#booleanfunctions — Public Fediverse posts

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 • 18

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.

The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe   What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.

Reading the arrows off the map produces the following data.

In short, is a constant field, having the value at each cell.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 17

Enlargement and Difference Maps

Continuing with the example the following venn diagram shows the enlargement or shift map in the same style of field picture we drew for the tacit extension

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension and the enlargement are in a sense dual to each other.  The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true.  The only arc they have in common is the no‑change loop at   If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of shown in the following venn diagram.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 15

Differential Fields

The structure of a differential field may be described as follows.  With each point of there is associated an object of the following type:  a proposition about changes in that is, a proposition   In that frame of reference, if is the universe generated by the set of coordinate propositions then is the differential universe generated by the set of differential propositions   The differential propositions and may thus be interpreted as indicating and respectively.

A differential operator of the first order type we are currently considering, takes a proposition and gives back a differential proposition   In the field view of the scene, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to feel a change in the felt value of the field

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 14

Field Picture

Let us summarize the outlook on differential logic we’ve reached so far.  We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse to considering a larger universe of discourse   An operator of that general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe

The operators we’ve examined so far are the enlargement or shift operator and the difference operator   The operators and act on propositions in that is, propositions of the form which amount to propositions about the subject matter of and they produce propositions of the form which amount to propositions about specified collections of changes conceivably occurring in

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.

One good picture comes to us by way of the field concept.  Given a space a field of a specified type over is formed by associating with each point of an object of type   If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.

In the field picture a proposition becomes a scalar field, that is, a field of values in

For example, consider the logical conjunction shown in the following venn diagram.

Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what in real analysis are usually called vector fields over

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 13

Transforms Expanded over Ordinary and Differential Variables

Two views of how the difference operator acts on the set of sixteen functions are shown below.  Table A5 shows the expansion of over the set of ordinary variables and Table A6 shows the expansion of over the set of differential variables.

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

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

Logical Graphs • Formal Development 1
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-1-a/

Recap —

A first approach to logical graphs was outlined in the article linked below.

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/

That introduced the initial elements of logical graphs and hopefully supplied the reader with an intuitive sense of their motivation and rationale.

Formal Development —

Logical graphs are next presented as a formal system by going back to the initial elements and developing their consequences in a systematic manner.

The next order of business is to give the precise axioms used to develop the formal system of logical graphs. The axioms derive from C.S. Peirce's various systems of graphical syntax via the “calculus of indications” described in Spencer Brown's “Laws of Form”. The formal proofs to follow will use a variation of Spencer Brown's annotation scheme to mark each step of the proof according to which axiom is called to license the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

Resources —

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

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Logical Graphs • First Impressions 1
• https://inquiryintoinquiry.com/2024/08/30/logical-graphs-first-impressions-1/

Moving Pictures of Thought —

A logical graph is a graph‑theoretic structure in one of the systems of graphical syntax Charles S. Peirce developed for logic.

Introduction —

In numerous papers on qualitative logic, entitative graphs, and existential graphs, C.S. 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 their line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph‑theoretic structures. The posts to follow explore the common basis of those 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.

Resources —

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

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

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

Charles Sanders Peirce • Bibliography
• https://mywikibiz.com/Charles_Sanders_Peirce
• https://mywikibiz.com/Charles_Sanders_Peirce_%28Bibliography%29

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Transformations of Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/22/transformations-of-logical-graphs-discussion-1/

Re: Laws of Form
• https://groups.io/g/lawsofform/topic/transformations_of_logical/105927945

Mauro Bertani
• https://groups.io/g/lawsofform/message/3204

Dear Mauro,

The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

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/

Additional Resources —

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

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

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 2.2
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that's aways down the road from here.

Resources —

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

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/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-1/

Re: Mathematical Duality in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

Re: Laws of Form • Lyle Anderson
• https://groups.io/g/lawsofform/message/109

Re: Brading, K., Castellani, E., and Teh, N., (2017), “Symmetry and Symmetry Breaking”, The Stanford Encyclopedia of Philosophy (Winter 2017), Edward N. Zalta (ed.).
• https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/

Dear Lyle,

Thanks for the link to the article on symmetry and symmetry breaking. I did once take a Master's in Mathematics, specializing in combinatorics, graph theory, and group theory. When it comes to the bearing of symmetry groups on logical graphs and the calculus of indications, it will take careful attention to the details of the relationship between the two interpretations singled out by Peirce and Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality, if they differed in what they found most convenient to use in their development and exposition, and most of us will emphasize one interpretation or the other as a matter of facility or taste in a chosen application, so it requires a bit of effort to keep the underlying unity in focus. I recently made another try at taking a more balanced view, drawing up a series of tables in parallel columns the way one commonly does with dual theorems in projective geometry, so I will shortly share more of that work.

Resources —

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

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/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Mathematical Duality in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

It was in this context that Peirce's systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as “entitative graphs” and “existential graphs”, respectively. He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared more natural in that case, but whether there is any logical or mathematical reason for the symmetry to break at that point is a good question for further research.

Resources —

Duality Indicating Unity
• https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/

C.S. Peirce • Logic of Number
• https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/

C.S. Peirce • Syllabus • Selection 1
• https://inquiryintoinquiry.com/2014/08/24/c-s-peirce-syllabus-selection-1/

References —

• Peirce, C.S., [Logic of Number — Le Fevre] (MS 229), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 2, 592–595.

• Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.1
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

“All other sciences without exception depend upon the principles of mathematics; and mathematics borrows nothing from them but hints.”

— C.S. Peirce • “Logic of Number”

“A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re‑align them with mathematics.”

— G. Spencer Brown • “Laws of Form”

The duality between entitative and existential interpretations of logical graphs tells us something important about the relation between logic and mathematics. It tells us the mathematical forms giving structure to reasoning are deeper and more abstract at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding a calculus of forms whose expressions can be read in alternate ways by switching the meanings assigned to a pair of primitive terms. Spencer Brown's mathematical approach to “Laws of Form” and the whole of Peirce's work on the mathematics of logic shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on analogy with the dualities in projective geometry. This gave rise to formal systems where the initial constants, and thus their geometric and graph‑theoretic representations, had no uniquely fixed meanings but could be given dual interpretations in logic.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Operator Variables in Logical Graphs • Discussion 2
• https://inquiryintoinquiry.com/2024/04/09/operator-variables-in-logical-graphs-discussion-2/

Re: Operator Variables in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

Re: Cybernetics List • Lou Kauffman
• https://groups.google.com/g/cybcom/c/XKT76QI_OnI/m/3u9P2Ir5AgAJ

LK:
❝I am writing to comment that there are some quite interesting situations that generalize the DeMorgan Duality.

❝One well-known one is this. Let R* denote the real numbers with a formal symbol @, denoting infinity, adjoined so that:

• @ + @ = @
• @ + 0 = @
• @ + x = @ when x is an ordinary real number
• 1 ÷ @ = 0

❝(Of course you cannot do anything with @ or the system collapses. One can easily give the constraints.)

❝Define ¬x = 1/x.

• x + y = usual sum otherwise.

❝Define x ∗ y = xy/(x + y) = 1/((1/x) + (1/y)).

❝Then we have x ∗ y = ¬(¬x + ¬y), so that the system (R*, ¬, +, ∗) satisfies DeMorgan duality and it is a Boolean algebra when restricted to {0, @}.

❝Note also that ¬ fixes 1 and -1. This algebraic system occurs of course in electrical calculations and also in the properties of tangles in knot theory, as you can read in the last part of my included paper “Knot Logic”. I expect there is quite a bit more about this kind of duality in various (categorical) places.❞

Thanks, Lou, there's a lot to think about here, so I'll need to study it a while. Just off hand, the embedding into reals brings up a vague memory of the very curious way Peirce defines negation in his 1870 “Logic of Relatives”. I seem to recall it involving a power series, but it's been a while so I'll have to look it up again.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Operator Variables in Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/04/08/operator-variables-in-logical-graphs-discussion-1/

Re: Operator Variables in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

Re: Academia.edu • Stephen Duplantier
• https://www.academia.edu/community/Lxn1Ww?c=yq1Rxy

SD:
❝The best way for me to read Peirce is as if he was writing poetry. So if his algebra is poetry — I imagine him approving of the approach since he taught me abduction in the first place — there is room to wander. With this, I venture the idea that his “wide field” is a local algebraic geography far from the tended garden. There, where weeds and wild things grow and hybridize are the non‑dichotomic mathematics.❞

Stephen,

“Abdeuces Are Wild”, as they say, maybe not today, maybe not tomorrow, but soon …

As far as my own guess, and a lot of my wandering in pursuit of it goes, I'd venture Peirce's field of vision opens up not so much from dichotomic to trichotomic domains of value as from dyadic to triadic relations, and all that with particular significance into the medium of reflection afforded by triadic sign relations.

Resources —

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

Semeiotic
• https://oeis.org/wiki/Semeiotic

Sign Relations
• https://oeis.org/wiki/Sign_relation

Triadic Relations
• https://oeis.org/wiki/Triadic_relation

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

Operator Variables in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

Consider De Morgan's rules:

• ¬(A ∧ B) = ¬A ∨ ¬B

• ¬(A ∨ B) = ¬A ∧ ¬B

The common form exhibited by the two rules could be captured in a single formula by taking “o₁” and “o₂” as variable names ranging over a family of logical operators, then asking what substitutions for o₁ and o₂ would satisfy the following equation.

• ¬(A o₁ B) = ¬A o₂ ¬B

We already know two solutions to this “operator equation”, namely, (o₁, o₂) = (∧, ∨) and (o₁, o₂) = (∨, ∧). Wouldn't it be just like Peirce to ask if there are others?

Having broached the subject of “logical operator variables”, I will leave it for now in the same way Peirce himself did:

❝I shall not further enlarge upon this matter at this point, although the conception mentioned opens a wide field; because it cannot be set in its proper light without overstepping the limits of dichotomic mathematics.❞ (Peirce, CP 4.306).

Further exploration of operator variables and operator invariants treads on grounds traditionally known as second intentional logic and “opens a wide field”, as Peirce says. For now, however, I will tend to that corner of the field where our garden variety logical graphs grow, observing the ways in which operative variations and operative themes naturally develop on those grounds.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

Operator Variables in Logical Graphs • 1.1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

In lieu of a field study requirement for my bachelor's degree I spent two years in various state and university libraries reading everything I could find by and about Peirce, poring most memorably through reels of microfilmed Peirce manuscripts Michigan State had at the time, all in trying to track down some hint of a clue to a puzzling passage in Peirce's “Simplest Mathematics”, most acutely coming to a head with that bizarre line of type at CP 4.306, which the editors of Peirce's “Collected Papers”, no doubt compromised by the typographer's reluctance to cut new symbols, transmogrified into a script more cryptic than even the manuscript's original hieroglyphic.

I found one key to the mystery in Peirce's use of “operator variables”, which he and his students Christine Ladd‑Franklin and O.H. Mitchell explored in depth. I will shortly discuss that theme as it affects logical graphs but it may be useful to give a shorter and sweeter explanation of how the basic idea typically arises in common logical practice.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

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

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 • Overview
• https://inquiryintoinquiry.com/2023/11/12/differential-propositional-calculus-overview-a/

❝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

Here's the outline of a sketch I wrote on “differential propositional calculi”, which extend propositional calculi by adding 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. I wrote this as an intuitive introduction to differential logic, which is my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms. I'll be looking at ways to improve this draft as I serialize it to my blog.

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

Survey of Differential Logic • 6
• https://inquiryintoinquiry.com/2023/11/10/survey-of-differential-logic-6/

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/

Differential Logic
• https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

Differential Logic and Dynamic Systems
• https://inquiryintoinquiry.com/2023/03/04/differential-logic-and-dynamic-systems-overview-a/

cc: https://www.academia.edu/community/lQX66L

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

Logical Graphs • Discussion 9
• http://inquiryintoinquiry.com/2023/10/12/logical-graphs-discussion-9/

Re: Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/
Re: Laws of Form • Lyle Anderson
• https://groups.io/g/lawsofform/message/2511

LA:
❝The Gestalt Switch from parenthesis to graphs is stimulating. There are probably things in Laws of Form that we didn't see because we were blinded by the crosses.❞

That has been my experience. Viewing a space of mathematical objects from a new angle and changing the basis of representation can bring out new and surprising aspects of their form and even expand the field of view to novel directions of generalization.

One of the first things I learned in the early years of computing with logical graphs is how essential it is to “slip the surly bonds” of the planar embedding and work with free trees in a space of their own.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Logical Graphs • Discussion 7
• https://inquiryintoinquiry.com/2023/10/01/logical-graphs-discussion-7/

Re: Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-2/
Re: Laws of Form • Alex Shkotin
• https://groups.io/g/lawsofform/message/2461

AS:
❝When we look at undirected graph it is usual, before describing a rules of graph transformation, to describe exactly what kind of graphs we are working with ...❞

Hi Alex,

I am traveling this week, with limited internet. There's a quickie glossary under the heading “Painted And Rooted Cacti” on the following blog page.

Theme One Program • Exposition 2
• https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/

Regards,
Jon

P.S. Back home now ... with access to books ... will attempt to fill in some of the blanks in last week's sketchy vacation messages. —JA

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Logical Graphs • Formal Development 1
• https://inquiryintoinquiry.com/2023/09/15/logical-graphs-formal-development-1/

Recap —

A first approach to logical graphs can be found in the article linked below.

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

That introduces the initial elements of logical graphs and hopefully supplies the reader with an intuitive sense of their motivation and rationale.

Formal Development —

Logical graphs are next presented as a formal system by going back to the initial elements and developing their consequences in a systematic manner.

The next order of business is to give the precise axioms used to develop the formal system of logical graphs. The axioms derive from C.S. Peirce's various systems of graphical syntax via the “calculus of indications” described in Spencer Brown's “Laws of Form”. The formal proofs to follow will use a variation of Spencer Brown's annotation scheme to mark each step of the proof according to which axiom is called to license the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Logical Graphs • Discussion 6
• https://inquiryintoinquiry.com/2023/08/29/logical-graphs-discussion-6/

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

Logical Graphs • Figures 1 and 2
• https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-figures-1-2-framed.png

Re: Academia.edu • Robert Appleton
• https://www.academia.edu/community/lavbw5?c=Q4jlVy

RA:
❝As a professional graphic designer and non-mathematician reading your two diagrams, I need to ask for a simpler statement of their purpose. What do Fig 1 and Fig 2 represent to you? And what insight do they provide us?❞

My Comment —

Figures 1 and 2 are really just a couple of “in medias res” pump‑primers or ice‑breakers. This will all be explained in the above linked blog post, where I'm revising the text and upgrading the graphics of some work I first blogged in 2008 based on work I did even further back. I'll be taking a fresh look at that as I serialize it here.

Those two Figures come from George Spencer Brown's 1969 book Laws of Form, where he called them the Law of Calling and the Law of Crossing. GSB revived and clarified central aspects of Peirce's systems of logical graphs and I find it helpful to integrate his work into my exposition of Peirce. For now you can think of those as exemplifying two core formal principles which go to the root of the mathematical forms underlying logical reasoning.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

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 #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus

Survey of Differential Logic • 5
• https://inquiryintoinquiry.com/2023/04/25/survey-of-differential-logic-5/

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 Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2
• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2
3 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3
• https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2
3 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3
4 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4
5 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5
• https://inquiryintoinquiry.com/2023/03/04/differential-logic-and-dynamic-systems-overview-2/

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

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2023/03/28/survey-of-animated-logical-graphs-5/

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#Boole #BooleanAlgebra #BooleanFunctions #ModelTheory #ProofTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #LogicAsSemiotics

Differential Logic and Dynamic Systems • Review and Transition 1
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Review_and_Transition

This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.

Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable \(k\)-ary scope.

• A bracketed list of propositional expressions in the form \(\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\) indicates that exactly one of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) is false.

• A concatenation of propositional expressions in the form \(e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k\) indicates that all of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) are true, in other words, that their logical conjunction is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

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

Differential Logic and Dynamic Systems • Overview
• https://inquiryintoinquiry.com/2023/03/04/differential-logic-and-dynamic-systems-overview-2/
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

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

In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade-off between dynamic paradigms and symbolic paradigms. Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time. Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information. Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus. The work laid out in this report is intended to address that lack.

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

@bblfish @josd @semwebpro @hochstenbach

One thing I found out early on is how critical it is to get #AlphaGraphs (#BooleanFunctions, #PropositionalCalculus, #ZerothOrderLogic) down tight. If you do that it changes how you view #FOL (#PredicateCalculus, #QuantificationalLogic). That tends to rub people who view FOL as #GOL (#GodsOwnLogic) the wrong way so you have watch out for that if you go down this road.

Here's a primer on \(\alpha\) #LogicalGraphs as I see them —
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

#LogicalGraphs • 14
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

The procedure just described is called “traversing” the tree and the string read off is called the “#TraversalString” of the tree. The reverse operation of going from the string to the tree is called “parsing” the string and the tree constructed is called the “#ParseGraph” of the string.

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

#DifferentialPropositionalCalculus • 4.5
• https://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-4/

Each of the families — #LinearPropositions, #PositivePropositions, #SingularPrpositions — is naturally parameterized by the coordinate \(n\)-tuples in \(\mathbb{B}^n\) and falls into \(n+1\) ranks, with a #BinomialCoefficient \(\tbinom{n}{k}\) giving the number of propositions having rank or weight \(k\) in their class.

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions