[CG] Re: Differential Propositional Calculus

Differential Propositional Calculus • 9 • https://inquiryintoinquiry.com/2024/12/08/differential-propositional-calcul… 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 [a₁, …, aₙ] is one of the propositions in the set {a₁, …, aₙ}. There are of course exactly n of these. Depending on context, basic propositions may also be called “coordinate propositions” or “simple propositions”. Among the 2^(2ⁿ) propositions in [a₁, …, aₙ] are several families numbering 2ⁿ propositions each which take on special forms with respect to the basis {a₁, …, aₙ}. Three of those 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 a₁, …, aₙ 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 (a₁)(a₂)(a₃), that is, ¬a₁ ∧ ¬a₂ ∧ ¬a₃. The basic propositions a_i : Bⁿ → B are both linear and positive. So those two families of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. It is important to note that all the above distinctions are relative to the choice of a particular logical basis †A† = {a₁, …, aₙ}. 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 that entire determination is tantamount to the purely conventional choice of a cell as origin. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Survey of Differential Logic • https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7 Regards, Jon cc: https://www.academia.edu/community/VjKj3b cc: https://www.researchgate.net/post/Differential_Propositional_Calculus