An effective criterion of deductivity of a sets of logical formulas

R. Wojcicki introduced the notion of well-defined logic [1]. A propositional logic is called well-determined if it satisfies conjunction property and deductive theorem. Well-determined logics are interested because their logical consequence may be certainly represented by means of the logic. Note that R. Wojcicki proved a criterion of deductivity for a set of formulas in which he claimed the testing set to contain some certain infinite set of formulas. Therefore, the proposed criterion is not effective (not algorithmic). In this paper some effective criterions for deductivity of sets of formulas in languages with implication and conjunction are proved. Also it is proved that minimal deductive sets in languages of well-determined logics are finitely axiomatizable.