An Unexpected Boolean Connective

We consider a 2-valued non-deterministic connective $${\wedge \!\!\!\!\!\vee }$$ defined by the table resulting from entry-wise union of tables conjunction and disjunction. Being half disjunction we named it platypus. The value is not completely determined input, contrasting with usual notion Boolean connective. call any based on multi-functions over set. In this way, non-determinism allows for an extended truth-functional Unexpectedly, very simple logic defines, illustrate various key advantages in working generalized notions semantics (by incorporating non-determinism), calculi allowing multiple-conclusion rules) even (moving Tarskian to Scottian consequence relations). show that associated cannot be characterized finite set matrices, whereas two values suffice. Furthermore, finitely axiomatizable using single-conclusion rules, however provide analytic axiomatization only rules. Finally, deciding $$\mathbf {coNP}$$ -complete, but its fragment $${\mathbf {P}}$$ .