Erratum to "The Ricean Objection: An Analogue of Rice's Theorem for First-Order Theories" . Logic Journal of the IGPL, 16(6): 585-590(2008)

There is an error in the proof of the main theorem on page 6 (''Rice's Theorem for Logic''). If φ is undecidable in Q, the claim that Ax1 and Ax2 are tautologies is not valid, and this fact compromises the rest of the proof. Furthermore, the following decidable property is a counterexample to our result: Definition 1. Let T be a theory. A theory T + is an extension of T if Th(T) ⊆ Th(T +). Definition 2. Sub(T) denotes the set {Th(T −) : T is an extension of T − }. Proposition 1. If T is a consistent decidable theory, then Sub(T) is a non-trivial decidable property. Proof. By the consistency of T , Sub(T) is non-trivial. For each finite A ⊆ L , the following holds: Th(A) ∈ Sub(T) ⇐⇒ T is an extension ofA ⇐⇒ T φ∈A φ. Since T is decidable, there exists an algorithm which decides for given finite A ⊆ L whether T φ∈A φ or not. That is, Sub(T) is decidable. For instance, Sub({∀ x ∀ y (x = y)}) is a counterexample to our result. This negative result, which is contrary to our initial intuition, led us to consider the existence of these ''Ricean'' undecidability results in a more general sense.