Well-determined logic

Abstract

It is known that the deductive property can be used to define logics. If the deduction theorem holds for a logic, then, under certain conditions, the relation of logical consequence is reduced to the set of tautologies of the logic. This allows us to use the set of tautologies of the logic to define the corresponding derivability relation. Logics defined by sets of tautologies were called by R.Wojcicki well-defined. He also noticed that, in some cases, in order for the logic to be well-defined, it is enough that the weak deduction theorem is true for it. (The weak deduction theorem concerns derivability over an empty set of hypotheses.)

The sets of formulas that allow us to define the relation of the standard logical consequence were called, by R.Wojcicki, implication systems or deductive sets. Note that R.Wojcicki studied the concepts of well-defined logic and deductive set for logics in languages containing conjunction and implication only.

In this paper, we call a set of formulas weak-deductive if it defines a logical consequence relation on a set of pairs of formulas. A criterion of weak-deductivity is found. A minimal weak-deductive logic is constructed. Also, in this paper, the concepts of well-defined logic and deductive set were extended to languages without conjunction. A criterion of deductivity of sets in such languages is found. In the language with implication only, a minimal well-defined logic is constructed. It is proved that the deduction theorem does not hold for this logic.