What is modal logic

This paper considers logical formulas built on the single binary connector of implication and a finite number of variables. When the number of variables becomes large, we prove the following quantitative results: asymptotically, all classical tautologies are simple tautologies. It follows that asymptotically, all classical tautologies are intuitionistic.

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Classical and intuitionistic logic are asymptotically identical

We present quantitative analysis of various (syntactic and behavioral) properties of random \lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the \lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator.

Asymptotically almost all \lambda-terms are strongly normalizing

We present a quantitative analysis of various (syntactic and behavioral) prop- erties of random �-terms. Our main results show that asymptotically, almost all terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the �-calculus into combi- nators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator.

/pdf/asymptotically-almost-all-lambda-terms-are-strongly-3cfvsjhlfy.pdf

We examine how we can define several probability distributions on the set of Boolean functions on a fixed number of variables, starting from a representation of Boolean expressions by trees. Analytic tools give us a systematic way to prove the existence of probability distributions, the main challenge being the actual computation of the distributions. We finally consider the relations between the probability of a Boolean function and its complexity.

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Random Boolean expressions

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.

Statistics of Intuitionistic versus Classical Logics

In this paper we construct a continuum of logics, extensions of the modal logic T2 = KTB ⊕ □2p □3p, which are non-compact (relative to Kripke frames) and hence Kripke incomplete. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On non‐compact logics in NEXT(KTB)

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.

Asymptotic Densities in Logic and Type Theory

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.

http://mathinfo06.iecn.u-nancy.fr/papers/dmAG161-170.pdf

On the density of truth in modal logics

We prove that the density of truth exists for a large class of locally finite (locally tabular) propositional logics. We are primarily interested in classical and intuitionistic logic and show that their implicational fragments have the same density. There are also given some locally finite logics without the density of truth.

Zofia Kostrzycka

Papers

Statistics of Intuitionistic versus Classical Logics

On non‐compact logics in NEXT(KTB)

Asymptotic Densities in Logic and Type Theory

On the density of truth in modal logics

On the Density of Truth of Locally Finite Logics