Some anomalies in Fitch's system QD

Abstract

This note shows that the argument used in the proof of the inconsistency of Curry's system (see [1]) can also be applied to Fitch's system QD (see [3, Chapter 6]). As one vital rule of is not present in QD this argument does not lead to an actual contradiction, but it does lead to a theorem which is not a proposition if the system is consistent.The method used below is that of [2], which is simpler than that of [1].The axioms and rules of QD that we require are also present in [1] and [2] provided that Fitch's D is replaced by H. These axioms and rules are the following:DD int: ⊦ D(Da),m p: If ⊦ a ⊃ b2 and ⊦ a then ⊦ b, andres imp int: If a ⊦ b then Da ⊦ a ⊃ b.Let a be arbitrary and let G be [x](Dx ⊃ (x ⊃ a)). Then let X be BWBG(BWBG). We then have the following proof:Thus, we have ⊦ X. Now suppose we also had ⊦ DX. Then by the method of the innermost subproof in the above proof, we would have ⊦ a, and since a isarbitrary the system would be inconsistent. Hence, if QD is consistent, we do not have ⊦ DX, and so X is a theorem which is not a proposition.