The disjunction and existence properties for axiomatic systems of truth
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.
truth predicate for extensions of Peano arithmetic
scientific article; zbMATH DE number 4095444