Axiomatization of the infinite-valued predicate calculus

Hay, Louise

doi:10.2307/2271339

Cited by 61 publications

(30 citation statements)

References 4 publications

(5 reference statements)

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“…Nonetheless, Fact 4.2 has been proved in [BC63] (indeed, for Lukasiewicz first-order logic; see also [Háj98]) and Fact 4.4 has a counterpart in [Hay63] (again, see also [Háj98]), while Fact 4.5 has been proved for rational Pavelka propositional logic [Háj98] and for Lukasiewicz propositional logic [CDM00, Háj98].…”

Section: Black Box Theoremsmentioning

confidence: 99%

A proof of completeness for continuous first-order logic

Yaacov

Pedersen²

2010

J. symb. log.

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Abstract. 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 arbitrary 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 is uniformly recursive from ϕ, where ϕ • T 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.

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Section: Black Box Theoremsmentioning

confidence: 99%

A proof of completeness for continuous first-order logic

Yaacov

Pedersen²

2010

J. symb. log.

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“…By contrast, Łukasiewicz's infinitely-valued logic Ł ∞ as is commonly understood (i.e., defined syntactically by taking the tautologies as axioms and Modus Ponens as inference rule) is not one of the logics we have considered, and does not fall under our group of examples. By Hay's theorem [25] it coincides with Ł [0,1] on finite sets, but Ł ∞ = Ł [0,1] since the latter is not finitary.…”

Section: Lemma 40mentioning

confidence: 99%

Leibniz filters and the strong version of a protoalgebraic logic

Font¹,

Jansana²

2001

Arch. Math. Logic

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A filter of a sentential logic S is Leibniz when it is the smallest one among all the S-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic S + defined by the class of all S-matrices whose filter is Leibniz, which is called the strong version of S, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of S + and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from S to S + . For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics.

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“…Basic operations in Fuzzy Logic apply to fuzzy sets include negation, intersection, union, and implication. Today, in the broader sense, Fuzzy Logic is actually a family of fuzzy operations [35] [13] divided into different classes, among which, the most widely known include Zadeh Logic [37], Lukasiewicz Logic [16], Product Logic [13], Gödel Logic [10,1], and Yager Logic [36]. For example, in Zadeh Logic, the membership function of the union of two fuzzy sets is defined as the maximum of the two membership functions for the two fuzzy sets (the maximum criterion); the membership function of the intersection of two fuzzy sets is defined as the minimum of the two membership functions (the minimum criterion); while the membership function of the complement of a fuzzy set is defined as the negation of the specified membership function (the negation criterion).…”

Section: Encoding Uncertainty In Rifmentioning

confidence: 99%

A Fuzzy Logic-Based Approach to Uncertainty Treatment in the Rule Interchange Format: From Encoding to Extension

Zhao

Boley

Jing

2013

Uncertainty Reasoning for the Semantic Web II

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Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n'arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. NRC Publications Record / Notice d'Archives des publications de CNRC:http://nparc.cisti-icist.nrc-cnrc.gc.ca/eng/view/object/?id=d016e61a-7236-421e-8ee1-60d96cc1008d http://nparc.cisti-icist.nrc-cnrc.gc.ca/fra/voir/objet/?id=d016e61a-7236-421e-8ee1-60d96cc1008d Abstract. The Rule Interchange Format (RIF) is a W3C recommendation that allows rules to be exchanged between rule systems. Uncertainty is an intrinsic feature of real world knowledge, hence it is important to take it into account when building logic rule formalisms. However, the set of truth values in the RIF Basic Logic Dialect (RIF-BLD) currently consists of only two values (t and f), although the RIF Framework for Logic Dialects (RIF-FLD) allows for more. In this paper, we first present two techniques of encoding uncertain knowledge and its fuzzy semantics in RIF-BLD presentation syntax. We then propose an extension leading to an Uncertainty Rule Dialect (RIF-URD) to support a direct representation of uncertain knowledge. In addition, rules in Logic Programs (LP) are often used in combination with the other widely-used knowledge representation formalism of the Semantic Web, namely Description Logics (DL), in many application scenarios of the Semantic Web. To prepare DL as well as LP extensions, we present a fuzzy extension to Description Logic Programs (DLP), called Fuzzy DLP, and discuss its mapping to RIF. Such a formalism not only combines DL with LP, as in DLP, but also supports uncertain knowledge representation. 4

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Axiomatization of the infinite-valued predicate calculus

Cited by 61 publications

References 4 publications

A proof of completeness for continuous first-order logic

A proof of completeness for continuous first-order logic

Leibniz filters and the strong version of a protoalgebraic logic

A Fuzzy Logic-Based Approach to Uncertainty Treatment in the Rule Interchange Format: From Encoding to Extension

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