An asymptotic similarity of some fragments of two fuzzy logics is proved. We focus on two 3-valued fuzzy logics: the Gödel-Dummett one and the Łukasiewicz one and we consider their purely implicative fragments of two variables. This paper shows the existence of the densities of truth of these logics and determines their values. For this purpose we build the appropriate Tarski-Lindenbaum algebra and use extensively generating functions. Our method can be generalized to n-valued logics, n > 3, but all computations will be extremely complicated.

Abstract We prove that $\textbf {K4}^{\textbf {n+}}\textbf {B}^{\textbf {k+}}$ has projective unification, for any $k,n\geq 1$. It means, in particular, that any weakly transitive and weakly symmetric modal logic is unitary. Some consequences of projective unification concerning structural completeness and passive rules are provided.

In this paper, we focus on two main 3-valued logics used by the fuzzy logic community. The Gödel-Dummett logic and the Łukasiewicz one. Both are based on the same language of implication and negation. In both, we consider fragments consisting of formulas formed with one variable. We investigate the proportion of the number of true (or satisfiable) formulas of a certain length n to the number of all formulas of such length. We are especially interested in the asymptotic behavior of this fraction when length n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which is called the density of truth or the density of SAT. Using the powerful theory of analytic combinatorics, we state several results comparing the density of truth and the density of satisfiable formulas for both Gödel-Dummett and Łukasiewicz 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.

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

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.