Formal fuzzy logic has developed into an extensive, rigorous, and exciting discipline since it was first proposed by Joseph Goguen and Lotfi Zadeh in the midtwentieth century, and it is a wonderful topic for introducing students to the richness and fascination of formal logic and the philosophy thereof. This textbook grew out of an interdisciplinary course on fuzzy logic that I've taught at Smith College, a course that attracts philosophy, computer science, and mathematics majors. I taught the course for several years with only a course reader because the few existing texts devoted to fuzzy logic were too advanced for my undergraduate audience (and probably for some graduate audiences as well). Finally, after writing voluminous supplements for the course, I decided to write an accessible introductory textbook on many-valued and fuzzy logic. It is my hope that after working through this textbook, students will have the necessary background to tackle more advanced texts, such as Gottwald (2001), Hájek (1998b), and Novák, Perfilieva, and Močkoř (1999), along with the rest of the vast fuzzy literature.

This book opens with a discussion of the philosophical issues that give rise to fuzzy logic – problems and paradoxes arising from vague language – and returns to those issues as new logical systems are presented. There is a two-chapter review of classical logic to familiarize students and instructors with my terminology and notation, and to introduce formal logic to those who have no prior background.

It's time to face two problems that we sidestepped while exploring three-valued logical systems for vagueness.

Although the Sorites argument is valid in all of the systems we've presented, we claimed that the paradox can nevertheless be dissolved in three-valued logic because the Principle of Charity premise is not true on any reasonable interpretation. The first problem concerns the exact nature of the principle's nontruth. Our sample interpretations rendered the premise false in Bochvar's external system, which didn't sound right because its negation – which states that 1/8″ does make a difference – must then be true. However, the situation looked more promising in the other three systems, where the Principle of Charity and its negation were neither true nor false. But now let us recall that the Principle of Charity is so called by virtue of the colloquial reading, One-eighth of an inch doesn't make a difference. Put that way, the Principle of Charity seems true, or close to it, doesn't it? If you shrink a tall person by 1/8″, surely that person will still be tall. (If you disagree, change the shrinking to 1/100″ – we'll still get the paradox, but surely 1/100″ doesn't make a difference.) Three-valued accounts can avoid the paradox by claiming that the Principle of Charity is either false or neither true nor false, but that leaves another puzzle: why does the principle seem to be true?

Kleene's “strong” three-valued logic

We began Chapter 1 by noting that sentences concerning borderline cases of vague predicates pose counterexamples to the Principle of Bivalence. For example, the sentence Mary Middleford is tall appears to be neither true nor false. We begin our exploration of logics for vagueness by dropping the Principle of Bivalence and allowing sentences to be either true (T), false (F), or neither true nor false (N – if you like, you may also say that N is neutral). This gives rise to three-valued (trivalent) systems of logic. We use the same language as classical propositional logic. Truth-value assignments can now assign N (as well as T or F) to atomic formulas, and we'll use this value to signal the application of a vague predicate to a borderline case.

How are the truth-functions for the standard propositional connectives defined over the three values? There are several plausible choices, and the set of truth-functions we choose will define a specific system of three-valued logic. In this chapter we present four well-known systems of three-valued logic. Many others have been developed, but these four systems are sufficient to explore the flavor of three-valued logics and how they might be used to tackle problems associated with vagueness.

Issues of vagueness

Some people, like 6′ 7″ Gina Biggerly, are just plain tall. Other people, like 4′ 7″ Tina Littleton, are just as plainly not tall. But now consider Mary Middleford, who is 5′ 7″. Is she tall? Well, kind of, but not really – certainly not as clearly as Gina is tall. If Mary Middleford is kind of but not really tall, is the sentence Mary Middleford is tall true? No. Nor is the sentence false. The sentence Mary Middleford is tall is neither true nor false. This is a counterexample to the Principle of Bivalence, which states that every declarative sentence is either true, like the sentence Gina Biggerly is tall, or false, like the sentence Tina Littleton is tall (bivalence means having two values). The counterexample arises because the predicate tall is vague: in addition to the people to whom the predicate (clearly) applies or (clearly) fails to apply, there are people like Mary Middleford to whom the predicate neither clearly applies nor clearly fails to apply. Thus the predicate is true of some people, false of some other people, and neither true nor false of yet others. We call the latter people (or, perhaps more strictly, their heights) borderline or fringe cases of tallness.

Vague predicates contrast with precise ones, which admit of no borderline cases in their domain of application. The predicates that mathematicians typically use to classify numbers are precise.

The language of classical first-order logic

First-order logic (sometimes called predicate logic) includes all of the connectives of propositional logic. Unlike propositional logic, however, first-order logic analyzes simple sentences into terms and predicates. We use uppercase roman letters as predicates, lowercase roman letters a through t as (individual) constants, and lowercase roman letters u through z as (individual) variables. Predicates, constants, and variables may be augmented with subscripts if necessary, thus guaranteeing an infinite supply of each.

Constants function like names in English, and variables function like pronouns. Together constants and variables count as terms. Predicates have arities, where an arity is the number of terms to which a predicate applies. In English, for example, the arity of the predicate runs in John runs is 1 – it combines with a single term, John in this case – while the arity of the predicate loves in John loves Sue is 2 – it combines with two terms. Atomic formulas are formed by writing predicates in initial position followed by an appropriate number of terms (determined by the predicate's arity). John runs and John loves Sue might thus be symbolized as Rj and Ljs.

There are two standard quantifiers in first-order logic, the universal and the existential quantifiers. We'll use ∀ as the universal quantifier symbol and ∃ as the existential quantifier symbol.