
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 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 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…
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