Linear Logic Properly Displayed

Abstract

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination, and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic, and applies the guidelines of the multi-type methodology in the design of display calculi. Keywords : Linear logic, substructural logics, algebraic proof theory, sequent calculi, cut elimination, display calculi, multi-type calculi, proof calculi for categorial grammar. 2010 Math. Subj. Class. CCS, Theory of computation, Logic, Proof theory.

Non-normal modal logics and conditional logics: Semantic analysis and proof theory

“…To argue that the calculi introduced in Section 7 conservatively extend their corresponding Hilbert systems, we follow the standard proof strategy discussed in [35,36]. Let L denote the syntactic consequence relation arising from Hilbert systems presented in Section 2.1, and |= H denote the semantic consequence relation arising from heterogeneous Kripke frames and their complex (heterogeneous) algebras.…”

“…This refinement would build a bridge between the literature in non-normal modal logics and the literature on so-called modal logics for structural control in linguistics and logic (see e.g. [46,52,30,36]).…”

“…Thirdly, the multi-type reformulation of non-normal logics facilitates establishing connections with other areas of investigation in logic and neighbouring fields in which logical frameworks connecting entities of different types have already been studied and exploited. One such area is structural control, which has given rise to a rich literature both in substructural logic [30,36,42,16,65] and in formal linguistics [53,45,54,55,39,3,66]. 3 It is perhaps worth stressing that, by their definition, the (interpretation of) non-normal connectives lack the minimum order-theoretic properties necessary for any non-normal logic to be properly displayable, i.e.…”