The disjunction and existence properties for axiomatic systems of truth (Q1118582)

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.