By Anant Pai
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Such an adjective, like Russell's barber, cannot exist. Yet on an ordinary language level, the adjectival phrases true-ofself and not-true-of-self would seem to apply in the situation described. The paradox is difficult to dismiss unless one also makes a distinction between language and metalanguage. Just as there is a hierarchy of sets, one can also pose a hierarchy of metastatements. Descriptive statements about the real world-"All cats are animals" and "My cat is black"-are considered examples of object language, the lowest level of language in the hierarchy.
In fact, the barber belongs to neither set because, as shown above, his existence would produce the contradictory conclusion that he shaves himself if and only if he does not. In fact, as the American philosopher Willard V. Quine has observed, the paradox itself can be considered a valid proof for the fact that the barber cannot exist: it appears to be a classic case of a reductio ad absurdum. However, the matter is not so simple, for the paradox is exactly parallel in structure to another of Russell's paradoxes, that of the set of all sets that do not contain themselves as members.
27. The vanishing line paradox, II. A more sophisticated type of geometric vanish involves the apparent loss of area. Consider, for example, the arrangement of squares shown in Figure 28. The entire figure contains sixty-five smaller squares arranged to form a rectangle that is five units high by thirteen units wide . r--... -...... r---.. r--... -r--... ~ .......... ~ ........ ~ ~ 28. The vanishing square paradox, I. Suppose we dissected the square and rearranged its parts to form the square shown in Figure 29.