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

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 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. 1 Truth-Values: Many, versus Gaps and Gluts Everyone knows and loves the two “classical” truth-values, True and False. In “classical” logic, every sentence is either T or F but not both. Before getting to the detailed issues of this paper, we pause to make four comments of a “philosophical nature” with the intent of setting the issues aside. First, there is the long-standing issue of whether a logic contains sentences, statements, or propositions. We take no position on this issue: what we think of as the logical point is neutral on this issue. Related to this, perhaps, is the issue of whether a logic should be considered a set of its theorems or defined instead by the valid inferences it accepts. This topic does mark a distinction in the three-valued cases we consider, and we will remark on this as the occasion arises. Second, there are logics – perhaps not “classical” ones – that would prefer to claim the objects of interest are (e.g.) obligations, or commands, or ethical/epistemic items, rather than T and F. Perhaps the items that have such values are not to be considered ‘sentences’ but rather exhortations or promises There are those (e.g., Priest, 2006) who claim that this common terminology of ‘classical’ betrays a bias and is in fact incorrect as an account of the history of thought about logic. We won’t attempt to deal with this, and will just use the common terminology. For a more comprehensive review of topics and issues relevant to truth values see Shramko and Wansing (2017)