A weak completeness theorem for infinite valued first-order logic

Belluce, L. P.; Chang, C. C.

doi:10.2307/2271335

Cited by 42 publications

(49 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.

View full text Add to dashboard Cite

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.

show abstract

Section: Black Box Theoremsmentioning

confidence: 99%

A proof of completeness for continuous first-order logic

Yaacov

Pedersen²

2010

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…It is obvious that this function is the left m-adjoint to the canonical embedding h : A 0 ,→ A. Then (2) is equivalent to (3). The proof is now complete.…”

Section: M-relatively Complete MV -Algebrasmentioning

confidence: 76%

“…Moreover, it is routine to check that (MLPC) ⊆ QL. For a detailed consideration of Lukasiewicz predicate calculus we refer to [11,16,2,3].…”

Section: Introductionmentioning

confidence: 99%

On monadic MV-algebras

Nola

Grigolia

2004

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

show abstract

“…The axioms of L include all the axioms for Łukasiewicz propositional logic, and we have modus ponens, so we can form the Lindenbaum MV-algebra L. An element from L shall be denoted by [α], with α a formula from L. Proof See ([5, Corollary 10, p.14] and [3]). …”

Section: Definition 30mentioning

confidence: 99%

“…Hence perfect MV-algebras are directly connected with the very important phenomenon of incompleteness in Łukasiewicz first order logic (see [3,12]). …”

mentioning

confidence: 99%