David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 48 (1):41 - 65 (1989)
The purpose of this paper is to connect the proof theory and the model theory of a family of prepositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK 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, Hilbert-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.Below is the sequel to the first part of the paper, which appeared in the previous issue of this journal (vol. 47 (1988), pp. 353–386). The first part contained sections on sequent-systems and Hilbert-formulations, and here is the third section on groupoid models. This second part is meant to be read in conjunction with the first part.
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References found in this work BETA
Kit Fine (1974). Models for Entailment. Journal of Philosophical Logic 3 (4):347 - 372.
Hiroakira Ono & Yuichi Komori (1985). Logics Without the Contraction Rule. Journal of Symbolic Logic 50 (1):169-201.
Helena Rasiowa (1963). The Mathematics of Metamathematics. Warszawa, Państwowe Wydawn. Naukowe.
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Citations of this work BETA
Eunsuk Yang (2014). Algebraic Kripke-Style Semantics for Relevance Logics. Journal of Philosophical Logic 43 (4):803-826.
Heinrich Wansing (1993). Informational Interpretation of Substructural Propositional Logics. Journal of Logic, Language and Information 2 (4):285-308.
Robert Goldblatt (2011). Grishin Algebras and Cover Systems for Classical Bilinear Logic. Studia Logica 99 (1-3):203-227.
Hiroakira Ono (2012). Crawley Completions of Residuated Lattices and Algebraic Completeness of Substructural Predicate Logics. Studia Logica 100 (1-2):339-359.
Tomoyuki Suzuki (2011). Canonicity Results of Substructural and Lattice-Based Logics. Review of Symbolic Logic 4 (1):1-42.
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