On interpretation of inconsistent theories (Q1123189)

The paraconsistent predicate calculus Cont, proposed by the author [Avtom. Telemekh. 1983, No.6, 113-124 (1983; Zbl 0532.03007); ibid. No.7, 908-914 (1983; Zbl 0532.03008)], is considered as a base for formal theories of first order. This calculus is as close as possible to the classical one, but the principle ``everything follows from an inconsistency'' is excluded. A three-valued semantics is proposed for theories based on Cont, and the basic metatheorems (of completeness, compactness, etc.) are proved. Relations of such theories to corresponding classical theories are established. In particular it is proved that a theory based on Cont is inconsistent iff a corresponding classical theory is inconsistent. The semantics of the equality in the frame of Cont is considered. It is proved that the pure predicate calculus Cont and Cont with equality are undecidable.