The purpose of logic is to distinguish valid forms of argument from faulty ones, called "fallacies." If you conduct me from premises that I accept to a conclusion that I dislike by means of a fallacious argument, I am under no obligation to embrace that conclusion. If I, on the other hand, take you from premises you accept to a conclusion you dislike by means of a logically valid argument, then you are compelled to embrace the conclusion.

"Compelled by what?" you ask. By the court of rationality, I respond. The fact that my argument is logically valid means that the premises cannot be true without the conclusion being true; so if you believe the premises, you have to believe the conclusion, or you are being irrational. "What's so bad about being irrational?" you ask defiantly. I then dig myself in deeper by giving you reasons for accepting my reasons, which leave you similarly unmoved. What I'd like to do, though, is turn my logic into a club and use it to beat you into submission.

The impotence of logic was celebrated by Robert Nozick in his 1981 book Philosophical Explanations. "Why are philosophers intent on forcing others to believe things?" he asked. "Is that a nice way to behave toward someone?" What logicians really wish they had, Nozick darkly suggests, is an argument that sets up reverberations in the brain so that if the person refuses to accept the conclusion, he dies.

One basic logical principle is the law of noncontradiction, which says that a proposition and its negation cannot both be true. In fact, there are many people who violate this law without realizing it. They believe p, q, r, and s, while unaccountably failing to notice that q, r, and s logically entail not-p. Should you bring this to their attention, they might eject p from their creed. More likely, they will try to dodge the charge of inconsistency by quibbling about meanings. ("What I said was that neutral countries should not be invaded. That was an incursion, not an invasion!")

But what if your interlocutor invokes the spirit of Walt Whitman and says, "Do I contradict myself? Very well, I contradict myself!" (The physicist Niels Bohr once came close to this. A colleague, seeing a horseshoe over Bohr's office door, said, "You don't really believe in that stuff, do you?" Bohr replied, "No, but I hear it works even for those who don't believe.") What do you say to such a person?

"This would vitiate all science," is what W.V. Quine, the world's preeminent living logician, would say: "Any conjunction of the form p and not-p logically implies every sentence whatever; therefore acceptance of one sentence and its negation as true would commit us to accepting every sentence as true, and thus forfeiting all distinction between true and false." To see what Quine means, suppose you believe both p and not-p. Since you believe p, you must also believe p or q, where q is any arbitrary proposition. But from p or q and not-p, it obviously follows that q. Hence any arbitrary proposition is true.

The idea that a contradiction is bad because absolutely anything follows from it is often alien to the average person. Bertrand Russell was once trying to get this very point across at a public lecture when a heckler interrupted him. "So prove to me that if two plus two is five, I'm the Pope," the heckler said. "Very well," Russell replied. "From 'two plus two equals five' it follows, subtracting three from each side, that two equals one. You and the Pope are two, therefore you are one."

"Coercive" formal logic, as philosophy departments have traditionally taught it, seems to be getting shoved aside these days by other concerns. Ruth Ginzberg, a philosopher at Wesleyan University, has taken issue with the law of modus ponens, which sanctions inferences of the form if p, then q; but p; therefore q. Ginzberg maintains that modus ponens is used by males to marginalize women--who are less likely to recognize its supposed validity--as "irrational." In a different vein, some would-be reformers have suggested that fallacies be thought of as "failures of cooperation" rather than as errors of reasoning. Others advocate the adoption of "principles of charity": If, for example, your interlocutor's argument is a muddle, try to reconstrue it in a way that makes it valid.

Whether this approach is a better route to truth is arguable. But the new niceness in logic certainly threatens to take some of the zest out of polemics. Coercion by logic--or its simulacrum--can be a marvelous blood sport. One thinks, for example, of the great confrontation between Diderot and the Swiss mathematician Leonhard Euler before the court of Catherine the Great in 1773. Diderot, an atheist, was almost entirely ignorant of mathematics. Euler, a devout Christian, approached the philosophe, bowed, and said very solemnly, "Sir, (a + b)n/n=x, hence God exists. Reply!" Delighted laughter broke out on all sides as Diderot crumpled before this stunning inference.

The next day Diderot asked Catherine for permission to return to France, a request to which the empress graciously consented.

JIM HOLT