Gödel logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise different (...) sets of tautologies, all these sets being non-arithmetical, whereas under the witnessed semantics all the logics have the same set of tautologies and it is Π2-complete. Further similar results are obtained. (shrink)

First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete (...) axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized. (shrink)

Inspired by the work done by Baaz et al. (Ann Pure Appl Log 147(1–2): 23–47, 2007; Lecture Notes in Computer Science, vol 4790/2007, pp 77–91, 2007) for first-order Gödel logics, we investigate Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra, establishing also a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, we analyze axiomatizability, undecidability and (...) the monadic fragments. (shrink)

Axiomatization of Gödel-Dummett predicate logics S2G, S3G, and PG, where PG is the weakest logic in which all prenex operations are sound, and the relationships of these logics to logics known from the literature are discussed. Examples of non-prenexable formulas are given for those logics where some prenex operation is not available. Inter-expressibility of quantifiers is explored for each of the considered logics.

Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider (...) scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction. (shrink)