Multiple Conclusion Rules in Logics with the Disjunction Property

TL;DR

This paper demonstrates that in certain intermediate and modal logics with the disjunction property, bases of admissible rules and m-rules are interconvertible, simplifying the understanding of their rule structures.

Contribution

It establishes the equivalence between bases of admissible rules and m-rules in a wide class of logics with the disjunction property, including positive, Johansson, and modal logics.

Findings

Bases of admissible rules can be reduced to bases of admissible m-rules.

Bases of admissible m-rules can be reduced to bases of admissible rules.

Results apply to various logics including positive, Johansson, and S4 extensions.

Abstract

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics.