Harmonization, Dualization and Globalization of Categorical Propositions
Yinsheng Zhang
The paper uncovers three so far unsolved problems with Aristotelian categorical propositions (ACPs, formatted as “𝑄𝑥 be/be not 𝑦”, where 𝑄 is a universal or particular quantifier), which puzzle modern logic to assimilate ACPs in representing and deducting knowledge: 1) the inconsistency of particular quantifier resulted from confusing universe-restrictive and -unrestrictive readings, 2) the lack of a manifested quantifier on y, and 3) localization, i.e., there has not been yet such a logic to generate a system of categorical propositions, which completely describe quantitative and qualitative relations complying with a real situation of the categories {x} and {y}, and ubiquitously being linked to various contexts and knowledge systems. Further, the paper has offered solutions to overcome the three problems by reforming the forms of ACPs to create the “fully expanded categorical propositions (FECPs)” with consistent and dyadic fuzzy quantifiers, which makes FECPs compatible to globally be infused in, and transformed into, various knowledge representations, chiefly of classical (first-order and high-order) logic, including set theory, type theory, model theory, and recursion theory (Turning machines), and of modern algebra, probability theory, fuzzy mathematics, informatics, formal languages (Chomsky 0-type grammar) and natural languages. These efforts show a turn of proposition logic and predicate logic in traditional logic into modern logic, expecting knowledge representation, acquisition, and computation of the reformed categorical propositions — FECPs in more exact and ubiquitous manners.
Keywords: Fuzzy quantifiers, generalized quantifier, partial quantifier, particular quantifier, categorical propositions, type theory