Logical Graphs • Discussion 6
• https://inquiryintoinquiry.com/2023/08/29/logical-graphs-discussion-6/

Re: Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Figures 1 and 2
• https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-figures-1-2-framed.png

Re: Academia.edu • Robert Appleton
• https://www.academia.edu/community/lavbw5?c=Q4jlVy

RA:
❝As a professional graphic designer and non-mathematician reading your two diagrams, I need to ask for a simpler statement of their purpose. What do Fig 1 and Fig 2 represent to you? And what insight do they provide us?❞

My Comment —

Figures 1 and 2 are really just a couple of “in medias res” pump‑primers or ice‑breakers. This will all be explained in the above linked blog post, where I'm revising the text and upgrading the graphics of some work I first blogged in 2008 based on work I did even further back. I'll be taking a fresh look at that as I serialize it here.

Those two Figures come from George Spencer Brown's 1969 book Laws of Form, where he called them the Law of Calling and the Law of Crossing. GSB revived and clarified central aspects of Peirce's systems of logical graphs and I find it helpful to integrate his work into my exposition of Peirce. For now you can think of those as exemplifying two core formal principles which go to the root of the mathematical forms underlying logical reasoning.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Logical Graphs • First Impressions 1
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Introduction • Moving Pictures of Thought —

A “logical graph” is a graph-theoretic structure in one of the systems of graphical syntax Charles Sanders Peirce developed for logic.

In numerous papers on “qualitative logic”, “entitative graphs”, and “existential graphs”, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.

In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on the aspects of form shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistensialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2023/08/22/differential-logic-%ce%b1/

Differential logic is the logic of variation — the logic of change and difference.

Differential logic is the component of logic whose object is the description of variation, for example, the aspects of change, difference, distribution, and diversity, in universes of discourse subject to qualitative logical description. In its formalization, differential logic treats the principles governing the use of a “differential logical calculus”, in other words, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple case of a differential logical calculus is furnished by a differential propositional calculus. This augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources —

Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
• Part 1 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 )
• Part 2 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 )
• Part 3 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 )

Differential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
• Part 1 ( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1 )
• Part 2 ( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2 )

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
• Part 1 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1 )
• Part 2 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2 )
• Part 3 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3 )
• Part 4 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4 )
• Part 5 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5 )

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#Leibniz #BooleanFunctions #BooleanDifferenceCalculus #QualitativeDynamics
#DifferentialPropositions #MinimalNegationOperators #NeuralNetworkSystems

Cactus Rules
• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules

With an eye toward the aims of the NKS Forum I've begun to work out a translation of the “elementary cellular automaton rules” (ECARs), in effect, just the boolean functions of abstract type \(f : \mathbb{B}^3 \to \mathbb{B},\) into cactus language, and I'll post a selection of my working notes here.

#Logic #LogicalGraphs #BooleanFunctions #PropositionalCalculus
#CactusCalculus #CactusLanguage #CactusSyntax #CellularAutomata

Logic Syllabus • 5
• https://inquiryintoinquiry.com/logic-syllabus/

Related Articles
• https://oeis.org/wiki/Logic_Syllabus#Related_articles

Cactus Language • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Futures Of Logical Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs
Differential Propositional Calculus • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
Differential Logic • https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Propositions As Types Analogy • https://oeis.org/wiki/Propositions_As_Types_Analogy
Propositional Equation Reasoning Systems • https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems
Prospects for Inquiry Driven Systems • https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems
Introduction to Inquiry Driven Systems • https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems
Inquiry Driven Systems • Inquiry Into Inquiry • https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

#Logic #LogicSyllabus #CactusLanguage #LogicalGraphs #DifferentialLogic
#DifferentialPropositionalCalculus #DifferentialLogicAndDynamicSystems
#PropositionsAsTypesAnalogy #PropositionalEquationReasoningSystems
#Inquiry #InquiryDrivenSystems #InquiryIntoInquiry #DynamicalSystems

Logic Syllabus • 5
• https://inquiryintoinquiry.com/logic-syllabus/

Related Articles
• https://oeis.org/wiki/Logic_Syllabus#Related_articles

Cactus Language • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview
Futures Of Logical Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs
Differential Propositional Calculus • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
Differential Logic • https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Propositions As Types Analogy • https://oeis.org/wiki/Propositions_As_Types_Analogy
Propositional Equation Reasoning Systems • https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems
Prospects for Inquiry Driven Systems • https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems
Introduction to Inquiry Driven Systems • https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems
Inquiry Driven Systems • Inquiry Into Inquiry • https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

#Logic #LogicSyllabus #CactusLanguage #LogicalGraphs #DifferentialLogic
#DifferentialPropositionalCalculus #DifferentialLogicAndDynamicSystems
#PropositionsAsTypesAnalogy #PropositionalEquationReasoningSystems
#Inquiry #InquiryDrivenSystems #InquiryIntoInquiry #DynamicalSystems

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 Differential Logic • 5
• https://inquiryintoinquiry.com/2023/04/25/survey-of-differential-logic-5/

This is a Survey of work in progress on Differential Logic, resources under development toward a more systematic treatment.

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models. To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical
calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

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

Differential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2
• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2
3 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3
• https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2
3 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3
4 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4
5 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5
• https://inquiryintoinquiry.com/2023/03/04/differential-logic-and-dynamic-systems-overview-2/

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DynamicSystems
#BooleanFunctions #BooleanDifferenceCalculus #QualitativePhysics
#CactusCalculus #MinimalNegationOperators #NeuralNetworkSystems

Survey of Theme One Program
• https://inquiryintoinquiry.com/2023/03/30/survey-of-theme-one-program-5/

This is a Survey of resources for the Theme One Program I worked on all through the 1980s. The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning. I had earlier developed separate programs for basic components of those tasks, namely, 2-level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce's logical graphs as a syntax for propositional logic. Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modes of operation to cooperate. The long-term vision is the design and implementation of an Automated Research Tool able to double as a platform for Inquiry Driven Education.

Please follow the above link for the full set of resources. A sample of basics are liked below.

Wiki Hub —

Theme One Program • Overview
• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Documentation —

Theme One Program • Pascal Source Code
• https://www.academia.edu/5210987/Theme_One_Program_Pascal_Source_Code

Theme One Program • User Guide
• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Theme One Program • Exposition
• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Applications —

Applications of a Propositional Calculator • Constraint Satisfaction Problems
• https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

Exploratory Qualitative Analysis of Sequential Observation Data
• https://oeis.org/wiki/User:Jon_Awbrey/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

References —

An Architecture for Inquiry • Building Computer Platforms for Discovery
• https://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery

Exploring Research Data Interactively • Theme One : A Program of Inquiry
• https://www.academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry

#Peirce #Logic #LogicalGraphs #ThemeOneProgram #IdeaProcessor
#BooleanSatisfiability #CactusLanguage #DeclarativeProgramming

Theme One Program • Discussion 10
• https://inquiryintoinquiry.com/2023/04/17/theme-one-program-discussion-10/

Re: Seamus Bradley ( https://mathstodon.xyz/@Scmbradley/110198310724722597 )
❝I thought of a programming language where every function can only return one type: the return type. The return type is just a wrapper around a struct that contains the actual return value, but also a reference to the called function and arguments, and possibly an error code.❞

My flashback —

Way back in the last millennium I started work on a programming style I called an “idea processor”, where an “idea” is a pointer to a “form” and a form is a minimal type of record containing 1 character, 1 number, and 4 more ideas.

I implemented a functional style where all the main functions are transformations of one or more ideas to a return idea. The principal data type is an “idea-form flag” which serves a role analogous to a “cons cell” in Lisp.

Here's one entry point —

Theme One Program • Exposition 1
• https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

#Peirce #Logic #LogicalGraphs #ThemeOneProgram #IdeaProcessor
#CactusLanguage #DeclarativeProgram #FunctionalProgramming

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

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

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#Boole #BooleanAlgebra #BooleanFunctions #Semiotics #Semeiotics
#SpencerBrown #LawsOfForm #PropositionalCalculus #SignRelations

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, \([t].\)❞

#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.

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