The aim of this article is to examine the class of Gentzen-style sequent calculi where Cut is admissible but not derivable that prove all the (finite) inferences that are usually taken to characterize Classical Logic---conceived with conjunctively-read multiple premises and disjunctively-read multiple conclusions. We'll do this starting from two different calculi, both counting with Identity and the Weakening rules in unrestricted form. First, we'll start with the usual introduction rules and consider what expansions thereof are appropriate. Second, we'll start with the usual elimination or inverted rules and consider what expansions or subsets thereof are appropriate. Expansions, in each case, may or may not consist of additional standard or non-standard introduction or elimination rules, as well as of restricted forms of Cut.
We explore certain algebraic structures that naturally emerge within the framework of logics of synonymy, analytic containment, and hyperintensionality. In particular, we argue that Angell's logic AC, one of the earliest and most successful attempts to analyse the properties of logical constants with a topic-transformative character, can be better understood through a direct algebraic study of De Morgan bisemilattices. Inter alia, we show that a certain 9-element algebra introduced by Ferguson generates De Morgan bisemilattices as a quasivariety, making it the most adequate semantics for AC, as opposed to other 7-element and 16-element algebras considered in the literature.
Our goal is to characterize the regularization, conditionally balanced regularization, and balanced regularization of De Morgan lattices and some of their subvarieties, namely, Kleene lattices and Boolean algebras. For this purpose, (i) we study some of their finite generators, (ii) we provide representations of these varieties within the theory of Plonka sums and De Morgan Plonka sums (recently introduced by Randriamahazaka), (iii) we present axiomatizations for these systems relative to each other, (iv) we characterize the lattice of covers of these varieties in terms of the direct product of the lattice of subvarieties of the original classes and the 4-element distributive lattice.