Abstract

In this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple mathbf{G1}-system employing rules of contraposition. On the other hand we present a mathbf{G3}-system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

Highlights

  • Leitgeb [4] proposed a logic for hyperintensional contexts that is called HYPE.1 In his paper Leitgeb worked with an axiomatic calculus

  • In the second section we introduce the propositional fragment of the logic of HYPE

  • In the last section we show how the cut-elimination can be used to establish the conservativity of HYPE over IL as well as FDE

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Summary

Introduction

Leitgeb [4] proposed a logic for hyperintensional contexts that is called HYPE. In his paper Leitgeb worked with an axiomatic calculus. It would be desirable to have a calculus that allows for cut-elimination and with it to be able to establish results, such as the conservativity over intuitionistic logic, in a proof-theoretic way. We first present problems for the standard strategy of G3-systems in the case of our introduction rules for the conditional and present a counterexample for cut-elimination in G3hp. We consider an extension cGh+p of the system G1hp by connections and more liberal rules This is based on the solution for cutelimination in the case of constant domains in IL as suggested by Kashima and Shimura in [3].

The Propositional Logic of HYPE
Intersubstitutivity
Properties of G3hp
The Equivalence of G1hp and G3hp
Equivalence with the Axiomatic Calculus of HYPE
A Counterexample for Cut-Elimination in G3hp
Connections
Conservativity

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