Corrigendum to "The logic of tied implications, part 2: Syntax" [Fuzzy Sets and Systems 157(2006)2030-2057]

This derivation of (2) is not valid, because this duality applies to inferences and theorems only, not to definitions. In fact, (1) should be treated as an axiom. Accordingly, the duality of Lemma 3.3.1 would not be valid until the dual (2) of (1) is proved independently, as we do below. Our new proof will make use of Proposition 3.3.3, the original proof of which is based on duality. So, we must begin by supplying an alternative proof for that proposition. Note that in the following two proofs, none of the quoted inferences and theorems was based on the duality of Lemma 3.3.1.