not. So,

\[\mathit{m} ~-\!\!\!< \mathit{l}\]

❝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 \(\mathrm{f} ~-\!\!\!< \mathrm{m}\) and \(\mathrm{m} ~-\!\!\!< \mathrm{a}\) we can infer that \(\mathrm{f} ~-\!\!\!< \mathrm{a},\) that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.❞

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870

Peirce's 1870 “Logic of Relatives” • Selection 2.2
• https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-selection-2/

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 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.

• Peirce follows the mathematician's usual practice of making the status of being an individual or a universal relative to a discourse in progress.

• Peirce takes the plural denotation of terms for granted. Indeed. what's the number of a term for, if it could not vary apart from being one or nil?

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

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

We continue with §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 (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, \([t].\)❞

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