Studia Logica

, Volume 48, Issue 1, pp 41–65 | Cite as

Sequent-systems and groupoid models. II

  • Kosta Došen


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.


Mathematical Logic Model Theory Modal Logic Logical Operation Computational Linguistic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    A. R. Anderson and N. D. Belnap, Jr, Entailment: The Logic of Relevance and Necessity, vol. I, Princeton University Press, Princeton, 1975.Google Scholar
  2. [2]
    G. Birkhoff and O. Frink, Jr, Representation of lattices by sets, Transactions of the American Mathematical Society 64 (1948), pp. 299–316.Google Scholar
  3. [3]
    W. Buszkowski, Completeness results for Lambek syntactic calculus, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32 (1986), pp. 13–28.Google Scholar
  4. [4]
    D. van Dalen, Logic and Structure, 2nd edition, Springer, Berlin, 1983.Google Scholar
  5. [5]
    K. Fine, Models for entailment, Journal of Philosophical Logic 3 (1974), pp. 347–372.Google Scholar
  6. [6]
    L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963.Google Scholar
  7. [7]
    H. Ono and Y. Komori, Logics without the contraction rule, The Journal of Symbolic Logic 50 (1985), pp. 169–201.Google Scholar
  8. [8]
    H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, Państwowe Wydawnictwo Naukowe, Warsaw, 1963.Google Scholar
  9. [9]
    R. Routley and R. K. Meyer, The semantics of entailment. I, in: H. Leblanc (ed.), Truth, Syntax, Modality, North-Holland, Amsterdam, 1973, pp. 199–243.Google Scholar
  10. [10]
    R. Routley and R. K. Meyer, The semantics of entailment III, Journal of Philosophical Logic 1 (1972), pp. 192–208.Google Scholar
  11. [11]
    A. Urquhart, Semantics for relevant logics, The Journal of Symbolic Logic 37 (1972), pp. 159–169.Google Scholar

Copyright information

© Polish Academy of Sciences 1989

Authors and Affiliations

  • Kosta Došen
    • 1
  1. 1.Matematički InstitutBelgradeYugoslavia

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