Independence of the Dual Axiom in Modal K with Primitive ◊
Richmond H. Thomason · Notre Dame Journal of Formal Logic · 2017
Explicit axioms relating ˘ and appear to be needed if ˘is taken to be primitive.We prove that such axioms are in fact indispensable. IntroductionBlackburn, de Rijke, and Venema [1] formulated systems of propositional modal logic with ˘as primitive, and with p defined as :˘:p.In axiomatizing these logics, the authors resorted to an axiom that is not needed when is the modal primitive.This is the dual axiom: ˘p $ : :p:The purpose of this article is to show that such an axiom is indispensable: in fact, both ˘p !˘::p and ˘::p !˘p can be invalidated in a modal logic with ˘as primitive and with the usual Boolean axioms, the necessitation rule, and the K axiom.Of course, these axioms cannot be invalidated in Kripke frames, or even in Boolean propositional logic.So the models used in this article are somewhat exotic. Eight-Valued Models for ModalityWe will use many-valued models with eight values.It is best to think of these values as made up out of two 4-element Boolean algebras B and B 0 .See Figure 1 for a picture.The units of the two Boolean algebras, _ and _ 0 , are the only designated values: a formula is valid if it only receives values in ¹_; _ 0 º.Negation is nonstandard.Within B, it is as expected, but the "complement" of an element of B 0 is the complement