[CG] Re: Differential Propositional Calculus

Differential Propositional Calculus • 18 • https://inquiryintoinquiry.com/2024/12/16/differential-propositional-calcul… The Extended Universe of Discourse — The “extended basis” E†A† of a universe of discourse [†A†] is formed by taking the initial basis †A† together with the differential basis d†A†. Thus we have the following formula. • E†A† = †A† ∪ d†A† = {a₁, …, aₙ, da₁, …, daₙ} This supplies enough material to construct the “differential extension” EA of the space A, also called the “tangent bundle” of A, in the following fashion. • EA = ⟨E†A†⟩ = ⟨†A† ∪ d†A†⟩ = ⟨a₁, …, aₙ, da₁, …, daₙ⟩ and also • EA = A × dA = A₁ × … × Aₙ × dA₁ × … × dAₙ That gives EA the type Bⁿ × Dⁿ. Finally, the “extended universe” EA• = [E†A†] is the full collection of points and functions, or interpretations and propositions, based on the extended set of features E†A†, a fact summed up in the following notation. • EA• = [E†A†] = [a₁, …, aₙ, da₁, …, daₙ] That gives the extended universe EA• the following type. • (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, (Bⁿ × Dⁿ → B)) A proposition in the extended universe [E†A†] is called a “differential proposition” and forms the logical analogue of a system of differential equations, constraints, or relations in ordinary calculus. With these constructions, the differential extension EA and the space of differential propositions (EA → B), we arrive at the launchpad of our space explorations. Table 11 summarizes the notations needed to describe the first order differential extensions of propositional calculi in a systematic manner. Table 11. Differential Extension • Basic Notation • https://inquiryintoinquiry.files.wordpress.com/2020/03/differential-extensi… The adjective “differential” or “tangent” is systematically attached to every construct based on the differential alphabet d‡A‡, taken by itself. In like fashion, the adjective “extended” or the substantive “bundle” is systematically attached to any construct associated with the full complement of 2n features. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… The Extended Universe of Discourse • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/l76pGk cc: https://www.researchgate.net/post/Differential_Propositional_Calculus