1,721,377 research outputs found
Lindström theorems in graded model theory
Full text linkStemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the Löwenheim–Skolem property, the Tarski union property, and the Robinson property, among others. As necessary technical restrictions, we assume that the models are valued on finite MTL-chains and the language has a constant for each truth-value
Arithmetical Complexity of First-Order Predicate Logics over Distinguished Semantics
No full textAll promiment examples of first-order predicate fuzzy logics are undecidable. This leads to the problem of the arithmetical complexity of their sets of tautologies and satisfiable sentences. This article is a contribution to the general study of this problem. We propose the classes of first-order core and Delta-core fuzzy logics as a good framework to address these arithmetical complexity issues. We obtain general results providing lower bounds for the complexities associated with arbitrary semantics, and we compute upper bounds and exact positions in the arithmetical hierarchy for distinguished semantics: general semantics given by all chains, finite-chain semantics, standard semantics and rational semantics
Logic and Implication: An Introduction to the General Algebraic Study of Non-Classical Logics
No full textThis monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to (abstract) algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions.
Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics.
The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics
Fraïssé classes of graded relational structures
Full text linkWe study classes of graded structures satisfying the properties of amalgamation, joint embedding and hereditariness. Given appropriate conditions, we can build a graded analogue of the Fraïssé limit. Some examples such as the class of all finite weighted graphs or the class of all finite fuzzy orders (evaluated on a particular countable algebra) will be examined
A Note on Natural Extensions in Abstract Algebraic Logic
No full textTransfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider setting (including all finitary logics and those with countable language) in which they can still be used
A General Omitting Types Theorem in Mathematical Fuzzy Logic
No full textThis article is a contribution to the theoretical study of weighted structures in fuzzy logic. We consider an important item from classical model theory: the construction of models that do not have any collection satisfying certain prescribed properties, that is, an omitting types theorem. We generalize the work done by Cintula and Diaconescu (Omitting Types Theorem for Fuzzy Logics, IEEE Transactions on Fuzzy Systems 27(2):273-277, 2019), who solved the problem for standard one-sided types. Instead, we introduce types for fuzzy structures as pairs of sets of formulas with free variables (expressing, respectively, properties to be satisfied and those to be avoided) and prove the corresponding omitting types theorem in the framework of uninorm-based logics
The Proof by Cases Property and its Variants in Structural Consequence Relations
No full textThis paper is a contribution to the study of the rôle of disjunction in Abstract Algebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic. © 2013 Springer Science+Business Media Dordrecht
Neighborhood semantics for modal many-valued logics
Full text linkThe majority of works on modal many-valued logics consider Kripke-style possible worlds frames as the principal semantics despite their well-known axiomatizability issues when considering non-Boolean accessibility relations. The present work explores a more general semantical picture, namely a many-valued version of the classical neighborhood semantics. We present it in two levels of generality. First, we work with modal languages containing only the two usual unary modalities, define neighborhood frames over algebras of the logic FLew with operators, and show their relation with the usual Kripke semantics (this is actually the highest level of generality where one can give a straightforward definition of the Kripke-style semantics). Second, we define generalized neighborhood frames for arbitrary modal languages over a given class of algebras for an arbitrary protoalgebraic logic and, assuming certain additional conditions, axiomatize the logic of all such frames (which generalizes the completeness theorem of the classical modal logic E with respect to classical neighborhood frames)
Extension Properties and Subdirect Representation in Abstract Algebraic Logic
No full textThis paper continues the investigation, started in Lávička and Noguera (Stud Log 105(3): 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, (completely) intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics
A 0-1 Law in Mathematical Fuzzy Logic
No full textThis paper continues the theoretical study of weighted structures in mathematical fuzzy logic focusing on the finite model theory of fuzzy logics valued on arbitrary finite MTL-chains. We show that for any first-order (or infinitary with finitely many variables) formula phi, there is a unique truth-value that phi takes almost surely in every finite many-valued model and such that every other truth-value is almost surely not taken. This generalizes a theorem in the fuzzy setting due to Robert Kosik and Christian G. Fermuller
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