Paradox of Self-Amendment by Peter Suber of its component p. A contradiction implies its own negation, but is not implied by its own negation. A contradiction is a formal falsehood, e.g. "Both proposition p and ~p are true." This proposition is false by virtue of its form. A contingency neither implies nor is implied by its own negation. Its truth is not a function of its form, but only of its content, e.g. "Either proposition p is true or proposition q is true." Following this pattern we may define the Grelling and Liar type of paradox as a statement that both implies and is implied by its negation. In the most common notation: p is a tautology if and only if ¬(p ⊃ ¬p) · (¬p ⊃ p) contradiction (p ⊃ ¬p) · ¬ (¬p ⊃ p) contingency ¬ (p ⊃ ¬ p) · ¬ (¬ p ⊃ p) paradox, Liar type (p ⊃ ¬p) · (¬p ⊃ p) (This notation is to clarify matters for those who already know it; it will not be used in the body of the book.) But there is another type of paradox. Logically it is closer to a contradiction, by the definitions above, than a paradox. It includes Russell's paradox, but the type is best illustrated by the popular paradox of the Barber. Suppose a town has a barber who shaves all and only those men in the town who do not shave themselves. If the barber is a shaved man who lives in the town, then does the barber shave himself?[Note 6] At first this looks like the type of paradox already described, for the answer is: if he does, then he doesn't (because he does not shave those who shave themselves), and if he doesn't, then he does (because he shaves all those who do not shave themselves). But there is an important difference between the Barber and the Liar paradoxes. We can evade all contradiction in the Barber paradox by concluding that the Barber does not exist as defined. The contradictions only arise from the assumption that such a peculiar Barber exists, and we are free to reject that assumption; in fact, the contradictions give us good reason to do so. There is no comparable assumption in the Liar paradox to reject, unless it is the belief that the words say what they seem to say. "Liar-type" paradoxes, then, make contradiction uncomfortably unavoidable; they demand radical remedies. "Barber-type" paradoxes, by contrast, simply prove that something cannot exist as defined on pain of contradiction. They are "paradoxical" only at first sight; in the last analysis they are proofs. By the definitions set out above, the statement that the Barber shaves himself is a Liar-type paradox; but the statement that the Barber exists is a contradiction, not a paradox. If we knew nothing about the Barber but his ambition to shave all and only those men in a certain town who did not shave themselves, then we could avoid contradiction by concluding that the Barber was a woman, or did not live in the town, or both. But if the Barber is a man in the town, then we can avoid contradiction only by denying his existence. This turns out to be a perfectly consistent proposition, even if we are surprised at the necessity of concluding it. Russell's paradox is of the Barber-type, not the Liar-type. It proves with finality that there is no such thing as a set of all sets that are not members of themselves. Russell's paradox is more difficult to cope with than the Barber, not because it is logically different, but because so much has depended on the broad notion of a set as any collection of any elements, and so little has depended on belief in the Barber. This shows how a paradox may require revision of our most fundamental concepts. Paradoxes of the Barber-type are often called "veridical" paradoxes because they establish the truth of a proposition. The opposite of a veridical paradox is not the Liar-type but a "falsidical" paradox which attempts to prove the falsehood of a proposition by deriving contradictions from its affirmation. Zeno's paradoxes of motion are the most common examples. They purport to prove that motion is impossible, or to falsify the belief that motion is real.[Note 7] Some writers distinguish both veridical and falsidical paradoxes from the Liar-type paradoxes by calling the latter "antinomies". However, that usage has not been widely adopted. I shall call the Liar-type paradoxes simply "paradoxes" or "genuine paradoxes" for emphasis, and the Barber-type, "veridical paradoxes", but only when the context requires that the two types be distinguished. If the Liar-type or genuine paradox is symbolized, (p ⊃ ¬p) · (¬p ⊃ p) (p implies its negation and its negation implies p), then the veridical paradox should be symbolized, {[p ⊃ (q · ¬q)] ⊃ ¬p} · ¬(¬p · p) (the fact that p implies a contradiction implies that p is false, and it is not the case that the falsehood of p implies the truth of p). But this formulation is equivalent to the definition of a contradiction. Hence, acceptable alternate terminology is that the Liar-type statement is a genuine paradox, and the Barber-type statement is merely a contradiction, although a surprising one, and one which leads to genuine paradox if affirmed.[Note 8] B. "Solving" paradoxes in logic and law 4