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

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

Before moving on, let's unpack some of the assumptions, conventions, and implications involved in the array of concepts and notations introduced above.

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to B, accordingly pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → B evaluates to 1 or 0 on each element of A. The analogy between logical propositions and boolean-valued functions is close enough to adopt the latter as models of the former and simply refer to the functions f : A → B as propositions about the elements of A.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 7.2
• https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calculus-7-b/

Formal Development (cont.)

Table 7 summarizes the basic notations needed to describe ordinary propositional calculi in a systematic fashion.

Table 7. Propositional Calculus • Basic Notation
• https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calculus-basic-notation.png

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 6.2
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/

Cactus Calculus (cont.)

The briefest expression for logical truth is the empty word, denoted ε or λ in formal languages, where it forms the identity element for concatenation. It may be given visible expression in textual settings by means of the logically equivalent form (()), or, especially if operating in an algebraic context, by a simple 1. Also when working in an algebraic mode, the plus sign “+” may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions.

• x + y = (x, y)

• x + y + z = ((x, y), z) = (x, (y, z))

It is important to note the last expressions are not equivalent to the triple bracket (x, y, z).

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 6.1
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/

Cactus Calculus —

Table 6 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable k‑ary scope.

• A bracketed sequence of propositional expressions (e₁, e₂, …, eₖ) is taken to mean exactly one of the propositions e₁, e₂, …, eₖ is false, in other words, their “minimal negation” is true.

• A concatenated sequence of propositional expressions e₁ e₂ … eₖ is taken to mean every one of the propositions e₁, e₂, …, eₖ is true, in other words, their “logical conjunction” is true.

Table 6. Syntax and Semantics of a Calculus for Propositional Logic
• https://inquiryintoinquiry.files.wordpress.com/2022/10/syntax-and-semantics-of-a-calculus-for-propositional-logic-4.0.png

All other propositional connectives may be obtained through combinations of the above two forms. As it happens, the concatenation form is dispensable in light of the bracket form but it is convenient to maintain it as an abbreviation for more complicated bracket expressions. While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for bracket forms. In contexts where parentheses are needed for other purposes “teletype” parentheses (…) or barred parentheses (|…|) may be used for logical operators.

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 5
• https://inquiryintoinquiry.com/2024/12/03/differential-propositional-calculus-5-b/

Casual Introduction (concl.)

Table 5 exhibits the rules of inference responsible for giving the differential proposition dq its meaning in practice.

Table 5. Differential Inference Rules
• https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-differential-inference-rules.png

If the feature q is interpreted as applying to an object in the universe of discourse X then the differential feature dq may be taken as an attribute of the same object which tells it is changing “significantly” with respect to the property q — as if the object bore an “escape velocity” with respect to the condition q.

For example, relative to a frame of observation to be made more explicit later on, if q and dq are true at a given moment, it would be reasonable to assume ¬q will be true in the next moment of observation. Taken all together we have the fourfold scheme of inference shown above.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 4
• https://inquiryintoinquiry.com/2024/12/02/differential-propositional-calculus-4-b/

Casual Introduction (cont.)

In Figure 3 we saw how the basis of description for the universe of discourse X could be extended to a set of two qualities {q, dq} while the corresponding terms of description could be extended to an alphabet of two symbols {“q”, “dq”}.

Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Salient among those propositions in the present setting are the four which single out the individual sample points at the initial moment of observation. Table 4 lists the initial state descriptions, using overlines to express logical negations.

Table 4. Initial State Descriptions
• https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-initial-state-descriptions.png

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 3.2
• https://inquiryintoinquiry.com/2024/12/01/differential-propositional-calculus-3-b/

Casual Introduction (cont.)

Figure 1 represents a universe of discourse X together with a basis of discussion {q} for expressing propositions about the contents of that universe. Once the quality q is given a name, say, the symbol “q”, we have the basis for a formal language specifically cut out for discussing X in terms of q. That language is more formally known as the “propositional calculus” with alphabet {“q”}.

In the context marked by X and {q} there are just four distinct pieces of information which can be expressed in the corresponding propositional calculus, namely, the constant proposition False, the negative proposition ¬q, the positive proposition q, and the constant proposition True.

For example, referring to the points in Figure 1, the constant proposition False holds of no points, the negative proposition ¬q holds of a and d, the positive proposition q holds of b and c, and the constant proposition True holds of all points in the sample.

Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, {q, dq}. In corresponding fashion, the initial propositional calculus is extended by means of the enlarged alphabet, {“q”, “dq”}.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 3.1
• https://inquiryintoinquiry.com/2024/12/01/differential-propositional-calculus-3-b/

Casual Introduction (cont.)

Figure 3 returns to the situation in Figure 1, but this time interpolates a new quality specifically tailored to account for the relation between Figure 1 and Figure 2.

Figure 3. Back, To The Future
• https://inquiryintoinquiry.files.wordpress.com/2023/11/differential-propositional-calculus-e280a2-figure-3.png

The new quality, dq, is marked as a “differential quality” on account of its absence or presence qualifying the absence or presence of change occurring in another quality. As with any quality, it is represented in the venn diagram by means of a “circle” distinguishing two halves of the universe of discourse, in this case, the portions of X outside and inside the region dQ.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 2
• https://inquiryintoinquiry.com/2024/11/30/differential-propositional-calculus-2-b/

Casual Introduction (cont.)

Now consider the situation represented by the venn diagram in Figure 2.

Figure 2. Same Names, Different Habitations
• https://inquiryintoinquiry.files.wordpress.com/2023/11/differential-propositional-calculus-e280a2-figure-2.png

Figure 2 differs from Figure 1 solely in the circumstance that the object c is outside the region Q while the object d is inside the region Q.

Nothing says our encountering the Figures in the above order is other than purely accidental but if we interpret the sequence of frames as a “moving picture” representation of their natural order in a temporal process then it would be natural to suppose a and b have remained as they were with regard to the quality q while c and d have changed their standings in that respect. In particular, c has moved from the region where q is true to the region where q is false while d has moved from the region where q is false to the region where q is true.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 1
• https://inquiryintoinquiry.com/2024/11/29/differential-propositional-calculus-1-b/

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.

Casual Introduction —

Consider the situation represented by the venn diagram in Figure 1.

Figure 1. Local Habitations, And Names
• https://inquiryintoinquiry.files.wordpress.com/2023/11/differential-propositional-calculus-e280a2-figure-1.png

The area of the rectangle represents the universe of discourse X. The universe under discussion may be a population of individuals having various additional properties or it may be a collection of locations occupied by various individuals. The area of the “circle” represents the individuals with the property q or the locations in the corresponding region Q. Four individuals, a, b, c, d, are singled out by name. As it happens, b and c currently reside in region Q while a and d do not.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • Overview 2
• https://inquiryintoinquiry.com/2024/11/27/differential-propositional-calculus-overview-b/

What follows is the outline of a sketch on differential propositional calculus intended as an intuitive introduction to the larger subject of differential logic, which amounts in turn to my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms.

Note. I'll give just the links to the main topic heads below. Please follow the link at the top of the page for the full outline.

Part 1 —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1

Casual Introduction
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction

Cactus Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Cactus_Calculus

Part 2 —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2

Formal_Development
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Formal_Development

Elementary Notions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Elementary_Notions

Special Classes of Propositions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Special_Classes_of_Propositions

Differential Extensions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions

Appendices —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices

References —
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • Overview 1
• https://inquiryintoinquiry.com/2024/11/27/differential-propositional-calculus-overview-b/

❝The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.❞

— W. Ross Ashby • An Introduction to Cybernetics

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. 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.

In accord with the strategy of approaching logical systems in stages, first gaining a foothold in propositional logic and advancing on those grounds, we may set our first stepping stones toward differential logic in “differential propositional calculi” — propositional calculi extended by sets 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.

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #Mathematics

Survey of Differential Logic • 7
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/

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 Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2023/08/22/differential-logic-the-logic-of-change-and-difference/

Differential Propositional Calculus
• https://inquiryintoinquiry.com/2023/11/12/differential-propositional-calculus-overview-a/
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview

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

Differential Logic and Dynamic Systems
• https://inquiryintoinquiry.com/2023/03/04/differential-logic-and-dynamic-systems-overview-a/
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

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

Differential Propositional Calculus • Discussion 9
• https://inquiryintoinquiry.com/2024/01/21/differential-propositional-calculus-discussion-9/

❝Consider what effects that might conceivably have
practical bearings you conceive the objects of your
conception to have. Then, your conception of those
effects is the whole of your conception of the object.❞

— C.S. Peirce • The Maxim of Pragmatism

Re: Facebook Discussion
• https://www.facebook.com/JonnyCache/posts/pfbid022WSZBCe85n6KZ35ZumgABL3SgDgotwgT9RYiLxiYwAizJjQNeLyAUWyCSMzKP12fl

Re: Tim Browning
• https://www.facebook.com/JonnyCache/posts/pfbid0shLb5x7y9j7CuPSorRMXVBjXiaqEKnR7bb37uN9xxFfVnsEVhcjXPcPtXi97hJNyl?comment_id=1321064655185834

Tim Browning wrote:
Makes me wonder if all that is the case, i.e. the universe, is the existence of objects (materialism) or information (idealism).

“Objects of your conception” seems to imply a transcendent perspective that can distinguish between concept and object. Am I overthinking this?
[end quote]

Hi Tim,

It helps to read “object” in a fuller sense than we often do in billiard‑ball philosophies, as a lot gets lost in the translation from the Greek “pragma” from which pragmatism naturally takes it cue. For a sample of that fuller sense see the following lexicon entry.

πρᾶγμα • Liddell, H.G., and Scott, R. (1925), A Greek-English Lexicon (1940 edition)
• https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dpra%3Dgma

Perseus Digital Library
• https://www.perseus.tufts.edu/hopper/

Resources —

Pragmatic Maxim
• https://inquiryintoinquiry.com/2008/08/07/pragmatic-maxim/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3

Differential Logic • Analytic Expansions
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Analytic_Expansions

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics
#Pragma #Pragmata #PragmaticMaxim #PracticalBearings #ConceptionOfEffects

Differential Propositional Calculus • 37
• https://inquiryintoinquiry.com/2024/01/01/differential-propositional-calculus-37/

Foreshadowing Transformations • Extensions and Projections of Discourse —

❝And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well‑conducted shadow should.❞

— Gaston Leroux • The Phantom of the Opera

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the type \([\mathcal{X}] \to [\mathcal{Y}]\) is implied any time we make use of one basis \(\mathcal{X}\) which happens to be included in another basis \(\mathcal{Y}.\) When discussing differential relations we usually have in mind the extended alphabet \(\mathfrak{Y}\) has a special construction or a specific lexical relation with respect to the initial alphabet \(\mathfrak{X},\) one which is marked by characteristic types of accents, indices, or inflected forms.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3

Differential Logic • Foreshadowing Transformations
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Foreshadowing_Transformations

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • Discussion 8
• https://inquiryintoinquiry.com/2023/12/30/differential-propositional-calculus-discussion-8/

Re: Differential Propositional Calculus • 33
• https://inquiryintoinquiry.com/2023/12/27/differential-propositional-calculus-33/

Re: Laws of Form • Lyle Anderson
• https://groups.io/g/lawsofform/message/2797

LA: ❝Some of your diagrams, specifically Figure 16. A Couple of Fourth Gear Orbits, are beginning to look like Heim's sketches for the structure of the photon. […] I can't quite see the connection, yet, but maybe you can.❞

Lyle,

There is a curious analogy between the primitive operations which lie at the basis of logical graphs and basic themes of quantum mechanics, for example, the evaluation of a minimal negation operator proceeds in a manner reminiscent of the way a wave function collapses. That's something I noticed early on in my work on logical graphs but I haven't got much further than the mere notice so far.

I confess I've never gotten around to tackling Heim's work — Peirce and Spencer Brown have loaded more than enough on my plate for any one lifetime — I do see lots of partial derivatives so maybe there's a connection there — if I had to guess I would imagine any structure generated by a differential law as simple as what we have here is bound to find itself inhabiting all sorts of mathematical niches.

Regards,

Jon

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Drives and Their Vicissitudes
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics