Improvements of Categorical Propositions on Consistency and Computability
The author asserts that Aristotelian categorical propositions (ACPs), formalized as the structure “QX apply to/ not to Y” where Q is either universal or particular, and X, Y are terms, remain the two limitations: 1) the particular quantifier is ambiguous for its restricted reading by Euler as “non-empty but not universal”, and the unrestricted reading by Gergonne as “non-empty and possibly universal”; 2) Y formally lacks a quantifier to be modified, rendering vaguely, at least insufficiently expressing its quantity. So, expanded categorical propositions (ECPs) with dyadic (in two places in left-hand of both X and Y) and generalized quantifiers are proposed for overcoming the two limitations. ECPs are proved to be calculated logically and operated mathematically; moreover, these two kinds of computations are corresponding, i.e., ECPs and their mathematical models are in homomorphic algebraic structures, which enable ECPs’ logical calculi to be consistent and running in precisely mathematical significance.
Keywords: Categorical proposition, particular quantifier, partial quantifier, existential quantifier, syllogism