BOUNDED-ANALYTIC SEQUENT CALCULI AND EMBEDDINGS FOR HYPERSEQUENT LOGICS

Agata Ciabattoni,Revantha Ramanayake,Timo Lang

doi:10.1017/jsl.2021.42

Abstract

AbstractA sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory.

Highlights

What concepts are essential to the proof of a given statement? This is a fundamental and long-debated question in logic since the time of Leibniz, Kant, and Frege, when a broad notion of analytic proof was formulated to mean truth by conceptual containments, or purity of method in mathematical arguments.The familiar Hilbert-style proof systems are excellent for many purposes including defining a logic and a notion of proof
Enriching the logical language with a structural language enabled him to state and prove the famous cut-elimination theorem—this is a constructive procedure that eliminates all applications of the cut rule from a given proof
From cut-elimination it follows that every provable formula has a proof respecting the subformula property

Summary

Introduction

The familiar Hilbert-style proof systems are excellent for many purposes including defining a logic and a notion of proof. They offer few insight concerning analyticity because of their reliance on the inference rule of modus ponens. In 1935, Gentzen [20] showed how to address this weakness of the Hilbert systems by placing the logical language within a meta-logical (structural) language He introduced a new type of proof system called the sequent calculus built from sequents Γ ⇒ Δ where Γ and Δ are lists/multisets of logical formulas. From cut-elimination it follows that every provable formula has a proof respecting the subformula property (i.e., every formula in the proof is a subformula of the theorem) Gentzen later used this property to give a proof of the consistency of arithmetic.

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