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Formalized Deduction of Quantifier-Expanded Syllogisms
Yinsheng Zhang
The paper solves the problem how to reform Aristotelian syllogisms (ASs), making them compatible with classic logic, and further deducts the reformed syllogisms in terms of formal languages. It asserts that there exist two challenging defaults in Aristotelian categorical propositions (ACPs) making up ASs. One is inconsistently to regard the particular quantifier as the existential and as the partial simultaneously. Another one is lacking a quantifier binding the second term. To overcome the two defaults, new forms of categorical propositions (called expanded categorical propositions, ECPs, term-bound by dyadic fuzzy quantifiers with membership invariants) are introduced. Spontaneously, the quantifier-expanded syllogisms (QESs), made up of ECPs, are constructed, To deduct QESs, a formal system, also a Turing machine, is designed by deciding and symbolically generating valid conclusions drawn from a massive solution space. The system has removed the bane of the inconsistency of the particular quantifier, and remedied the imperfectness in quantifying the second term, realizing decisions and symbolical generations of QESs, complying with first-order based logic that must be semantics-consistent and entirely binding the terms of propositions for representing knowledge. Especially, the reform of ASs and creation of QES show a way to overcome the logic defaults by increasing the symmetry of semantics and of formal structures of logic.
Keywords: Quantifier-expanded syllogism, categorical proposition, particular quantifier, partial quantifier, formal system, automated reasoning, Turing machine, 0-type grammar, inconsistency