Universal Proof Theory: Semi-analytic Rules and Uniform Interpolation

Abstract

In [7] and [8], Iemhoff introduced a connection between the existence of a terminating sequent calculi of a certain kind and the uniform interpolation property of the super-intuitionistic logic that the calculus captures. In this paper, we will generalize this relationship to also cover the substructural setting on the one hand and a much more powerful class of rules on the other. The resulted relationship then provides a uniform method to establish uniform interpolation property for the logics FL e , FL ew , CFL e , CFL ew , IPC, CPC and their K and KD-type modal extensions. More interestingly though, on the negative side, we will show that no extension of FL e can enjoy a certain natural type of terminating sequent calculus unless it has the uniform interpolation property. It excludes almost all super-intutionistic logics and the logics K4 and S4 from having such a reasonable calculus.

Universal Proof Theory: Feasible Admissibility in Intuitionistic Modal Logics

“…Universal proof theory [1,2] is a recent research project designed to investigate the generic behavior of proof systems considered as the independent mathematical objects standing on their own feet. It opposes the usual attitude in proof theory that sees the proof systems as the auxiliary objects by which we investigate the logical systems they capture.…”

“…The focused rules are very limited in their form and it seems that even the intuitionistic propositional logic cannot have a system only consisting of these focused rules. To solve this problem, later, in [1,2], the focussed rules were strengthened to a more general form (called semi-analytic), where both Craig and uniform interpolation properties were studied. Moreover, the scope of the logics were also extended to cover the substructural (modal) logics.…”

“…They form a basis for the admissible rules of intuitionistic logic and any intermediate logic in which they are admissible. In this paper, we follow the universal proof theory program introduced and developed in [1,2,24,25] to establish a connection between the form of the rules in a sequent calculus for an intuitionistic modal logic and the admissibility of Visser's rules in that logic. More precisely, by investigating the form of the constructively acceptable rules, we first introduce a very general family of rules called the constructive rules.…”

Uniform Lyndon Interpolation for Basic Non-normal Modal Logics

“…Over the last years a method to prove uniform (Lyndon) interpolation has been developed by the authors that applies to various (intuitionistic) modal and intermediate logics [9,10,12,1,2,3]. Uniform interpolation is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication.…”

“…Uniform interpolation has applications in computer science, in particular in description logics [14], but our interest in the property stems from a project in universal proof theory where we aim to develop methods to prove that certain (classes of) logics cannot have certain well-behaved proof systems, in our case sequent calculi [3,10]. Here we make use of the fact that uniform interpolation seems to be a rare property among logics.…”