Hilbert's \(\varepsilon{}\)-operator and classical logic
type: Item
instance of: scholarly article
title: Hilbert's \(\varepsilon{}\)-operator and classical logic
zbMATH Open document ID: 0784.03008
DOI: 10.1007/BF01049178
published in: Journal of Philosophical Logic
publication date: 1993-05-16T00:00:00Z
review text: In the framework of the first-order language the \(\varepsilon\)-axiom corresponding to the famous Hilbert's \(\varepsilon\)-operator can be formulated as follows: \(A(x)\to A(\varepsilon_ A)\), where \(A(t)\) is obtained as the result of substituting term \(t\) for each free occurrence of \(x\) in \(A(x)\). Namely, \(\varepsilon_ A\) can be considered as an ideal object associated with \(A\) such that all one knows about it is that, if anything satisfies \(A\), it does. Logical relationships of the \(\varepsilon\)-axiom with some basic superintuitionistic axioms are analyzed and the roles of the extensionality principle and principle of bivalence are discussed. An algebraic type semantics for the extension of the Heyting first-order predicate calculus by the \(\varepsilon\)-axiom is described with the corresponding soundness theorem, but the provision of a complete semantics for this calculus remains an open problem.
Mathematics Subject Classification ID: 03B10
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03-03
zbMATH DE Number: 166038
zbMATH Keywords: \(\varepsilon\)-axiom
zbMATH Keywords: Hilbert's \(\varepsilon\)-operator
zbMATH Keywords: superintuitionistic axioms
zbMATH Keywords: extensionality principle
zbMATH Keywords: principle of bivalence
zbMATH Keywords: Heyting first-order predicate calculus
reviewed by: Branislav Boričić
MaRDI profile type: Publication
cites work: Q3679172
cites work: Q4133603
cites work: Q5582304
cites work: Q3960633
cites work: Q5734410
cites work: Q5573940
full work available at URL: https://doi.org/10.1007/bf01049178
OpenAlex ID: W2000295225
author: John L. Bell
description: scientific article; zbMATH DE number 166038