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On the asymptotic density of tautologies in logic of implication and negation

Zaionc M.

Reports on Mathematical Logic

2005

Nr 39

67-87

The paper solves the problem of finding the asymptotic probability of the set of tautologies of classical logic with one propositional variable, implication and negation. We investigate the proportion of tautologies of the given length n among the number of all formulas of length n. We are especially interested in asymptotic behavior of this proportion when $n \impl \infty$. If the limit exists it represents the real number between 0 and 1 which we may call density of tautologies for the logic investigated. In the paper [2] the existence of this limit for classical (and at the same time intuitionistic) logic of implication built with exactly one variable is proved. The present paper answers the question "how much does the introduction of negation influence the "density of tautologies" in the propositional calculus of one variable?" While in the case of implicational calculus the limit exists and is about 72.36\% (see Theorem 4.6 in the paper [2] ), in our case the limit exists as well, but negation lowers the density of tautologies to the level of about 42.32%.

Asymptotic Properties of Logics

Zaionc M.

Schedae Informaticae

2003

Vol. 12

129-138

This paper presents the number of results concerning problems of asymptotic densities in the variety of propositional logics. We investigate, for propositional formulas, the proportion of tautologies of the given length n against the number of all formulas of length n. We are specially interested in asymptotic behavior of this fraction. We show what the relation between a number of premises of an implicational formula and asymptotic probability of finding a formula with this number of premises is. Furthermore we investigate the distribution of this asymptotic probabilities. Distribution for all formulas is contrasted with the same distribution for tautologies only.