[CG] Re: Differential Propositional Calculus

Differential Propositional Calculus • 9 • https://inquiryintoinquiry.com/2023/11/24/differential-propositional-calcul… All. Note. As always, please see the blog post linked above for the proper mathematical formatting. 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 [a_1, ..., a_n] is one of the propositions in the set {a_1, ..., a_n}. There are of course exactly n of these. Depending on the context, basic propositions may also be called “coordinate propositions” or “simple propositions”. Among the 2^(2ⁿ) propositions in [a_1, ..., a_n] are several families numbering 2ⁿ propositions each which take on special forms with respect to the basis {a_1, ..., a_n}. Three of these 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 a_1, ..., a_n 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 (a_1)(a_2)(a_3), that is, ¬a_1 ∧ ¬a_2 ∧ ¬a_3. The basic propositions a_i : Bⁿ → B are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions. Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis †A† = {a_1, ..., a_n}. 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 this whole determination is tantamount to the purely conventional choice of a cell as origin. Resources — Logic Syllabus • https://inquiryintoinquiry.com/logic-syllabus/ Differential Logic and Dynamic Systems • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Special Classes of Propositions • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Basis Relativity and Type Ambiguity • https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part… Regards, Jon cc: https://www.academia.edu/community/V1nYdL