Functional Logic • Inquiry and Analogy • Preliminaries
• https://inquiryintoinquiry.com/2021/11/14/functional-logic-inquiry-and-analogy-preliminaries/

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

This report discusses C.S. Peirce's treatment of analogy, placing it in relation to his overall theory of inquiry. We begin by introducing three basic types of reasoning Peirce adopted from classical logic. In Peirce's analysis both inquiry and analogy are complex programs of logical inference which develop through stages of these three types, though normally in different orders.

Note on notation. The discussion to follow uses logical conjunctions, expressed in the form of concatenated tuples \(e_1 \ldots e_k,\) and minimal negation operations, expressed in the form of bracketed tuples \(\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},\) as the principal expression-forming operations of a calculus for boolean-valued functions, that is, for propositions. The expressions of this calculus parse into data structures whose underlying graphs are called “cacti” by graph theorists. Hence the name “cactus language” for this dialect of propositional calculus.

Resources —

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

Boolean Function
• https://oeis.org/wiki/Boolean_function

Boolean-Valued Function
• https://oeis.org/wiki/Boolean-valued_function

Logical Conjunction
• https://oeis.org/wiki/Logical_conjunction

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

#Peirce #Logic #Abduction #Deduction #Induction #Analogy #Inquiry
#BooleanFunction #LogicalConjunction #MinimalNegationOperator
#LogicalGraph #CactusLanguage #PropositionalCalculus

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 • Discussion 1
• https://inquiryintoinquiry.com/2023/06/02/logic-syllabus-discussion-1/

Re: Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
Re: Laws of Form ( https://groups.io/g/lawsofform/topic/logic_syllabus/99218507 )
❝_❞ John Mingers ( https://groups.io/g/lawsofform/message/2326 )

❝In a previous post you mentioned the minimal negation operator. Is there also the converse of this, i.e. an operator which is true when exactly one of its arguments is true? Or is this just XOR?❞

Yes, the “just one true” operator is a very handy tool. We discussed it earlier under the headings of “genus and species relations” or “radio button logic”. Viewed as a venn diagram it describes a partition of the universe of discourse into mutually exclusive and exhaustive regions.

Reading \(\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_m \texttt{)}\) to mean just one of \(x_1, \ldots, x_m\) is false, the form \(\texttt{((} x_1 \texttt{),} \ldots \texttt{,(} x_m \texttt{))}\) means just one of \(x_1, \ldots, x_m\) is true.

For two logical variables, though, the cases “condense” or “degenerate” and saying “just one true” is the same thing as saying “just one false”.

\[\texttt{((} x_1 \texttt{),(} x_2 \texttt{))} = \texttt{(} x_1 \texttt{,} x_2 \texttt{)} = x_1 + x_2 = \textsc{xor} (x_1, x_2).\]

There's more information on the following pages.

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

Related Truth Tables
• https://oeis.org/wiki/Minimal_negation_operator#Truth_tables

Genus, Species, Pie Charts, Radio Buttons
• https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/

Related Discussions
• https://inquiryintoinquiry.com/?s=Radio+Buttons

#Logic #LogicSyllabus #BooleanDomain #BooleanFunction #BooleanValuedFunction
#Peirce #LogicalGraph #MinimalNegationOperator #ExclusiveDisjunction #XOR
#CactusLanguage #PropositionalCalculus #RadioButtonLogic #TruthTable

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

@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 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

@mpiedrav

Cf. #CactusLanguage
• https://oeis.org/wiki/Cactus_Language_•_Overview

#MinimalNegationOperators
• https://oeis.org/wiki/Minimal_negation_operator

#Peirce #LogicalGraphs #MinimalNegations
#BooleanFunctions #PropositionalCalculus
#CactusGraphs #CactusLanguage #CactusSyntax

@bblfish @pietercolpaert

I would suggest bumping up the genus from #Trees to #Cacti. They've got the #Lobes ...

#CactusLanguage
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

@somedonkey @NicoleCRust

#CactusLanguage #CactusSyntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

@albertcardona @NicoleCRust @complexsystems @cogneurophys @PessoaBrain @SussilloDavid @carlosbrody @Neurograce @neuralreckoning @tyrell_turing @DrYohanJohn @cian @WiringtheBrain @tdverstynen @neuralengine

I'll just tag this for later when I've had a chance to collect my thoughts.

#WinnerTakeAll #Grossberg #CompetitionCooperation
#McClelland #Rumelhart #ParallelDistributedProcessing
#CactusLanguage #LogicalGraphs #MinimalNegationOperators

@NicoleCRust @complexsystems @cogneurophys @PessoaBrain @SussilloDavid @carlosbrody @Neurograce @neuralreckoning @tyrell_turing @DrYohanJohn @cian @WiringtheBrain @tdverstynen @neuralengine

#LogicalGraphs
#DifferentialLogic
#CactusLanguage
#InquiryDrivenSystems

@charlient

Try Cacti —

#CactusLanguage
• https://oeis.org/wiki/Cactus_Language_•_Overview

#CactusGraphs #CactusSyntax

@panlepan

One of my favorite quotations.
Used several times in #Epitext,
#ForInstance ☟

#AnimatedLogicalGraphs
• https://inquiryintoinquiry.com/2021/03/04/animated-logical-graphs-65/

#CactusLanguage
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#The_Cactus_Patch

Survey of #DifferentialLogic
• https://inquiryintoinquiry.com/2022/11/20/survey-of-differential-logic-4/

This is a Survey of blog and wiki posts on Differential Logic, material I plan to develop toward a more compact and systematic account.

#Logic
#LogicalGraphs
#CactusLanguage
#QualitativeDynamics
#PropositionalCalculus
#LogicalTransformations
#MinimalNegationOperators
#DiscreteDynamicalSystems
#TransformationsOfDiscourse
#DifferentialPropositionalCalculus
#DifferentialAnalyticTuringAutomata

#DifferentialPropositionalCalculus • 3
• https://inquiryintoinquiry.com/2020/02/24/differential-propositional-calculus-3/

Formal Development —

The preceding discussion outlined the ideas leading to the #DifferentialExtension of #PropositionalLogic. The next task is to lay out the concepts and terminology needed to describe various orders of differential propositional calculi.

Related Subjects —
#Peirce #Semiotics #BooleanFunctions
#Logic #LogicalGraphs #DifferentialLogic
#GraphTheory #CactusGraphs #CactusLanguage
#MinimalNegationOperators

#DifferentialPropositionalCalculus • 2
https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/

#CactusCalculus —

Table 6 introduces a #Syntax for #PropositionalCalculus based on 2 families of #Multigrade #LogicalConnectives.

Table 6. #Syntax and #Semantics of a Calculus for #PropositionalLogic
https://inquiryintoinquiry.files.wordpress.com/2020/02/syntax-and-semantics-of-a-calculus-for-propositional-logic.png

Related Subjects —
#Peirce #Semiotics #BooleanFunctions
#Logic #LogicalGraphs #DifferentialLogic
#GraphTheory #CactusGraphs #CactusLanguage
#MinimalNegationOperators #PaintedAndRootedCacti