Pecularities of Some Three- and Four-Valued Second Order Logics

Logics that have many truth values—more than just True and False —have been argued to be useful in the analysis of very many philosophical and linguistic puzzles (as well, sometimes, in various computational-oriented tasks). In this paper, which is a followup to (Hazen and Pelletier in K3, Ł3, LP, RM3, A3, FDE, M: How to make many-valued logics work for you. Winning paper for the Canadian Schotch-Jennings Prize, one of the prizes of the Universal Logic competition in 2018 ; Notre Dame J Form Log 59, 2018 ), we will start with a particularly well-motivated four-valued logic that has been studied mainly in its propositional and first-order versions. And we will then investigate its second-order version. This four-valued logic has two natural three-valued extensions: what is called a “gap logic” (some formulas are neither True nor False ), and what is called a “glut logic” (some formulas are both True and False ). We mention various results about the second-order version of these logics as well. And we then follow our earlier papers, where we had added a specific conditional connective to the three valued logics, and now add that connective to the four-valued logic under consideration. We then show that, although this addition is “conservative” in the sense that no new theorems are generated in the four-valued logic unless they employ this new conditional in their statement, nevertheless the resulting second-order versions of these logics with and without the conditional are quite different in important ways. We close with a moral for logical investigations in this realm.