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In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\) , as well as for its extensions with analytic structural rules satisfying certain additional properties. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Analyticity, also known as the subformula property, typically guarantees decidability of derivability in propositional sequent calculi. To utilize this fact, two substantial gaps have to be addressed: (i) What makes a sequent calculus analytic? and (ii) How do we obtain an efficient decision procedure for derivability in an analytic calculus? In the first part of this article, we answer these questions for pure calculi--a general family of fully structural propositional sequent calculi whose rules allow arbitrary context formulas. We provide a sufficient syntactic criterion for analyticity in these calculi, as well as a productive method to construct new analytic calculi from given ones. We further introduce a scalable decision procedure for derivability in analytic pure calculi by showing that it can be (uniformly) reduced to classical satisfiability. In the second part of the article, we study the extension of pure sequent calculi with modal operators. We show that such extensions preserve the analyticity of the calculus and identify certain restricted operators (which we call "Next" operators) that are also amenable for a general reduction of derivability to classical satisfiability. Our proofs are all semantic, utilizing several strong general soundness and completeness theorems with respect to nondeterministic semantic frameworks: bivaluations (for pure calculi) and Kripke models (for their extension with modal operators). [ABSTRACT FROM AUTHOR]