Differential Propositional Calculus • 3 • https://inquiryintoinquiry.com/2023/11/17/differential-propositional-calcul… All, 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-proposi… This new quality, dq, is an example of a “differential quality”, since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other 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. 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. This 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”}. Regards, Jon cc: https://www.academia.edu/community/LZb2ZL

Differential Propositional Calculus • 5 • https://inquiryintoinquiry.com/2023/11/19/differential-propositional-calcul… All, Casual Introduction (concl.) Table 5 shows the rules of inference responsible for giving the differential quality dq its meaning in practice. Table 5. Differential Inference Rules • https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-proposi… Regards, Jon cc: https://www.academia.edu/community/5ANwNV

Differential Propositional Calculus • 7 • https://inquiryintoinquiry.com/2023/11/21/differential-propositional-calcul… All, Note. Please see the blog post linked above for the proper formats of the notations used below, as they depend on numerous typographical distinctions lost in the following transcript. Formal Development — The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology needed to describe various orders of differential propositional calculi. Elementary Notions — Logical description of a universe of discourse begins with a collection of logical signs. For simplicity in a first approach we assume the signs are collected in the form of a finite alphabet, ‡A‡ = {“a_1”, ..., “a_n”}. The signs are interpreted as denoting logical features, for example, properties of objects in the universe of discourse or simple propositions about those objects. Corresponding to the alphabet ‡A‡ there is then a set of logical features, †A† = {a_1, ..., a_n}. A set of logical features †A† = {a_1, ..., a_n} affords a basis for generating an n-dimensional universe of discourse, written A• = [†A†] = [a_1, ..., a_n]. It is useful to consider a universe of discourse as a categorical object incorporating both the set of points A = <a_1, ..., a_n> and the set of propositions A↑ = {f : A→B} implicit with the ordinary picture of a venn diagram on n features. Accordingly, the universe of discourse A• may be regarded as an ordered pair (A, A↑) having the type (Bⁿ, (Bⁿ → B)) and this last type designation may be abbreviated as Bⁿ +→ B, or even more succinctly as [Bⁿ]. For convenience, the data type of a finite set on n elements may be indicated by either one of the equivalent notations, [n] or *n*. Table 7 summarizes the 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-calcul… Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Functional Conception of Propositional Calculus • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/54Gadl

Differential Propositional Calculus • 9 • https://inquiryintoinquiry.com/2023/11/24/differential-propositional-calcul… All. Note. As always, please see the blog post linked above for the proper mathematical formatting. Special Classes of Propositions — The full set of propositions f : A → B contains a number of smaller classes deserving of special attention. A “basic proposition” in the universe of discourse [a_1, ..., a_n] is one of the propositions in the set {a_1, ..., a_n}. There are of course exactly n of these. Depending on the context, basic propositions may also be called “coordinate propositions” or “simple propositions”. Among the 2^(2ⁿ) propositions in [a_1, ..., a_n] are several families numbering 2ⁿ propositions each which take on special forms with respect to the basis {a_1, ..., a_n}. Three of these families are especially prominent in the present context, the “linear”, the “positive”, and the “singular” propositions. Each family is naturally parameterized by the coordinate n‑tuples in Bⁿ and falls into n + 1 ranks, with a binomial coefficient (n choose k) giving the number of propositions having rank or weight k in their class. Linear Propositions ℓ : Bⁿ → B may be written as sums. • https://inquiryintoinquiry.files.wordpress.com/2020/02/linear-propositions-… Positive Propositions p : Bⁿ → B may be written as products. • https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-proposition… Singular Propositions s : Bⁿ → B may be written as products. • https://inquiryintoinquiry.files.wordpress.com/2020/02/singular-proposition… In each case the rank k ranges from 0 to n and counts the number of positive appearances of the coordinate propositions a_1, ..., a_n in the resulting expression. For example, when n = 3 the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is (a_1)(a_2)(a_3), that is, ¬a_1 ∧ ¬a_2 ∧ ¬a_3. The basic propositions a_i : Bⁿ → B are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis †A† = {a_1, ..., a_n}. A singular proposition with respect to the basis †A† will not remain singular if †A† is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options {a_i}∪{¬a_i} to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Special Classes of Propositions • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Basis Relativity and Type Ambiguity • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/V1nYdL

Differential Propositional Calculus • 11 • https://inquiryintoinquiry.com/2023/11/26/differential-propositional-calcul… All, Special Classes of Propositions (cont.) Next we take up the family of positive propositions and follow the same plan as before, tracing the rule of their formation in the case of a 3‑dimensional universe of discourse. Positive Propositions — The positive propositions p : Bⁿ → B may be written as products: • https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-proposition… In a universe of discourse based on three boolean variables, p, q, r, there are 2³ = 8 positive propositions, taking the shapes shown in Figure 9. Figure 9. Positive Propositions on Three Variables • https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2… At the top is the venn diagram for the positive proposition of rank 3, corresponding to the boolean product or logical conjunction pqr. Next are the venn diagrams for the three positive propositions of rank 2, corresponding to the three boolean products, pr, qr, pq, respectively. Next are the three positive propositions of rank 1, which are none other than the three basic propositions, p, q, r. At the bottom is the positive proposition of rank 0, the everywhere true proposition or the constant 1 function, which may be expressed by the form (( )) or by a simple 1. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Differential Propositional Calculus • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1 • https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2 Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Special Classes of Propositions • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/ln1AjL

Differential Propositional Calculus • 13 • http://inquiryintoinquiry.com/2023/11/28/differential-propositional-calculu… All, Note. Please see the blog post linked above for the proper formats of the notations used below, as they depend on many typographical distinctions lost in the following transcript. Differential Extensions — An initial universe of discourse A• supplies the groundwork for any number of further extensions, beginning with the first order differential extension EA•. The construction of EA• can be described in the following stages. The initial alphabet ‡A‡ = {“a₁”, …, “aₙ”} is extended by a first order differential alphabet d‡A‡ = {“da₁”, …, “daₙ”} resulting in a first order extended alphabet E‡A‡ defined as follows. • E‡A‡ = ‡A‡ ∪ d‡A‡ = {“a₁”, …, “aₙ”, “da₁”, …, “daₙ”}. The initial basis †A† = {a₁, …, aₙ} is extended by a first order differential basis d†A† = {da₁, …, daₙ} resulting in a first order extended basis E†A† defined as follows. • E†A† = †A† ∪ d†A† = {a₁, …, aₙ, da₁, …, daₙ}. The initial space A = ⟨a₁, …, aₙ⟩ is extended by a first order differential space or tangent space dA = ⟨da₁, …, daₙ⟩ at each point of A, resulting in a first order extended space EA defined as follows. • EA = A × dA = ⟨E†A†⟩ = ⟨†A† ∪ d†A†⟩ = ⟨a₁, …, aₙ, da₁, …, daₙ⟩. Finally, the initial universe A• = [a₁, …, aₙ] is extended by a first order differential universe or tangent universe dA• = [da₁, …, daₙ] at each point of A•, resulting in a first order extended universe EA• defined as follows. • EA• = [E†A†] = [†A† ∪ d†A†] = [a₁, …, aₙ, da₁, …, daₙ]. This gives EA• a type defined as follows. • [Bⁿ × Dⁿ] = (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, Bⁿ × Dⁿ → B). A proposition in a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe EA• and the first order differential propositions f : EA → B, we arrive at the foothills of differential logic. Regards, Jon cc: https://www.academia.edu/community/lOvrKL