First Degree Entailment

Abstract

Introduction 22.1.1 The present chapter is devoted to another many-valued logic, one that will lead us into a discussion of relevant logic: First Degree Entailment ( FDE ). 22.1.2 We start with the relational semantics for FDE , and see that this is equivalent to a many-valued semantics. 22.1.3 We will then look at tableaux for quantified FDE , in the process obtaining tableau systems for the 3-valued logics of the last chapter. 22.1.4 A quick look at free logics in the context of relational semantics is next on the agenda. 22.1.5 After that, we move on to the * semantics and tableaux for FDE , and note their equivalence with the relational semantics. 22.1.6 Finally, we will look at the behaviour of identity in both semantics for FDE . 22.1.7 The philosophical issues that tend to be raised by quantification and identity in FDE are much the same as those which we met in connection with the three-valued logics of the last chapter. There is therefore no new philosophical discussion in this chapter. Relational and Many-valued Semantics 22.2.1 An interpretation for quantified FDE is a structure 〈 D , ν〉, where D is the non-empty domain of quantification. For every constant in the language, c , ν( c ) ∈ D , and for every n -place predicate, P , ν( P ) is a pair 〈e, A 〉, where e and A are subsets of D n .