Studia Logica 47 (4):353 - 385 (1988)
The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.
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References found in this work BETA
Entailment: The Logic of Relevance and Neccessity, Vol. I.Alan R. Anderson & Nuel D. Belnap - 1975 - Princeton University Press.
Logics Without the Contraction Rule.Hiroakira Ono & Yuichi Komori - 1985 - Journal of Symbolic Logic 50 (1):169-201.
Citations of this work BETA
External Curries.Heinrich Wansing & Graham Priest - 2015 - Journal of Philosophical Logic 44 (4):453-471.
Cut Elimination and Strong Separation for Substructural Logics: An Algebraic Approach.Nikolaos Galatos & Hiroakira Ono - 2010 - Annals of Pure and Applied Logic 161 (9):1097-1133.
Canonicity Results of Substructural and Lattice-Based Logics.Tomoyuki Suzuki - 2010 - Review of Symbolic Logic 4 (1):1-42.
A Brief Survey of Frames for the Lambek Calculus.Kosta Došen - 1992 - Mathematical Logic Quarterly 38 (1):179-187.
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