Logics with Gaps, Gluts and Worlds

Abstract

Introduction 9.1.1 In this chapter, we will see how the techniques of modal logic and many-valued logic can be combined. More specifically, we will look at logics that add some kind of strict conditional with world semantics on top of a many-valued base-logic, specifically, FDE . 9.1.2 The non-normal worlds of chapter 4 will also make a reappearance, giving us some basic relevant logics. This will allow us to discuss further what, exactly, non-normal worlds are. 9.1.3 We will end the chapter with a brief look at so called logics of constructible negation, which have close connections with intuitionist logic; and an even briefer look at connexive logics. Adding → 9.2.1 FDE has no conditional operator. The material conditional, A ⊃ B , does not even satisfy modus ponens , as we saw in 8.6.5. In any case, as we have seen, using possible-world semantics provides a much more promising approach to the logic of conditional operators. Thus, an obvious thing to do is to build a possible-world semantics on top of the relational semantics of FDE . 9.2.2 To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with K ν , a relational interpretation for such a language is a pair 〈 W , ρ〉, where W is a set of worlds, and for every w ∈ W , ρ w is a relation between propositional parameters and the values 1 and 0.