Introduction

21.1.1 In this chapter we leave world-semantics for the time being, and turn to many-valued logics.

21.1.2 We will start with a brief look at the general situation concerning many-valued logics, before turning to the special cases of the 3-valued logics of chapter 7 for more detailed consideration.

21.1.3 Free versions of these logics are next on the agenda – in particular, now that we have the machinery of truth value gaps at our finger tips, the neutral free logics mentioned in 13.4.7. This will occasion a discussion of the behaviour of the existence predicate in a many-valued logic, and the question of whether it might make good philosophical sense for a statement of existence to have a non-classical value.

21.1.4 Next, we turn to the behaviour of identity in many-valued logics, and particularly the 3-valued logics of chapter 7. This will occasion a discussion of whether identity statements may plausibly be taken to have non-classical values.

21.1.5 We will finish with a few comments on supervaluations and subvaluations in the context of quantificational logic.

Quantified Many-valued Logics

21.2.1 As we saw in 7.2.2, a propositional many-valued logic is characterised by a structure 〈V,D,{fc : c ∈ C}〉, where V is the set of truth values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes.