Quantitative Study of Fuzzy Logics

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.