Non-Reflexive Nonsense: Proof Theory of Paracomplete Weak Kleene Logic

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic \(\textbf{K}_{\textbf{3}}^{\textbf{w}}\) ${\mathbf{K}}_{\mathbf{3}}^{\mathbf{w}}$ by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules.