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Show Chapter III. The Concept of a Genuine Preference 162 we see that (T3) and (Asy) together, when joined with (C'), generate a straightforward logical contradiction. This explains why (T3) and (C') cannot logically be true together. Finally, we can show how (C') violates the requirements of logical consistency that (T3) satisfies, by recurring to the case considered in Volume I, Chapter IV.3.1, of Rex, who has and applies the concept of ranking superiority to z - hence gives a cyclical ordering - because he has forgotten the relation of x and y to z established by his two previous rankings. I argued that in that case, every alternative is preferred to every other, hence that none of the three is superior in ranking to any of the others, and so none superior to x. I concluded that Rex's application of the concept of some one thing's ranking superiority to z at t3 therefore had involved him in a logical contradiction, i.e. that z both was and was not preferred to y. However, the restrictions of conventional preference notation gave us no way to express this conclusion formally. With the aid of the variable term calculus I have suggested here, we are now in a better position to express formally the thought that a cyclical ranking is logically contradictory. Rex ranks x, y and z as follows: (7) t1: Pw(x.~y) (8) t2: Pw(y.~z) (9) t3: Pw(z.~x) From (8) and (9), (T3) permits the inference to (10) Pw(y.~x). From (9) and (7), (T3) permits the inference to (11) Pw(z.~y). And from (7) and (8), (T3) permits the inference to (12) Pw(x.~z). This much simply translates Savage's notation into mine. But before, in Volume I, Chapter IV, we could state the crucial conclusion to logical inconsistency only in natural language and not symbolically in Savage's notation. We can now, however, state it symbolically in the variable term calculus. From (7) and (10), (T3) permits us to infer (13) Pw(x.~x), © Adrian Piper Research Archive Foundation Berlin |
