[CG] Re: Differential Propositional Calculus

Differential Propositional Calculus • 29 • https://inquiryintoinquiry.com/2024/12/25/differential-propositional-calcul… Back to the Feature — ❝I guess it must be the flag of my disposition, out of hopeful green stuff woven.❞ — Walt Whitman • Leaves of Grass Let's assume the sense intended for differential features is well enough established in the intuition for now to continue outlining the structure of the differential extension [E†X†] = [A, dA]. Over the extended alphabet E†X† = {x₁, dx₁} = {A, dA} of cardinality 2ⁿ = 2 we generate the set of points EX of cardinality 2²ⁿ = 4 which bears the following chain of equivalent descriptions. • EX = ⟨A, dA⟩ = {(A), A} × {(dA), dA} = {(A)(dA), (A)dA, A(dA), A∙dA}. The space EX may be given the nominal type B×D, at root isomorphic to B×B = B². An element of EX may be regarded as a “disposition at a point” or a “situated direction”, in effect, a singular mode of change occurring at a single point in the universe of discourse. In practice the modality of those changes may be interpreted in various ways, for example, as expectations, intentions, or observations with respect to the behavior of a system. To construct the extended universe of discourse EX° = [A, dA] the basic dispositions in EX need to be extended to the full set of differential propositions EX↑ = {g : EX→B}, each of type B×D → B. There are 2^{2²ⁿ} = 16 propositions in EX↑, as detailed in the following Table. Differential Propositions • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… Aside from changing the names of variables and shuffling the order of rows, the Table follows the format previously used for boolean functions of two variables. The rows are grouped to reflect natural similarity classes holding among the propositions. In a future discussion the classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for X↑. Thus the first set of propositions {fₖ} is automatically embedded in the present set {gₘ} and the corresponding inclusions are indicated at the far left margin of the Table. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Back to the Feature • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/VBqD3R cc: https://www.researchgate.net/post/Differential_Propositional_Calculus