Zofia Kostrzycka

Statistics of Intuitionistic versus Classical Logics

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 Dummetts 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.

On non-compact logics in NEXT(KTB)

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)

Asymptotic Densities in Logic and Type Theory

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.

On asymptotic divergency in equivalential logics

In this paper we characterise the equivalential reducts of classical and intuitionistic logics over a language with two propositional variables. We then investigate the size of the fraction of the tautologies of these logics against all formulas. Some methods from complex analysis are used to achieve this goal.

On a Finitely Axiomatizable Kripke Incomplete Logic Containing KTB

We construct a finite extension of T2 =KTB⊕ 2p→ 3p which is Kripke incomplete.

On linear Brouwerian logics

We define a special family of Brouwerian logics determined by linearly ordered frames. Then we prove that all logics of this family have the finite model property and are Kripke complete.

On the Density of Truth of Locally Finite 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.

On Halldén Completeness of Modal Logics Determined by Homogeneous Kripke Frames

Halldén complete modal logics are defined semantically. They have a nice characterization as they are determined by homogeneous Kripke frames.