K3, L3, LP, RM3, A3, FDE: How to Make Many-Valued Logics Work for You

TL;DR

This paper explores various three- and four-valued logics, enhancing their usefulness through added features, and reveals surprising results and relationships among these logics, including a natural deduction system for augmented FDE.

Contribution

It introduces modifications to well-known many-valued logics to improve their logical utility and presents new insights into the relationships and properties of these systems.

Findings

Augmented FDE shows surprising logical properties.

Certain many-valued logics can be made more useful with added features.

A natural deduction system for augmented FDE is developed and proven sound and complete.

Abstract

We investigate some well-known (and a few not-so-well-known) many-valued logics that have a small number (3 or 4) of truth values. For some of them we complain that they do not have any \emph{logical} use (despite their perhaps having some intuitive semantic interest) and we look at ways to add features so as to make them useful, while retaining their intuitive appeal. At the end, we show some surprising results in the system FDE, and its relationships with features of other logics. We close with some new examples of "synonymous logics." An Appendix contains a natural deduction system for our augmented FDE, and proofs of soundness and completeness.