The disjunction and existence properties for axiomatic systems of truth
type: Item
instance of: scholarly article
title: The disjunction and existence properties for axiomatic systems of truth
zbMATH Open document ID: 0669.03026
DOI: 10.1016/0168-0072(88)90038-3
author: Michael Sheard
author: Harvey M. Friedman
published in: Annals of Pure and Applied Logic
publication date: 1988-01-01T00:00:00Z
review text: This is a sequel to the authors' earlier paper [ibid. 33, 1-21 (1987; Zbl 0634.03058)]. There they considered various extensions of Peano arithmetic enriched by the truth predicate T( ), and accompanying axioms and rules. Here they address themselves to the problem: in which of these extensions do the truth disjunction property and the truth existence property [if \(\vdash \exists x T\#A(x)\) then \(\vdash T\#A(n)\) for some numeral n] hold? Very roughly, their results are as follows. On the negative side, if the system satisfies U-inf[\(\forall x T(A(x))\to T(\forall x A(x))]\) or T-comp[TA\(\vee T(\neg A)]\), then it lacks both properties or T(A) is provable for all A. The positive side: For a system R, let \({\mathfrak M}(R)=<\underset \tilde{} N,\{\#A/R\vdash T(\#A)\}>\). If \(R\subseteq S\) and \({\mathfrak M}(R)\vDash S\), then S has both properties. And so, the authors work with rather simple systems R's (e.g. ``primitive recursive arithmetic'') and complicated S's. In the last section, they comment on the conjecture that the disjunction property implies the existence property. ``This conjecture is far more reasonable than it might at first appear'', they state. Indeed, they proved the corresponding facts for the intuitionistic and modal systems.
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03B30
zbMATH DE Number: 4095444
zbMATH Keywords: truth predicate for extensions of Peano arithmetic
zbMATH Keywords: truth disjunction property
zbMATH Keywords: truth existence property
MaRDI profile type: Publication
full work available at URL: 0168-0072(88)90038-3
OpenAlex ID: W2091412988
cites work: The disjunction property implies the numerical existence property
cites work: An axiomatic approach to self-referential truth
cites work: The equivalence of the disjunction and existence properties for modal arithmetic
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description: scientific article; zbMATH DE number 4095444