A Sequent Calculus for a Negative Free Logic

Abstract

This article presents a sequent calculus for a negative free logic with identity, called N. The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.

A More Unified Approach to Free Logics

“…In formulating the rules for the free logics discussed here we take as our starting point the sequent calculus for the negative variety first introduced in [6]. We then streamline it by transforming it into a G3-style calculus along the lines of [19], and to a lesser extent, [16,20].…”

“…The language L utilized in this paper is a standard first order language without functions, adapted from [6], with the vocabulary defined as A formula of our language L is defined as…”

“…The sequent calculi for free logics offered so far, e.g. for either version of free logics in [2] (here simplified to be context-sharing) or specifically for negative free logic in [6] use the following versions of the quantifier rules for L∀ and R∃:…”

“…Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic [8,9]. The term is due to Karel Lambert [9][10][11] and is short for first order logic free of existential assumptions, and we take a logic to be free iff (1) it is free of existential presuppositions with respect to its singular terms, (2) it is free of existential presuppositions with respect to its general terms and finally (3) its quantifiers have existential import (the last claim is expressed by the axioms A3 and A4 later in the paper) [6]. The three (families of) free logics we will here distinguish depend on their decisions on the truth values of (atomic) statements containing empty terms.…”

Generality and Existence 1: Quantification and Free Logic

“…where α is globally replaced in that sequent by another term β in the category A. 10 In first order classical predicate logic, function terms are not deductively general singular terms, but primitive names are. If we consider the conseqence relation of Peano Arithmetic, defined by setting P A iff P A, over the language of predicate logic with 0, successor, addition, multiplication and identity, where P A is an axiomatization of Peano Arithmetic, then the constant term 0 is not deductively general, since we have (∃x)(0 = x ) P A (that is, it's inconsistent with the axioms of P A for 0 be the successor of some number), but we do not have (∃x)(0 = x ) P A .…”