In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standard tautologies coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies iff for any transfinite sequence of elements of B, the condition ∧i ∈ I = 2 holds in B.

This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective _N_ satisfying nucleus property, called here substructural _nuclear_ logics, and its subclass, called here substructural _nuclear image-based_ logics, where _N_ further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce _operational Kripke-style_ semantics for those logics and provide two sorts of completeness results for (...) them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based logics. (shrink)