A particular notion of an interpretation of a theory over a fuzzy predicate logic in another such theory is discussed. For interpretability with the domain defined by a provably crisp formula, which is of course a syntactical notion, a semantic characterization is established. In the last section, we discuss the question of whether the extension of a decidable theory by a single new axiom is decidable and present an erratum to the paper (Hájek, 2007, Fundamenta informaticae, 81, 155–163).

Avariant of Łukasiewicz logic defined in a remark in White's 1979 paper on the consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz is studied; it is shown that the logic in question is an interesting expansion of Łukasiewicz logic which deserves further investigation together with its further expansions. Several formulas are proved in it and some model theory is offered. In the Appendix A, it is shown that a set theory with full comprehension over this logic (claimed consistent by White) is contradictory.

In this article, the computational complexity of all axiomatic extensions of Łukasiewicz propositional logic Ł and the arithmetical complexity of both the general and standard semantics of their corresponding predicate logics is determined.

Abstract In the last few decades many formal systems of fuzzy logics have been developed. Since the main differences between fuzzy and classical logics lie at the propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this text we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate logics. First, we generalize the completeness theorem, then we use it to get results on conservative extensions of theories and on witnessed models.

In this paper we investigate the falsehood-free fragments of main residuated fuzzy logics related to continuous t-norms (Hájek's Basic fuzzy logic BL and some well-known axiomatic extensions), and we relate them to the varieties of 0-free subreducts of the corresponding algebras. These turn out to be classes of algebraic structures known as hoops. We provide axiomatizations of all these fragments and we call them hoop logics; we prove they are strongly complete with respect to their corresponding classes of hoops, and that each fuzzy logic is a conservative extension of the corresponding hoop logic. Analogously, we also study the falsehood-free fragment of a weaker logic than BL, called MTL, which is the logic of left-continuous t-norms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom (∀3) in these predicate logics is studied. Finally, computational complexity issues of the prepositional logics are also addressed.

Pseudo-BL-algebras were introduced by di Nola, Georgescu and Iorgulescu as a non-commutative generalization of BL-algebras. BL-algebras serve as general semantics of the basic fuzzy logic BL. Here we develop a corresponding generalization psBL (pseudo-basic fuzzy logic) of the logic BL and show the relation of psBL to pseudo-BL-algebras.

We add to Gödel propositional fuzzy logic an unary connective originating in intuitionistic logic and having a natural meaning in fuzzy logic. We prove completeness with respect to a natural many-valued semantics.

Abstract Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying “xis true” and satisfying the “dequotation schema” for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

Abstract Rational Pavelka logic extends Łukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0. 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Łukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

Belief function theory is one of broadly used numerical formalisms for uncertainty processing. It was also studied in its qualitative form. In this paper we study relationship between this qualitative belief function theory and modal logic. We present a sound system of axioms and show its completeness with respect to sematics considered.

Abstract It is shown that the low basis theorem is meaningful and provable in IΣ1 and that the priority-free solution to Post's problem formalizes in this theory.

Abstract An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping ƒ satisfying the following: (1) ƒ associates with each formula A of DPL a sentence ƒ(A) of Peano arithmetic (PA) and with each program α a formula ƒ(α) of PA with one free variable describing formally a supertheory of PA; (2) ƒ commutes with logical connectives; (3) ƒ([α]A) is the sentence saying that ƒ(A) is provable in the theory ƒ(α); (4) for each axiom A of DPL, ƒ(A) is provable in PA (and consequently, for each A provable in DPL, ƒ(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation ƒ, ƒ(A) is provable in PA. Various modifications of this result are considered.

In this paper we are going to consider experimental logics introduced by Jeroslow [4] as models of human reasoning proceeding by trial and error, i.e. admitting changes of axioms in time (some axioms are deleted, some new ones accepted). Jeroslow's notion is based on the idea that events which may cause changes in axioms and rules of reasoning are mechanical. Suppose a finite alphabet Γ to be fixed and let Γ* be the set of words in the alphabet Γ. N denotes the set of natural numbers. 0.1. Definition. An experimental logic is a recursive relation H ⊆ N × Γ*; H(t, φ) is read “the expression is accepted at the point of time t”. φ is recurring w.r.t. H (notation: RecH(φ)) if H(t, φ) holds for infinitely many t; is stable w.r.t. H (notation: StblH(φ)) if H(t,φ) holds for all but finitely many t. In symbols: H is convergent if every recurring expression is stable. 0.2. We have the following facts: Let X ∈ Γ*. (1) X ∈ iff there is an experimental logic H such that X = {φ; RecH(φ)}- (2) X ∈ iff there is an experimental logic H such that X = {φ; StblH(φ)}. (3) X ∈ iff there is a convergent experimental logic H such that X is the set of all expressions recurring ( = stable) w.r.t. H. (See [4], [3], [7]; cf. also [5].)