Multiple Conclusion Rules in Logics with the Disjunction Property

Abstract

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.

Lorenzen and Constructive Mathematics

“…Two highly original features of his argument are that the metatheory is constructive (with use of generalised inductive definitions) and that there is no ordinal analysis. At about the same time, and independently, P. S. Novikov (1943) had a similar analysis, and also introduced the notion of an admissible/derivable rule (Citkin 2016). For a historical presentation of Lorenzen's work on infinitary calculus, see Coquand and Neuwirth 2020. Apart from Novikov, most treatments in proof theory (Gentzen, Schütte, Takeuti) involve ordinal analysis.…”

On the Blok-Esakia theorem for universal classes

“…The situation has been changed with Jeřábek's paper [31] and his observation that multi-conclusion inference rules may be used for the canonical axiomatization of intermediate and modal logics. This topic was recently undertaken in many papers [2,4,3,5,6,13,14,15,16,24,30,29,32].…”