Many-valued Logics

Abstract

Introduction 7.1.1 In this chapter, we leave possible-world semantics for a time, and turn to the subject of propositional many-valued logics. These are logics in which there are more than two truth values. 7.1.2 We have a look at the general structure of a many-valued logic, and some simple but important examples of many-valued logics. The treatment will be purely semantic: we do not look at tableaux for the logics, nor at any other form of proof procedure. Tableaux for some many-valued logics will emerge in the next chapter. 7.1.3 We also look at some of the philosophical issues that have motivated many-valued logics, how many-valuedness affects the issue of the conditional, and a few other noteworthy issues. Many-valued Logic: The General Structure 7.2.1 Let us start with the general structure of a many-valued logic. To simplify things, we take, henceforth, A ≡ B to be defined as ( A ⊃ B ) ∧ ( B ⊃ A ). 7.2.2 Let C be the class of connectives of classical propositional logic {∧, ∨, ¬, ⊃}. The classical propositional calculus can be thought of as defined by the structure 〈 V, D, {f c ; c ∈ C} 〉. V is the set of truth values {1, 0}. D is the set of designated values {1}; these are the values that are preserved in valid inferences. For every connective, c, f c is the truth function it denotes.