In fuzzy predicate logic, assignment of truth values may be partial, i.e. the truth value of a formula in an interpretation may be undefined . A logic is supersound if each provable formula is true in each interpretation in which the truth value of is defined. It is shown that among the logics given by continuous t-norms, Gödel logic is the only one that is supersound; all others are not supersound. This supports the view that the usual restriction of semantics (...) to safe interpretations is very natural. (shrink)

This paper presents two classes of propositional logics (understood as a consequence relation). First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics. This class is broad enough to contain many “usual” logics, yet easily manageable with nice logical properties. Then we introduce its subclass–the class of weakly implicative fuzzy logics. It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for (...) both classes, demonstrating their usefulness and importance. (shrink)

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, (...) for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. (shrink)

Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too.

We prove that the sets of standard tautologies of predicate Product Logic and of predicate Basic Logic, as well as the set of standard-satisfiable formulas of predicate Basic Logic are not arithmetical, thus finding a rather satisfactory solution to three problems proposed by Hájek in [H01].

In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's (...) Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with $\Delta$ , and the Product and Gödel's Logics with $\Delta$ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to $L \Pi$ and $L \Pi \frac{1}{2}$ . For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z. (shrink)

The ŁΠ and logics were introduced by Godo, Esteva and Montagna. These logics extend many other known propositional and predicate logics, including the three mainly investigated ones (Gödel, product and Łukasiewicz logic). The aim of this paper is to show some advances in this field. We will see further reduction of the axiomatic systems for both logics. Then we will see many other logics contained in the ŁΠ family of logics (namely logics induced by the continuous finitely constructed t-norms and (...) Takeuti and Titani's fuzzy predicate logic). (shrink)

Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies and satisfiable sentences as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.