Reflective Interpretive Frameworks • Incident 1
• https://inquiryintoinquiry.com/2026/03/26/reflective-interpretive-frameworks-incident-1/

Re: William Waites • The Agent That Doesn't Know Itself
• https://johncarlosbaez.wordpress.com/2026/03/20/the-agent-that-doesnt-know-itself/

WW: ❝Why Has Nobody Done This?❞

People who study C.S. Peirce would say reflective reasoning requires triadic relations at core and there is work being done on that. One of the challenges is clarifying the role of triadic relations in category theory and raising them into higher relief as fundamental operations.

Note. I was looking for a word to describe a random encounter with something that jogs one's memory of a recurring theme — “incident” plays into the “reflection” theme and looked worth trying for now.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_Frameworks

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/
• https://inquiryintoinquiry.com/2021/07/31/notes-on-categories-2/

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/

“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 — 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.

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

See —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview

Differential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview

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

cc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2

#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences

Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/

Cactus Language for Propositional Logic (cont.)

The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in short, their “logical conjunction” is true. An expression of that form is associated with a cactus structure called a “node” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.

Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpg

All other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (…) may be used for the logical operators.

Resources —

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

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences

Differential Logic • 2.1
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/

Cactus Language for Propositional Logic —

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions. One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable k-ary scope. The syntactic formulas of that calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective is a parenthesized sequence of propositional expressions, written (e₁, e₂, …, eₖ₋₁, eₖ) to mean exactly one of the propositions e₁, e₂, …, eₖ₋₁, eₖ is false, in short, their “minimal negation” is true. An expression of that form is associated with a cactus structure called a “lobe” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.

Lobe Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-connective.jpg

Resources —

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

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences

Differential Logic • 1
• https://inquiryintoinquiry.com/2026/02/05/differential-logic-1-b/

Introduction —

Differential logic is the component of logic whose object is the description of variation — focusing on the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition that broad 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 governs the use of a “differential logical calculus”, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by “differential propositional calculi”. A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe. Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences

Differential Logic • Overview
• https://inquiryintoinquiry.com/2026/02/03/differential-logic-overview-b/

A reader once told me “venn diagrams are obsolete” and of course we all know how unwieldy they become as our universes of discourse expand beyond four or five dimensions. Indeed, one of the first lessons I learned when I set about implementing Peirce’s graphs and Spencer Brown’s forms on the computer is that 2‑dimensional representations of logic quickly become death traps on numerous conceptual and computational counts.

Still, venn diagrams do us good service at the outset in visualizing the relationships among extensional, functional, and intensional aspects of logic. A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice.

All things considered, then, it is useful to make the links between various styles of imagery in logical representation as visible as possible. The first few steps in that direction are set out in the sketch of Differential Logic to follow.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Sign Relations • Semiotic Equivalence Relations 2.3
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

The semiotic equivalence relation for interpreter A yields the following semiotic equations.

• [“A”]_A = [“i”]_A

• [“B”]_A = [“u”]_A

Display 4
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-4.png

or

• “A” =_A “i”

• “B” =_A “u”

Display 5
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-5.png

In this way the SER for A induces the following semiotic partition.

• {{“A”, “i”}, {“B”, “u”}}.

Display 6
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-6.png

The semiotic equivalence relation for interpreter B yields the following semiotic equations.

• [“A”]_B = [“u”]_B

• [“B”]_B = [“i”]_B

Display 7
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-7.png

or

• “A” =_B “u”

• “B” =_B “i”

Display 8
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-8.png

In this way the SER for B induces the following semiotic partition.

• {{“A”, “u”}, {“B”, “i”}}.

Display 9
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-9.png

Taken all together we have the following picture.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/semiotic-partitions-for-interpreters-a-b-2.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

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

Sign Relations • Semiotic Equivalence Relations 2.2
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If L is a sign relation whose connotative component L_SI is an equivalence relation on S = I, let [s]_L be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}.

A statement that the signs x and y belong to the same equivalence class under a semiotic equivalence relation L_SI is called a “semiotic equation” (SEQ) and may be written in either of the following forms.

• [x]_L = [y]_L

• x =_L y

Display 3
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-3.png

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple (o, s, i) in a sign relation L, it is permissible to let [o]_L be defined as [s]_L. This modifications is designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B will serve to illustrate their use and utility.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/connotative-components-con-la-con-lb-3.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

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