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 • 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

Differential Propositional Calculus • 17
• https://inquiryintoinquiry.com/2023/12/06/differential-propositional-calculus-17/

Differential Propositions • Tangent Spaces —

The “tangent space” to A at one of its points x, sometimes written Tₓ(A), takes the form dA = ⟨d†A†⟩ = ⟨da₁, …, daₙ⟩. Strictly speaking, the name “cotangent space” is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here.

Proceeding as we did with the base space A, the tangent space dA at a point of A may be analyzed as the following product of distinct and independent factors.

• dA = ∏ dAₖ = dA₁ × … × dAₙ.

Each factor dAₖ is a set consisting of two differential propositions, dAₖ = {(daₖ), daₖ}, where (daₖ) is a proposition with the logical value of ¬daₖ. Each component dAₖ has the type B, operating under the ordered correspondence {(daₖ), daₖ} ≅ {0, 1}. A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type D, whose sense may be indicated as follows.

• D = {(daₖ), daₖ} = {same, different} = {stay, change} = {stop, step}.

Viewed within a coordinate representation, spaces of type Bⁿ and Dⁿ may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Propositions • Tangent Spaces
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Tangent_Spaces

#Peirce #Logic #LogicalGraphs #DiscreteDynamicalSystems
#DifferentialLogic #DifferentialPropositionalCalculus

Differential Propositional Calculus • 16
• https://inquiryintoinquiry.com/2023/12/05/differential-propositional-calculus-16/

Differential Propositions • Qualitative Analogues of Differential Equations —

The differential extension of a universe of discourse $[\mathcal{A}]$ is constructed by extending its initial alphabet $\mathfrak{A}$ to include a set of symbols for “differential features”, or “basic changes” capable of occurring in $[\mathcal{A}].$ The added symbols are taken to denote primitive features of change, qualitative attributes of motion, or propositions about how items in the universe of discourse may change or move in relation to features noted in the original alphabet.

With that in mind we define the corresponding “differential alphabet” or “tangent alphabet” $\mathrm{d}\mathfrak{A}$ $=$ $\{‘‘\mathrm{d}{a}_1",\dots ,‘‘\mathrm{d}{a}_n"\},$ in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet $\mathfrak{A}$ $=$ $\{‘‘a_1",\dots ,‘‘a_n"\}$ and given the meanings just indicated.

In practice the precise interpretation of the symbols in $\mathrm{d}\mathfrak{A}$ is conceived to be changeable from point to point of the underlying space $A.$ Indeed, for all we know, the state space $A$ might well be the state space of a language interpreter, one concerned with the idiomatic meanings of the dialect generated by $\mathfrak{A}$ and $\mathrm{d}\mathfrak{A}.$

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Propositions • Qualitative Analogues of Differential Equations
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Differential_Propositions_:_Qualitative_Analogues_of_Differential_Equations

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics