25 - Fuzzy Logics

Summary

Introduction

25.1.1 In this chapter, we will look at the addition of quantifiers to the Łukasiewicz continuum-valued logic.

25.1.2 We will then look at the behaviour of identity in this logic.

25.1.3 This will occasion a discussion of some philosophical issues concerning fuzzy identity, connected, in particular, with the sorites paradox and with vague objects.

25.1.4 A technical appendix describes the addition of quantifiers and identity to the general class of t-norm logics.

Quantified Łukasiewicz Logic

25.2.1 In the language we are concerned with, the set of connectives, C, is {∧, ∨, ¬, →}, and the set of quantifiers, Q, is {∀, ∃}. (A ↔ B can be taken as defined as (A → B) ∧ (B → A).)

25.2.2 As we saw in 21.2, an interpretation for a quantified many-valued logic is a structure 〈D, V, D, {f_c : c ∈ C}, {f_q : q ∈ Q}, ν〉. D is a non-empty domain of quantification. For every constant, c, ν(c) ∈ D, and for every n-place predicate, P, ν(P) is an n-place function that maps members of D into the truth values, V. In Łukasiewicz continuum-valued logic, V = [0, 1], the set of real numbers between 0 and 1, ordered in the usual way. f_∨, f_∧, f_¬ and f_→ are as in the propositional case (11.4.2).