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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)
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 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 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)