Notes on the Model Theory of DeMorgan Logics

We make preliminary investigations into the model theory of DeMorgan logics, attempting to make a case for the worth of such investigations before tackling the plight of particular mathematical theorems in these logics. We demonstrate that Loś’ Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor’s theorem that the theory of dense linear orderings without endpoints (DLO−−) is א0-categorical and show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, Graham Priest’s Collapsing Lemma and J. Michael Dunn’s Theorem in 3-Valued Logic. Semantics for DeMorgan Logic We may suppose that the fundamental component to a logic λ is the relation λ that holds between sets of formulae and sets of formulae, indicating that the latter is derivable from the former. As each logic λ that we will be invoking is sound and complete, we may consider the relation λ associated with each λ and define it semantically. In so doing, we will sufficiently define the logic itself. The logics upon which we herein focus are the classical predicate calculus CL, the paraconsistent (inconsistency-tolerant) logics LP and RM, the paracomplete (incompleteness-tolerant) logics K and L, and the paraconsistent and paracomplete logic FDE. For a discussion of these logics’ origins and philosophical motivation, we refer the reader to [8]. These logics may be thought of as, to extend the nomenclature of [2], DeMorgan logics, insofar as for each logic λ in this class the DeMorgan Laws hold. This motivates our referencing the