Linear logic using negative connectives: extended version
Résumé
In linear logic, the invertibility of a connective's right-introduction rule is equivalent to the noninvertibility of its left-introduction rule. This duality motivates the concept of polarity: a connective is termed negative if its right-introduction rule is invertible, and positive otherwise. A two-sided sequent calculus for first-order linear logic featuring only negative connectives exhibits a compelling proof theory. Proof search in such a system unfolds through alternating phases of invertible (right-introduction) rules and non-invertible (left-introduction) rules, mirroring the processes of goal-reduction and backchaining, respectively. These phases are formalized here using the framework of multifocused proofs. We analyze linear logic by dissecting it into three sublogics: L0 (first-order intuitionistic logic with conjunction, implication, and universal quantification); L1 (an extension of L0 incorporating linear implication which preserves its intuitionistic nature); and L2 (which includes multiplicative falsity ⊥ and encompasses classical linear logic). It is worth noting that the single-conclusion restriction on sequents, a constraint imposed by Gentzen, is not a prerequisite for defining intuitionistic logic proofs within this framework, as it emerges naturally by restricting the formulas to those of L0 and L1. While multifocused proofs of L2 sequents can accommodate parallel applications of left-introduction rules, proofs of L0 and L1 sequents cannot leverage such parallel rule applications. This notion of parallelism within proofs enables a novel approach to handling disjunctions and existential quantifiers in the natural deduction system for intuitionistic logic.
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