A note on three-valued modal logic.
Jørgen B. Jensen, Peter F. Larsen, Edwin J. MacLellan, Peter Κ. Schotch · Notre Dame Journal of Formal Logic · 1978
One of the advantages of the strong completeness result in [1] is that it allows one to extend the usual apparatus for proving completeness of two-valued modal propositional logics to three-valued logics.In the sequel 1 we carry out this programme for two logics, the modal part of which closely resembles the fundamental two-valued normal logic usually called (unfortunately) K. 2 The non-modal part of the logic is, in both cases, the Lukasiewicz three-valued logic (which we call t. 3 ) as axiomatized by Wajsberg.Of course there have been other attempts (by Lukasiewicz e.g.) to construct many-valued modal logics, but almost all of these involve taking a truth-functional view of the modal operators.In the case of a threevalued base logic, this course is almost guaranteed to result in certain theses which upon interpretation are inconsistent with any intuitive reading of the modal operators.This should not be the occasion of despair, however, since precisely the same thing happens if one tries to take a truth-functional approach to modality on a two-valued base.The way out of these difficulties which seems to have enjoyed the best reception in the latter case, has been to abandon truth-functionality and to employ "possible worlds" semantics.Given the success of this strategy, it seems very natural to use it again to do modal logic in a three-valued setting.We must expect some differences, but these turn out to be not so substantial as might be anticipated.3 We employ the terminology of [1], except for some trivial 1. Partially supported by National Research Council of Canada grant # A4085.2. Unfortunately, because this terminology conflicts with that of Sobociήski and others who use K in the names of a family of extensions of S4.3. Especially by those who affect to find three-valued logic impossibly clumsy and lacking in the all-around "niceness" of its two-valued competitor.In this connection see [3], p. 153.