Abstract
The first part of the paper is a reminder of fundamental results connected with the adequacy problem for sentential logics with respect to matrix semantics. One of the main notions associated with the problem, namely that of the degree of complexity of a sentential logic, is elucidated by a couple of examples in the second part of the paper. E.g., it is shown that the minimal logic of Johansson and some of its extensions have degree of complexity 2. This is the first example of an exact estimation of the degree of natural complex logics, i.e. logics whose deducibility relation cannot be represented by a single matrix. The remaining examples of complex logics are more artificial, having been constructed for the purpose of checking some theoretical possibilities.
Similar content being viewed by others
References
S. L. Bloom andR. Suszko,Investigations into the sentential calculus with identity,Notre Dame Journal of Formal Logic 13 (1972), pp. 289–308.
E. Capińska,On standard consequence operations in the implicational language,Bulletin of the Section of Logic 8 (1979), pp. 202–205.
Dov M. Gabbay,Semantic proof of the Craig interpolation theorem for intuitionistic logic and extensions. Part I. in:Logic Colloquium '69, edited by R. O. Gandy and C. M. E. Yates, North-Holland Publ. Co., Amsterdam 1971, pp. 391–401.
E. Graczyńska andA. Wroński,Constructing denumerable matrices strongly adequate for pre-finite logics,Studia Logica 33 (1974), pp. 417–423.
J. Hawranek,A matrix strongly adequate for S5 with MP and RN,Bulletin of the Section of Logic 9 (1980), pp. 122–124.
L. L. Maksimova,The principle of separation of variables in sentential logics (in Russian),Algebra i Logika 15 (1976), pp. 168–184.
J. Łoś andR. Suszko,Remarks on sentential logics,Indagationes Mathematicae 20 (1958), pp. 177–183.
J. Perzanowski andA. Wroński,The deduction theorems for the system of Feys-von Wright,Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Prace z logiki 6 (1971), pp. 11–14.
K. Segerberg,Propositional logics related to Heyting's and Johansson's,Theoria 34 (1968), pp. 26–61.
D. J. Shoesmith andT. Smiley,Deducibility and many-valuedness,The Journal of Symbolic Logic 36 (1970), pp. 610–622.
R. Suszko,Adequate models for the non-Fregean sentential calculus (SCI), in:Logic, language, and probability. A selection of papers contributed to sections IV, VI, and XI of the Fourth International Congress for Logic, Methodology, and Philosophy of Science, Bucharest, September 1971, Edited by Radu J. Bogdan and Ilkka Niiniluoto, D. Reidel Publ. Co., Dordrecht, 1973, pp. 49–54.
M. Tokarz,The existence of matrices strongly adequate for E, R and their fragments,Studia Logica 38 (1979), pp. 75–85.
R. Wójcicki,Some remarks on the consequence operation in sentential logics,Fundamenta Mathematicae 58 (1970), pp. 269–279.
R. Wójcicki,Investigations into methodology of sentential calculi I, Institute of Philosophy and Sociology, Polish Academy of Science, Warsaw 1971, 77 pp. (Mimeographed).
A. Wroński,On cardinalities of matrices strongly adequate for the intuitionistic propositional logic,Reports on Mathematical Logic 3 (1974), pp. 67–72.
S. Zachorowski,Proof of a conjecture of Roman Suszko,Studia Logica 34 (1975), pp. 253–256.
Author information
Authors and Affiliations
Additional information
The paper was presented to the Polish Philosophical Society, Wrocław Branch, at its meeting on March 27th, 1980.
The authors wish to thank both the referees of Studia Logica for their helpful and very insightful remarks. Following their criticism, we have been able to improve the style and structure of our presentation. In particular, we are indebted to the referees for pointing out a gap in the original proof of Theorem 2, and we have incorporated into the revised text a corrected proof of step (2.1) which one of them was kind enough to supply in detail.
Rights and permissions
About this article
Cite this article
Hawranek, J., Zygmunt, J. On the degree of complexity of sentential logics. A couple of examples. Stud Logica 40, 141–153 (1981). https://doi.org/10.1007/BF01874705
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01874705