An unexpected Boolean connective

TL;DR

This paper introduces a novel non-deterministic Boolean connective called 'platypus', demonstrating its advantages in generalized semantics, calculi, and logic, with significant results on axiomatization and computational complexity.

Contribution

It defines the 'platypus' connective, explores its logical properties, and shows how non-determinism enhances the expressiveness and axiomatizability of Boolean logics.

Findings

The logic cannot be characterized by any finite set of finite matrices.

A simple two-rule multiple-conclusion axiomatization exists.

Deciding the logic is coNP-complete; single-conclusion fragment is in P.

Abstract

We consider a 2-valued non-deterministic connective $\wedge \hskip-5.5pt \vee$ defined by the table resulting from the entry-wise union of the tables of conjunction and disjunction. Being half conjunction and half disjunction we named it platypus. The value of $\wedge \hskip-5.5pt \vee$ is not completely determined by the input, contrasting with usual notion of Boolean connective. We call non-deterministic Boolean connective any connective based on multi-functions over the Boolean set. In this way, non-determinism allows for an extended notion of truth-functional connective. Unexpectedly, this very simple connective and the logic it defines, illustrate various key advantages in working with generalized notions of semantics (by incorporating non-determinism), calculi (by allowing multiple-conclusion rules) and even of logic (moving from Tarskian to Scottian consequence relations). We show that the associated logic cannot be characterized by any finite set of finite matrices, whereas with non-determinism two values suffice. Furthermore, this logic is not finitely axiomatizable using single-conclusion rules, however we provide a very simple analytical multiple-conclusion axiomatization using only two rules. Finally, deciding the associated multiple-conclusion logic is coNP-complete, but deciding its single-conclusion fragment is in P.