For more than two centuries, the mainstream rationalist tradition in philosophy took it for granted that its chief role was to respond to the skeptic challenge. It is not quite clear why, for the challenge rests on the assumption that there is nothing to skepticism, that it is not serious, that it is obviously answerable. If so, why bother with it? Moreover, if it is so obviously faulty, why is it so hard to answer? Why do so many philosophers see it as so terrible and threatening to our sanity?

Skepticism is dangerous, most philosophers explain, because it is paralyzing. It is easy to show that this observation is obviously false; skeptics suffer from paralysis no more than other people. Remarkably, critics of skepticism use this observation as an even stronger argument against skepticism. People who claim that they are skeptics are not paralyzed; hence, it follows logically that they only pretend to be skeptics. Yet their preaching, the critics of skepticism continue, however flippant it is, can nonetheless cause harm by spreading discouragement. It is obvious they conclude their argument, that doubt discourages.

All this is very convincing; we do not know why. For, obviously, it is far from the truth: skeptics are not paralyzed because skepticism does not always paralyze. In truth, every philosophy moves some people to action and others to inaction.

Skepticism in Aesthetics

In previous chapters, we discussed epistemology, ethics, and politics. In this chapter, we discuss another field in which skepticism is applicable: aesthetics. We left it to the last because there are fewer objections to skepticism in aesthetics than elsewhere, due to a combination of two factors: (1) nihilism being regrettably a respected contender in aesthetics, and (2) the confusion of nihilism with skepticism that is the root of the prevalent hostility to skepticism. Perhaps there is also less dogmatism about beauty than about other matters, possibly because, regrettably, thinkers do not take beauty sufficiently seriously. It is serious all the same. It influences our lives. People make great investments in it – in the arts and in personal grooming. We argue about paintings to hang in the museum or in the home and what kind of building to construct and which cosmetics fit which person best. In what follows, we offer a new fragmentary and tentative theory of the judgment of beauty. We suggest that this fragment is applicable to discussions about beauty in order to increase its rationality and reduce its unpleasant emotional aspect.

Traditional aesthetics comprises the traditional studies of the following question: What kinds of aesthetic judgments are valid – that is, certain or at least plausible? This question arises because people argue about aesthetics, and such arguments imply that there are aesthetical criteria, explicit or not, and that they are open to discussion.

In a deductively valid argument, if the premises are true, the conclusion must be true. Deductive validity is a very strict standard of argument. If an argument is deductively valid, it is impossible for the premises to be true while the conclusion is false. In an inductively strong argument, if the premises are true, it is probable or likely that the conclusion is true. If an argument is inductively strong and the premises are true, it is logically possible that the conclusion could be false. So inductive strength is a less strict standard of argument than deductive validity. Inductive strength is a matter of probability.

Probability and statistics have an acknowledged place in scientific reasoning and experimental methods, but even outside these specialized contexts, the use of inductive argument is an important part of most reasonable dialogue. For example, the use of statistical arguments seems to play an increasingly significant role in political decision making on virtually any subject of discussion.

Of the many different kinds of inductive arguments, we will single out three for discussion in this chapter. An inductive generalization is an argument from premises about a specific group or collection of individual persons or things to a more general conclusion, about a larger group or collection. Traditional logic textbooks have often stressed the perils of hasty generalizations, for it has been rightly perceived that inductive generalization is associated with significant and common fallacies.

The argumentum ad hominem, meaning “argument directed to the man,” is the kind of argument that criticizes another argument by criticizing the arguer rather than his argument. Basically, this type of argument is the type of personal attack of an arguer that brings the attacked individual's personal circumstances, trustworthiness, or character into question. The argumentum ad hominem is not always fallacious, for in some instances questions of personal conduct, character, motives, etc., are legitimate and relevant to the issue. However, personal attack is inherently dangerous and emotional in argument, and is rightly associated with fallacies and deceptive tactics of argumentation. Three basic categories of fallacy have often traditionally been associated with three types of argumentum ad hominem.

The abusive ad hominem argument is the direct attack on a person in argument, including the questioning or vilification of the character, motives, or trustworthiness of the person. Characteristically, the focus of the personal attack is on bad moral character generally, or bad character for truthfulness.

The circumstantial ad hominem argument is the questioning or criticizing of the personal circumstances of an arguer, allegedly revealed, for example, in his actions, affiliations, or previous commitments, by citing an alleged inconsistency between his argument and these circumstances. The charge, “You don't practise what you preach!” characteristically expresses the thrust of the circumstantial ad hominem argument against a person.

Thus, our brief survey comes to its conclusion. This final chapter summarizes the main ideas of the previous chapters with an accent on practical implications and offers a few parting thoughts.

Throughout this work, we presented radical skepticism, according to which no statement (or judgment) is certain or plausible (in the epistemological sense of these terms). We claimed that, contrary to popular criticism, skepticism is common sense, implies no absurdities, and permits alternative tentative theories of reasonability to discuss it critically, much in accord with common sense.

Complete error avoidance is impossible. This is practically important because under the influence of empiricism, many people still try to avoid error at all cost and then they find themselves doing so by clinging as much as possible to known observation-reports. This may lead them to present as few ideas as they can, limiting themselves to irrefutable ideas. Such ideas do not reduce unexpectedness; as such, they are uninteresting. As skeptics, we claim that aiming at error avoidance at all cost is itself a serious error. (As a popular edict goes, those who close their door to risk also close it to opportunity.) Sadly, we have to repeat the obvious: we do not recommend the conscious advocacy of falsehoods.

Skepticism in Ethics

In the previous chapters, we presented skeptical epistemology. In this chapter, we present skeptical ethics.

We follow the same approach adopted in the previous chapters. We start by endorsing skepticism, according to which there is no moral knowledge and no final justification of any moral judgment. While refusing to reduce ethics to psychology, we present a tentative psychological theory regarding the conditions under which a mode of conduct is considered moral in order to place it on the agenda for public discussion.

Philosophers concerned with ethics have traditionally studied the following questions:

1. 1. What renders a moral judgment valid moral knowledge – namely, certain or at least plausible?

2. 2. What should we do to acquire new valid moral judgments?

We endorse the skeptical answers to these questions, which are as follows:

1. 1. No moral judgment is certain or plausible; there is no moral knowledge; no moral judgment can be fully justified. (Again, conditional justifications are available, but they beg the question of validity of the outcome.)

2. 2. No method can fully guarantee the validity of any moral judgment. (Again, conditional guarantees are available, and most of them are poor.)

Skepticism was always unpopular – in epistemology and ethics alike.

The arguments against skepticism in ethics run parallel to those against skepticism in epistemology that we reviewed in previous chapters.

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.