-/- Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? -/- The primary purpose (...) of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ. -/- Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable. (shrink)

The last decade has seen an enormous development in infinite-valued systems and in particular in such systems which have become known as mathematical fuzzy logics. The paper discusses the mathematical background for the interest in such systems of mathematical fuzzy logics, as well as the most important ones of them. It concentrates on the propositional cases, and mentions the first-order systems more superficially. The main ideas, however, become clear already in this restricted setting.

This paper continues the investigation, started in Lávička and Noguera : 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, 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. (shrink)

Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus.

Rational Pavelka logic extends Lukasiewicz 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 Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules (...) in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal. (shrink)

We show that it is possible to base fuzzy control on fuzzy logic programming. Indeed, we observe that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs and we show that the fuzzy function associated with a fuzzy system of IF-THEN rules is the fuzzy Herbrand interpretation associated with a suitable fuzzy program.

An algebraic setting for the validity of Pavelka style completeness for some natural expansions of Łukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard MV-algebra on the real segment [0, 1] is an injective MV-algebra. In particular the logics associated with MV-algebras with product and with divisible MV-algebras are considered.

We describe an infinitary logic for metric structures which is analogous to Lω1,ω. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.

We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this (...) logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants. (shrink)

We study a class of [0,1][0,1]-valued logics. The main result of the paper is a maximality theorem that characterizes these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.

The paper studies closure properties of classes of fuzzy structures defined by fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and Horn theories by means of closedness under suitable algebraic constructions.

We present a logic for reasoning about attribute dependencies in data involving degrees such as a degree to which an object is red or a degree to which two objects are similar. The dependencies are of the form A ⇒ B and can be interpreted in two ways: first, in data tables with entries representing degrees to which objects have attributes ; second, in database tables where each domain is equipped with a similarity relation. We assume that the degrees form (...) a scale equipped with operations representing many-valued logical connectives. If 0 and 1 are the only degrees, the algebra of degrees becomes the two-element Boolean algebra and the two interpretations become well-known dependencies in Boolean data and functional dependencies of relational databases. In a setting with general scales, we obtain a new kind of dependencies with naturally arising degrees of validity, degrees of entailment, and related logical concepts. The deduction rules of the proposed logic are inspired by Armstrong rules and make it possible to infer dependencies to degrees—the degrees of provability. We provide a soundness and completeness theorem for such a setting asserting that degrees of entailment coincide with degrees of provability, prove the independence of deduction rules, and present further observations. (shrink)

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.

We study a predicate extension of an unbounded real valued propositional logic that has been recently introduced. The latter, in turn, can be regarded as an extension of both the abelian logic and of the propositional continuous logic. Among other results, we prove that our predicate extension satisfies the property of weak completeness (the equivalence between satisfiability and consistency) and, under an additional assumption on the set of premisses, the property of strong completeness (the equivalence between logical consequence and provability). (...) Eventually we discuss some topological properties of the space of types in our logic. (shrink)