On the Density of Truth of Locally Finite Logics

2003

For the given logical calculus we investigate the size of the fraction of true formulas of a certain length n against the number of all formulas of such length. We are especially interested in asymptotic behaviour of this fraction when n tends to infinity. If the limit of the fraction exists it represents a number which we may call the density of truth for the investigated logic. In this paper we apply this approach to the Dummett intermediate linear logic (see [?]). Actually, this paper shows the exact density of this logic and demonstrates that it covers a substantial part of classical propositional 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.

Reports on Mathematical Logic, 2005

A b s t r a c t. 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 → ∞. 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%.

2003

This paper presents the number of results concerning prob-lems 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 for-mula and asymptotic probability of finding a formula with this number of premises is. Furthermore we investigate the distribution of this as-ymptotic probabilities. Distribution for all formulas is contrasted with the same distribution for tautologies only.

Journal of Logic, Language and Information, 2007

This paper is an attempt to count the proportion of tautologies of some intermediate logics among all formulas. Our interest concentrates especially on Dummett's and Medvedev's logics and their {→, ∨, ¬} fragments over language with one propositional variable.

Mics, 2006

The aim of this paper is counting the probability that a random modal formula is a tautology. We examine {→, 2} fragment of two modal logics S5 and S4 over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic S5 and this is the key to count the proportion of tautologies of S5 among all formulas. Although the logic S4 does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.

Studia Logica - An International Journal for Symbolic Logic, 2008

This paper presents a systematic approach for obtaining results from the area of quantitative investigations in logic and type theory. We investigate the proportion between tautologies (inhabited types) of a given length n against the number of all formulas (types) of length n. We investigate an asymptotic behavior of this fraction. Furthermore, we characterize the relation between number of premises of implicational formula (type) and the asymptotic probability of finding such formula among the all ones. We also deal with a distribution of these asymptotic probabilities. Using the same approach we also prove that the probability that randomly chosen fourth order type (or type of the order not greater than 4), which admits decidable lambda definability problem, is zero.

Discrete Mathematics & Theoretical Computer Science, 2006

International audience The aim of this paper is counting the probability that a random modal formula is a tautology. We examine $\{ \to,\Box \}$ fragment of two modal logics $\mathbf{S5}$ and $\mathbf{S4}$ over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic $\mathbf{S5}$ and this is the key to count the proportion of tautologies of $\mathbf{S5}$ among all formulas. Although the logic $\mathbf{S4}$ does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.

2015

In this paper I propose the non-Archimedean multiple-validity. Further, I build an infinite-order predicate logical language in that predicates of various order are considered as fuzzy relations. Such a language can have non-Archimedean valued semantics. For instance, infinite-order predicates can have an interpretation in the set [0, 1] of hyperreal (hyperrational) numbers. Notice that there exists an effectively axiomatizable part of non-Archimedean valued predicate logic, namely the class of higher-order formulas such that all their predicate quantifiers are universal (or existential). There exist various many-valued logical systems (see [9]). However non-Archimedean valued predicate logic has not been constructed yet. The idea of non-Archimedean multiple-validity is that (1) the set of truth values is uncountable infinite and (2) this set isn’t well-ordered. This idea can have a lot of applications in probabilistic reasoning (see [7], [13], [8]). In this paper I show that infini...

Studia Logica, 2004

For the given logical calculus we investigate the size of the proportion of the number of true formulas of a certain length n against 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 of fractions exists it represents the 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 it to the same problem of Dummett's intermediate linear logic of one variable ( see [?]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more then 93%) of classical propositional 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.