#peirceslaw — Public Fediverse posts

Survey of Animated Logical Graphs • 7
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

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.
Articles and blog series on the core ideas are linked below.

Beginnings —

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

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Elements —

Logic Syllabus
• https://oeis.org/wiki/Logic_Syllabus

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Minimal Negation Operators
• https://oeis.org/wiki/Minimal_negation_operator

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Examples —

Peirce's Law
• https://inquiryintoinquiry.com/2023/10/18/peirces-law-a/
• https://oeis.org/wiki/Peirce%27s_law

Praeclarum Theorema
• https://inquiryintoinquiry.com/2023/10/05/praeclarum-theorema-a/
• https://oeis.org/wiki/Logical_Graphs#Praeclarum_theorema

Proof Animations
• https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

Excursions —

Cactus Language
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Futures Of Logical Graphs
• https://oeis.org/wiki/Futures_Of_Logical_Graphs

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

Differential Analytic Turing Automata
• https://oeis.org/wiki/Differential_Analytic_Turing_Automata_%E2%80%A2_Overview

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperator #PeircesLaw #TuringAutomata

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).

#Peirce #Logic #LogicalGraphs #PeircesLaw

Logic Syllabus • 3
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Concepts
• https://oeis.org/wiki/Logic_Syllabus#Logical_concepts

Ampheck • https://oeis.org/wiki/Ampheck
Boolean Domain • https://oeis.org/wiki/Boolean_domain
Boolean Function • https://oeis.org/wiki/Boolean_function
Boolean-Valued Function • https://oeis.org/wiki/Boolean-valued_function
Differential Logic • https://oeis.org/wiki/Differential_logic
Logical Graph • https://oeis.org/wiki/Logical_Graphs
Minimal Negation Operator • https://oeis.org/wiki/Minimal_negation_operator
Multigrade Operator • https://oeis.org/wiki/Multigrade_operator
Parametric Operator • https://oeis.org/wiki/Parametric_operator
Peirce's Law • https://oeis.org/wiki/Peirce%27s_law
Propositional Calculus • https://oeis.org/wiki/Propositional_calculus
Sole Sufficient Operator • https://oeis.org/wiki/Sole_sufficient_operator
Truth Table • https://oeis.org/wiki/Truth_table
Universe of Discourse • https://oeis.org/wiki/Universe_of_discourse
Zeroth Order Logic • https://oeis.org/wiki/Zeroth_order_logic

#Logic #LogicSyllabus #Ampheck #BooleanDomain #BooleanFunction #BooleanValuedFunction
#DifferentialLogic #LogicalGraph #MinimalNegationOperator #MultigradeOperator
#ParametricOperator #PeircesLaw #PropositionalCalculus #SoleSufficientOperator
#TruthTable #UniverseOfDiscourse #ZerothOrderLogic

@DavisSmith @philosophy

For that my (classical) favorites are #PeircesLaw and #Leibniz's #PraeclarumTheorema.

Here's how they look in a variant of #Peirce's #AlphaGraphs.

• https://inquiryintoinquiry.com/2008/10/06/peirces-law/

• https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/

Animated Logical Graphs • 6

Re: Peirce List • Jim Willgoose

At root we are dealing with a genre of very abstract formal systems.  They have grammars that determine their well-formed expressions and rules that determine the permissible transformations among expressions, but they lack all logical meaning until we supply an interpretation.

The formal system Peirce developed for propositional logic and Spencer Brown resurrected for his Laws of Form admits a formal duality which allows it to be fleshed out with logical meanings in two distinct ways.  The two interpretations are employed in Peirce’s entitative graphs and existential graphs, respectively.  It is clear from everything they write that both authors are well aware of both interpretations, but Peirce would come to found his later developments on the existential sense while Spencer Brown favored the entitative sense in his expositions.

See the following readings for further discussion.

• Logical Graphs • Formal Development ☞ This Blog • OEIS Wiki

References

• Theme One Program • User Guide
• Applications of a Propositional Calculator • Constraint Satisfaction Problems

Animated Logical Graphs • 3

Re: Peirce List • Helmut Raulien

I have a little more leisure now to start climbing back into the saddle, so let me see where we left off …

Try looking into the article I linked before:

• Logical Graphs

Or my first couple of blog posts on Logical Graphs:

• Logical Graphs • First Impressions
• Logical Graphs • Formal Development

There are literally decades of thought and work that went into those, and if they do not engage the reader in the excitement of possible future developments then I would sorely appreciate any feedback on where and why they fail to do so.

George Spencer Brown is one of the few writers I’ve run across in the time since my first encounter with Peirce’s logical graphs who truly grasped the full depth of Peirce’s insight into logic, a vision that pierced the veil of logical interpretations, entitative and existential, to the deep formal unity between them.  That is one of the reasons I’ve made an effort to treat the two interpretations in parallel as far as I was able.  It is an extremely enticing research question to me whether that symmetry is necessarily broken as we pass from propositional to quantificational logic, or whether there is some way it may be or need be maintained.

But that is a question for the future …

Resource

• Survey of Animated Logical Graphs

Animated Logical Graphs • 2

Re: Peirce List • Jim Willgoose

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away.  I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.

I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.

So I’ll try to keep focused on that.

Resources

• Logical Graphs • First Impressions
• Logical Graphs • Formal Development
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Animated Logical Graphs • 1

For Your Musement …

Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.

• Proof Animations
• Double Negation
• Peirce’s Law
• Praeclarum Theorema
• Two‑Thirds Majority Function

A full discussion of logical graphs can be found in the following article.

• Logical Graphs

Resources

• Logical Graphs • First Impressions
• Logical Graphs • Formal Development
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi