Charles Sanders Peirce, George Spencer Brown, and Me • 4
• https://inquiryintoinquiry.com/2017/08/06/charles-sanders-peirce-george-spencer-brown-and-me-4/
• https://bsky.app/profile/inquiryintoinquiry.bsky.social/post/3lh5fsszkmk23

Two things impacting my studies of Peirce and Spencer Brown over the years were my parallel studies in mathematics and computer science. In the overlap between those areas came courses in logic, mathematical linguistics, and the theory of formal languages, grammars, and automata.

My intellectual wanderings over a nine‑year undergraduate career would take me through a cycle of majors from math and physics, to communication, psychology, philosophy, and a cross‑cultural liberal arts program, then back to grad school in mathematics.

The puzzles Peirce and Spencer Brown beset my brain with were a big part of what drove me back to math, since I could see I had no chance of resolving them without learning a lot more algebra, logic, and topology than I had learned till then.

Peirce's Law • 1
• https://inquiryintoinquiry.com/2023/10/19/peirces-law-1/

A Curious Truth of Classical Logic —

Peirce's law is a propositional calculus formula which states a non‑obvious truth of classical logic and affords a novel way of defining classical propositional calculus.

Introduction —

Peirce's law is commonly expressed in the following form.

• ((p ⇒ q) ⇒ p) ⇒ p

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

History —

Here is Peirce's own statement and proof of the law:

❝A “fifth icon” is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:

• {(x ‒< y) ‒< x} ‒< x.

❝This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent x being false while its antecedent (x ‒< y) ‒< x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x ‒< y is false. But in the last case the antecedent of x ‒< y, that is x, must be true.❞ (Peirce, CP 3.384).

Peirce goes on to point out an immediate application of the law:

❝From the formula just given, we at once get:

• {(x ‒< y) ‒< α} ‒< x,

❝where the α is used in such a sense that (x ‒< y) ‒< α means that from (x ‒< y) every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of x follows the truth of x.❞ (Peirce, CP 3.384).

Logical Graphs • Discussion 5
• https://inquiryintoinquiry.com/2023/08/28/logical-graphs-discussion-5/

Re: Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/
Re: Facebook • Daniel Everett
• https://www.facebook.com/permalink.php?story_fbid=pfbid026jD3t75k69Wbs9q3qaCAvTA6zb1GXCqwu4ZWfssxgGGd1er7Wwz8PyygiQUmF6t3l&id=100093271525294

DE: Nice discussion. Development of icon-based reasoning.

My Comment —

As it happens, even though Peirce's systems of logical graphs do have iconic features, their real power over other sorts of logical diagrams (like venn diagrams) is due to their deeper symbolic character. Thereby will hang many tales to come …

#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
• 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

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

Differential Logic and Dynamic Systems • Overview
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

❝Stand and unfold yourself.❞
— Hamlet • Francisco • 1.1.2

This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.

The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DynamicSystems
#BooleanFunctions #BooleanDifferenceCalculus #QualitativeChange
#MinimalNegationOperators #NeuralNetworkSystems #Semiotics

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.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs