Abstract
For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences.
Similar content being viewed by others
References
COMTET, L., Advanced combinatorics. The art of finite and infinite expansions, Revised and enlarged edition, Reidel, Dordrecht, 1974.
DUMMETT, M., ‘A propositional calculus with a denumerable matrix’, Journal of Symbolic Logic 24 (1954), 96-107.
MOCZURAD, M., J. TYSZKIEWICZ, and M. ZAIONC, ‘Statistical properties of simple types’, Mathematical Structures in Computer Science 10 (2000), 575-594.
ROBINS, H., ‘A remark on Stirling formula’, Ammer. Math. Monthly 62 (1955), 26-29.
SZEGÖ, G., ‘Orthogonal polynomials’, AMS, Colloquium Publications, Fourth edition, 23, Providence, 1975.
WILF, H. S., Generatingfunctionology, Second edition, Academic Press, Boston, 1994.
ZAIONC, M., ‘On the asymptotic density of tautologies in logic of implication and negation’, submitted to Reports on Mathematical Logic.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kostrzycka, Z., Zaionc, M. Statistics of Intuitionistic versus Classical Logics. Studia Logica 76, 307–328 (2004). https://doi.org/10.1023/B:STUD.0000032101.88511.93
Issue Date:
DOI: https://doi.org/10.1023/B:STUD.0000032101.88511.93