Admissibility in Positive Logics

Abstract

The paper studies admissibility of multiple-conclusion rules in positive logics. Using modification of a method employed by M. Wajsberg in the proof of the separation theorem, it is shown that the problem of admissibility of multiple-conclusion rules in the positive logics is equivalent to the problem of admissibility in intermediate logics defined by positive additional axioms. Moreover, a multiple-conclusion rule $$\mathsf {r}$$ follows from a set of multiple-conclusion rules $$\mathsf {R}$$ over a positive logic $$\mathsf {P}$$ if and only if $$\mathsf {r}$$ follows from $$\mathsf {R}$$ over $$\mathbf {Int}+ \mathsf {P}$$ .