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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)

It is well known that classical propositional logic can be interpreted in intuitionistic propo-sitional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive sub-structural logic there exists a minimum… (More)

We extend the lattice embedding of the axiomatic extensions of the positive fragment of intuitionistic logic into the axiomatic extensions of intuitionistic logic to the setting of substructural logics. Our approach is algebraic and uses residuated lattices, the algebraic models for substructural logics. We generalize the notion of the ordinal sum of two… (More)