Asymptotic comparison of the implicative fragments of certain fuzzy logics
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.
Frames for fusions of modal logics
Let us consider multimodal logics and . We assume that is characterised by a class of connected frames, and there exists an -frame with a so-called -starting point. Similarly, the logic is characterised by a class of connected frames, and there exists an -frame with a -starting point. Using isomorphic copies of the frames and , we construct a connected frame which characterises the fusion . The frame thus obtained has some useful properties. Among others, is countable if both and are countable, and there is a special world of the frame such that any formula is valid in the frame if and only if it is valid at the point . We also describe a similar construction where we assume the existence o…
Finitary unification in locally tabular modal logics characterized
We provide necessary and sufficient conditions for finitary unification in locally tabular modal logics, solely in terms of Kripke models. We apply the conditions to establish the unification types of logics determined by simple finite frames. In particular, we show that unification is finitary (or unitary) in the logic determined by the fork (frame F4, see Fig. 6), the rhombus (frame F5), inGL.3m,GrzBd2,S4Bd2and other logics; whereas it is nullary in the logic of F6, and of the pentagon FN5. In Appendix analogous results are given for superintuitionistic logics.
Projective unification in transitive modal logics
We show that a transitive normal modal logic L enjoys projective unification (i.e. each unifiable formula is projective) if and only if L contains K4D1 ( D1 : ( x → y ) ∨ ( y → x ) ). It means, in particular, that K4D1 (and any of its extensions) is almost structurally complete, i.e. the logic is complete with respect to all non-passive admissible rules. We also characterize non-unifiable formulas and provide an explicit form of the basis for all passive rules over K4G + ( x → x )