Kamide is in America, Moisil and Leitgeb are in Australia

TL;DR

This paper explores different semantic interpretations of the expanded Belnap-Dunn logic (BD+) in an intuitionistic setting, comparing American and Australian approaches and their implications for related logical systems.

Contribution

It formulates a new Australian-style semantics for first-order BD+ and analyzes its connections with other logical systems, highlighting the divergence from the American approach.

Findings

Developed an Australian semantics for first-order BD+

Connected BDi with intermediate and Nelson-like logics

Clarified differences between American and Australian interpretations

Abstract

It is not uncommon for a logic to be invented multiple times, hinting at its robustness. This trend is followed also by the expansion BD+ of Belnap-Dunn logic by Boolean negation. Ending up in the same logic, however, does not mean that the semantic interpretations are always the same as well. In particular, different interpretations can bring us to different logics, once the basic setting is moved from a classical one to an intuitionistic one. For BD+, two such paths seem to have been taken; one (BDi) by N. Kamide along the so-called American plan, and another (HYPE) by G. Moisil and H. Leitgeb along the so-called Australian plan. The aim of this paper is to better understand this divergence. This task is approached mainly by (i) formulating a semantics for first-order BD+ that provides an Australian view of the system; (ii) showing connections of the less explored (first-order) BDi with neighbouring systems, including an intermediate logic and variants of Nelson's logics.