Kripke’s Fregean quantification logic FQ fails to formalize the usual first-order logic with identity due to the interpretation of the conditional operator. Motivated by Kripke’s syntax and semantics, the three-valued Fregean quantification logic FQ3 is proposed. This three valued logic differs from Kleene and Łukasiewicz’s three-valued logics. The logic FQ3 is decidable. A sound and complete Hilbert-style axiomatic system for the logic FQ3 is presented.

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional connectives of Łukasiewicz’s (...) logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete. (shrink)