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 〈ε, A〉, where ε and A are subsets of Dn.