We explore the philosophical, algebraic, and logical details of a 4-element algebra introduced by Humberstone in an article from 2003, dubbed by him an experiment in logic. There, a discussion takes place regarding the prospects of a framework to handle true and false propositions, though sometimes providing the qualification that they are either partly false or partly true, in a specific sense. Our goals are (i) to locate his philosophical developments in perspective within the literature on partial truth and its connected topics, (ii) to syntactically characterize the valid identities within the target algebra and link them to certain variable sharing structures, (iii) to scrutinze and describe the different notions of logical validity available to someone deploying said algebra in a logical matrix, relating them to certain logics of analytic containment.
This article investigates the thesis—famously associated with Wittgenstein—that contradictions and tautologies are meaningless or about nothing, together with the assumption that such expressions behave infectiously, transmitting their lack of significance to any compound in which they occur. Taking logical validity to be modeled as content inclusion---dually understood as encompassing both truth preservation and subject-matter preservation---we proceed by examining alethic structures, using the two-element Boolean algebra as our case study, alongside topical structures, focusing in particular on the 3-element involutive semilattice, whose operations provide a topical mereology well suited to our aims. Within this framework, negation always changes the topic of a formula and, more strikingly, can also function as a topic-cancellative operator: the combination of a proposition with its negation within a complex formula results in the cancellation of their subject matters, yielding a topically void expression that we call a bipolar formula. Building on these observations, we introduce what we term the bipolarly balanced right companion of classical logic and provide a syntactic characterization of logical consequence within this system.
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.
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.