Introduction

18.1.1 The techniques concerning quantification and identity in normal modal logics carry over in a natural way to other logics which have possible world semantics. In this chapter we will look at one of these, non-normal modal logics.

18.1.2 We will ignore identity to start with, and look at the constant and variable domain versions of non-normal modal logics (without descriptors).

18.1.3 We will then look at the addition of identity to these logics.

18.1.4 Non-normal worlds are important since, being worlds where logical truths may fail (as we saw in 4.4.7), they are harbingers of the impossible worlds of relevant logics (9.7). But the addition of quantifiers and identity to non-normal worlds appears to raise no novel philosophical issues. There is therefore no philosophical discussion in this chapter.

Non-normal Modal Logics and Matrices

18.2.1 In a non-normal modal logic, formulas of the form □A and ◇A are assigned truth values at non-normal worlds in a way that does not depend on the value of A. When quantification is involved, employing this strategy in the simple-minded way may cause a problem. Most obviously, □Pa and □Pb may be assigned different values at a world, even though a = b is true there. (More generally, the Denotation Lemma, which is integral to the correct functioning of quantifiers, breaks down.)

18.2.2 To overcome this problem, we have to treat formulas of the form □A and ◇A, with n free variables, effectively as n-place predicates.