Deductive Logics and Their Relation to Intuitionistic Logic

Abstract

R. Wojcicki introduced the notion of well-defined logic [5]. A propositional logic is called well-determined if it satisfies conjunction property and weak deduction theorem. The weak deduction theorem has the following form:$\alpha\vdash\beta\Leftrightarrow~\vdash\alpha\to\beta$ . Well-determined logics are interested because their logical consequence may be certainly represented by means of the logic.
We consider well-determined logics for which the following deductive theorem holds: for any set of formulas$ X$ and any formulas and it is true that $X,\alpha\vdash\beta\Leftrightarrow~X\vdash\alpha\to\beta$ . Logics with this property we call deductive. We call a set of formulas Lstrongly deductive if there exists a deductive logic $C$ such that $C(\varnothing)=L$.
DOI: 10.21146/2074-1472-2017-23-2-9-24
In this paper we introduce an operation of adding of consequences under a theory and study some its properties. We prove that any theory under a deductive logic is closed under modus ponens. The notion of minimal deductive logic is introduced. The main results are a criterion of strong deductivity for a set of formulas and the proof that the set of tautologies of minimal deductive logic coincides with the conjunctive and implicative fragment of intuitionistic logic.