A fixed point theorem for the weak Kleene valuation scheme
type: Item
instance of: scholarly article
title: A fixed point theorem for the weak Kleene valuation scheme
zbMATH Open document ID: 0543.03006
DOI: 10.1007/BF00453018
author: Robert L. Martin
author: Anil K. Gupta
published in: Journal of Philosophical Logic
publication date: 1984-01-01T00:00:00Z
review text: Tarski's theorem shows that no language with a classical interpretation can contain its own truth predicate. \textit{S. Kripke} [''Outline of a theory of truth'', J. Philos. 72, 690-716 (1975)] and others, have shown that if its interpretation allows for truth-value gaps, a language may contain a predicate whose extension is exactly the set of sentences true in the interpretation. This paper shows that if, in addition, the interpretation is a weak Kleene interpretation (distinguished, essentially, by the condition that a formula is truth-valueless in an interpretation iff some variant interpretation makes a sub-formula truth- valueless) then the language may also contain a predicate whose extension is exactly the set of sentences that are neither true nor false. However, even for such interpretations there will be important notions not, in general, definable in the language. For example, if there were a formula of one free variable, \(\phi\) (x), whose extension is exactly the complement of the set of true sentences, then, given sufficient diagonalising machinery, we could construct a sentence \(\phi\) (a), denoted by ''a''. The usual liar-paradox contradiction would then be forthcoming.
reviewed by: Graham Priest
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03C50
zbMATH DE Number: 3863165
zbMATH Keywords: expressive power
zbMATH Keywords: truth predicate
zbMATH Keywords: truth-value gaps
zbMATH Keywords: weak Kleene interpretation
zbMATH Keywords: liar-paradox
MaRDI profile type: Publication
cites work: Q3309807
cites work: Truth and paradox
cites work: Outline of a Theory of Truth
cites work: Q3760496
full work available at URL: https://doi.org/10.1007/bf00453018
OpenAlex ID: W1979831570
description: scientific article; zbMATH DE number 3863165