Proof Analysis of A Foundational Classical Singlesuccedent Sequent Calculus

TL;DR

This paper introduces a new classical singlesuccedent sequent calculus, G-Calculus, designed to be a foundational and robust system surpassing existing calculi in proof-theoretic capabilities.

Contribution

The paper formulates G-Calculus, a novel sequent calculus that aims to be the definitive classical singlesuccedent system, addressing limitations of prior calculi.

Findings

G-Calculus meets formal proof-theoretic expectations

Existing calculi lack the robustness of G-Calculus

G-Calculus includes novel rules for classical logic

Abstract

In this paper we investigate the question: 'How can A Foundational Classical Singlesuccedent Sequent Calculus be formulated?' The choice of this particular area of proof-theoretic study is based on a particular ground that is, to formulate a robust and foundational classical singlesuccedent sequent calculus that includes a number of novel rules with the ultimate aim of deriving the singlesuccedent sequent {\Gamma} sequent arrow C. To this end, we argue that among all standard sequent calculi (at least to the best of our knowledge) there is no classical singlesuccedent sequent calculus that can be considered the rightful successor to Gerhard Gentzen's (1935) original LK system. However, we also contend that while several classical singlesuccedent sequent calculi exist such as Sara Negri's and Jan von Plato's (2001 & 2011) G3ip+Gem-at and G0ip+Gem0-at calculi, none of these proof systems possess the classical proof-theoretic potential to meet the formal expectations of a dedicated classical proof theorist. Conversely, we shall demonstrate that our forthcoming system, namely G-Calculus through its classical division i.e. Gc has been entirely designed to meet these expectations. Prior to commencing our enquiry, a supplementary note must be made and that is in this work when discussing various sequent calculi, for proof-theoretic purposes, we are primarily concerned with their propositional components rather than their predicate divisions except in G-Calculus where we examine both aspects.