[CG] Re: Differential Propositional Calculus

Differential Propositional Calculus • 21 • https://inquiryintoinquiry.com/2024/12/18/differential-propositional-calcul… A One‑Dimensional Universe — There was never any more inception than there is now, Nor any more youth or age than there is now; And will never be any more perfection than there is now, Nor any more heaven or hell than there is now. — Walt Whitman • Leaves of Grass Let †X† = {A} be a logical basis containing one boolean variable or logical feature A. The basis element A may be regarded as a simple proposition or coordinate projection A : B→B. Corresponding to the basis †X† = {A} is the alphabet ‡X‡ = {“A”} which serves whenever we need to make explicit mention of the symbols used in our formulas and representations. The space X = ⟨A⟩ = {¬A, A} of points (cells, vectors, interpretations) has cardinality 2ⁿ = 2¹ = 2 and is isomorphic to B = {0, 1}. Moreover, X may be identified with the set of singular propositions {s : B→B}. The space of linear propositions X* = {ℓ : B→B} = {0, A} is algebraically dual to X and also has cardinality 2. Here, “0” is interpreted as denoting the constant function 0 : B→B, amounting to the linear proposition of rank 0, while A is the linear proposition of rank 1. Last but not least we have the positive propositions {p : B→B} = {A, 1} of rank 1 and 0, respectively, where “1” is understood as denoting the constant function 1 : B→B. All told there are 2^{2ⁿ} = 2^{2¹} = 4 propositions in the universe of discourse [†X†], collectively forming the set X↑ = {f : X→B} = {0, ¬A, A, 1}. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • One-Dimensional Universe • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/laY4w3 cc: https://www.researchgate.net/post/Differential_Propositional_Calculus