Uniform Interpolation and Admissible Rules: Proof-theoretic investigations into (intuitionistic) modal logics

Abstract

The thesis investigates classical and intuitionistic modal logics via proof-theoretic methods for two important and widely applied topics in logic: uniform interpolation and admissible rules. Both topics are treated in separate parts of the thesis. Part I studies uniform interpolation which is a stronger property than the well-known Craig interpolation property in which the interpolant only depends on the antecedent or the succedent of the implication. Our investigation is inspired by Iemhoff (2019a,b) who characterizes sufficient conditions for sequent calculi for proving uniform interpolation. We provide (terminating) sequent calculi for two intuitionistic modal logics, iGL and iSL (both have connections to provability). We give syntactic cutelimination proofs based on a non-trivial cut-elimination strategy for classical GL (Valentini, 1983). We establish the Craig interpolation property for both logics and we use the termination to develop a countermodel construction for iSL. In addition, we show that intuitionistic modal logics iK4 and iS4 do not have the uniform interpolation property. In light of the negative results from Iemhoff (2019a,b), we obtain that these logics cannot be described by certain terminating sequent calculi. In addition, we study uniform interpolation for classical modal logics via nested sequents and hypersequents. We develop a method to reprove uniform interpolation for logics K, T, D, and S5. We construct uniform interpolants via terminating nested sequent calculi and hypersequent calculi. To the best of our knowledge, this provides a first constructive definition of uniform interpolants for S5. Although the interpolants are defined constructively, our proof incorporates semantic reasoning based on so-called bisimulation quantifiers. Part II provides an investigation of admissible rules. The admissible rules of a logic are those rules that can be added to the logic without changing its valid formulas. This thesis provides a first study of admissible rules for intuitionistic modal logics. We are able to provide full characterizations in terms of bases for the admissible rules in six intuitionistic modal logics with the coreflection principle: iCK4, iCS4 ≡ IPC, iSL, KM, mHC, and PLL. In addition, we show decidability of admissibility for these logics. Our technique relies on a proof theory for admissibility based on (Iemhoff and Metcalfe, 2009b). This proof theory is special because it does not reason on the level of formulas, but it contains rules that reason about rules. The proof also relies on a semantic approach developed by Ghilardi (1999, 2000) about the interaction between so-called projective formulas and the extension property. We analyse their importance in the field of admissible rules.