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We carry out a unified investigation of two prominent topics in proof theory and order algebra: cut-elimination and completion, in the setting of substructural logics and residuated lattices. We introduce the substructural hierarchy — a new classification of logical axioms (algebraic equations) over full Lambek calculus FL, and show that a stronger form of(More)
We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of propositional nonclassical logics including intermediate , fuzzy and substructural logics. Our work encompasses many existing(More)
We develop a general algebraic and proof-theoretic study of sub-structural logics that may lack associativity, along with other structural rules. Our study extends existing work on (associative) substructural logics over the full Lambek Calculus FL (see e.g. [36, 19, 18]). We present a Gentzen-style sequent system GL that lacks the structural rules of(More)
Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential(More)
This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids (RSs). An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under finite joins. It is shown that a variety has(More)