[CG] Re: Differential Propositional Calculus

Differential Propositional Calculus • 30 • https://inquiryintoinquiry.com/2023/12/22/differential-propositional-calcul… Tacit Extensions — ❝I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.❞ — Michel Foucault • The Discourse on Language Re: Table of Differential Propositions • https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e… In viewing the previous Table of Differential Propositions it is important to notice the subtle distinction in type between a function fₖ : X → B and its inclusion as a function gₘ : EX → B, even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions representing the same proposition while appreciating the full diversity of a proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time. Because this special circumstance points to a broader theme, it's a good idea to discuss it more generally. Whenever there arises a situation like that above, where one basis †X† is a subset of another basis †Y†, we say any proposition f : X → B has a “tacit extension” to a proposition εf : Y → B and we say the space (X → B) has an “automatic embedding” within the space (Y → B). The “tacit extension operator” ε is defined in such a way that εf puts the same constraint on the variables of †X† within †Y† as the proposition f initially put on †X†, while it puts no constraint on the variables of †Y† beyond †X†, in effect, conjoining the two constraints. Indexing the variables as †X† = {x₁, …, xₙ} and †Y† = {x₁, …, xₙ, …, xₙ₊ₖ} the tacit extension from †X† to †Y† may be expressed by the following equation. • εf(x₁, …, xₙ, …, xₙ₊ₖ) = f(x₁, …, xₙ). On formal occasions, such as the present context of definition, the tacit extension from †X† to †Y† is explicitly symbolized by the operator ε : (X→B)→(Y→B), where the bases †X† and †Y† are set in context, but it's normally understood the “ε” may be silent. Resources — Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Differential Logic • Tacit Extensions • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/5wQD11 cc: https://www.researchgate.net/post/Differential_Propositional_Calculus