Umberto Rivieccio

We study an array of logics defined on a small set of connectives (including an implication $$\rightarrow $$ and a bottom particle $$\bot $$ ) by modularly considering subsets of a set of inference rules that we fix at the start of the game. We provide complete semantics based on possibly non-deterministic logical matrices and complexity upper bounds for the considered logics. As a consequence of the techniques applied, we also obtain completeness results for the negation-only fragments (obtained by defining the negation connective as $$\lnot p:=p\rightarrow \bot $$, as usual) of all the above-mentioned logics, and analyze their possible paraconsistent character.

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable.

The Belnap–Dunn logic is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view ofAlgebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.