Introduction

4.1.1 In this chapter we look at some systems of modal logic weaker than K (and so non-normal). These involve so-called non-normal worlds. Nonnormal worlds are worlds where the truth conditions of modal operators are different.

4.1.2 We are then in a position to return to the issue of the conditional, and have a look at an account of a modal conditional called the strict conditional.

Non-normal Worlds

4.2.1 Let us start by looking at the technicalities concerning non-normality. In due course we will be able to discuss what they mean.

4.2.2 A non-normal interpretation of a modal propositional language is a structure, 〈W, N, R, ν〉, where W, R and ν are as in previous chapters, and N ⊆ W. Worlds in N are called normal. Worlds in W – N (the worlds that are not normal) are called non-normal.

4.2.3 The truth conditions for the truth functions, ∧, ∨, ¬, etc. are the same as before (2.3.4). The truth conditions for □ and ◇ at normal worlds are also as before (2.3.5). But if w is non-normal:

1. νw(□A) = 0

2. νw(◇A) = 1

In a sense, at non-normal worlds, everything is possible, and nothing is necessary.

4.2.4 Note that at every world, w, ¬□A and ◇¬A still have the same truth value, as do ¬◇A and □¬A. We saw this to be the case for normal worlds in 2.3.9 and 2.3.10.