paradox a seemingly sound piece of reasoning based on seemingly true assumptions that leads to a contradiction (or other obviously false conclusion). A paradox reveals that either the principles of reasoning or the assumptions on which it is based are faulty. It is said to be solved when the mistaken principles or assumptions are clearly identified and rejected. The philosophical interest in paradoxes arises from the fact that they sometimes reveal fundamentally mistaken assumptions or erroneous reasoning techniques.
Two groups of paradoxes have received a great deal of attention in modern philosophy. Known as the semantic paradoxes and the logical or settheoretic paradoxes, they reveal serious difficulties in our intuitive understanding of the basic notions of semantics and set theory.
Other well-known paradoxes include the barber paradox and the prediction (or hangman or unexpected examination) paradox. The barber paradox is mainly useful as an example of a paradox that is easily resolved. Suppose we are told that there is an Oxford barber who shaves all and only the Oxford men who do not shave themselves. Using this description, we can apparently derive the contradiction that this barber both shaves and does not shave himself. (If he does not shave himself, then according to the description he must be one of the people he shaves; if he does shave himself, then according to the description he is one of the people he does not shave.) This paradox can be resolved in two ways. First, the original claim that such a barber exists can simply be rejected: perhaps no one satisfies the alleged description. Second, the described barber may exist, but not fall into the class of Oxford men: a woman barber, e.g., could shave all and only the Oxford men who do not shave themselves.
The prediction paradox takes a variety of forms. Suppose a teacher tells her students on Friday that the following week she will give a single quiz. But it will be a surprise: the students will not know the evening before that the quiz will take place the following day. They reason that she cannot give such a quiz. After all, she cannot wait until Friday to give it, since then they would know Thursday evening. That leaves Monday through Thursday as the only possible days for it. But then Thursday can be ruled out for the same reason: they would know on Wednesday evening. Wednesday, Tuesday, and Monday can be ruled out by similar reasoning. Convinced by this seemingly correct reasoning, the students do not study for the quiz. On Wednesday morning, they are taken by surprise when the teacher distributes it. It has been pointed out that the students’ reasoning has this peculiar feature: in order to rule out any of the days, they must assume that the quiz will be given and that it will be a surprise. But their alleged conclusion is that it cannot be given or else will not be a surprise, undermining that very assumption. Kaplan and Montague have argued (in ‘A Paradox Regained,’ Notre Dame Journal of Formal Logic, 1960) that at the core of this puzzle is what they call the knower paradox – a paradox that arises when intuitively plausible principles about knowledge (and its relation to logical consequence) are used in conjunction with knowledge claims whose content is, or entails, a denial of those very claims. See also DEONTIC PARADOXES , PARADOXES OF OMNIPOTENCE , SEMANTIC PARADOXES , SET -THEORETIC PARADOXES , ZENO ‘S PARA – DOXE. J.Et.