The disjunction and existence properties for axiomatic systems of truth

Abstract In a language for arithmetic with a predicate T ( x ), intended to mean “ x is the Godel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T ( \s#A ) ∨ T ( \s#B ), either S ⊢ T ( \s#A ) or S ⊢ T ( \s#B ). Similarly, S has the truth existence property if whenever S ⊢ ∃χ T ( \s#A(χ )), there is some n such that S ⊢ T ( \s#A ( n )). Continuing previous work, we establish whether these properties hold or fail for a large collection of possible axiomatic systems.