Moore's paradox
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G. E. Moore remarked once in a lecture on the peculiar inconsistency involved in saying something like "It's raining outside but I don't believe that it is." By contrast, "It's raining outside but he doesn't believe that it is," is a perfectly consistent statement. This paradox, sometimes known as Moore's paradox, might well have been forgotten if not for the fact that, supposedly, when Ludwig Wittgenstein first heard of the paradox he went to Moore's house in the middle of the night to insist Moore immediately repeat the lecture. In any case, it probably offers a decent entrance to Wittgenstein's later philosophy.
Both the consistent and the paradoxical sentences are conjunctions of two shorter sentences: (1) "It is raining" and (2a) "I don't believe that it is raining" or (2b) "He doesn't believe that it is raining." Furthermore both (2a) and (2b) are equally compatible with (1), in that they can be simultaneously true. However, any given person seems debarred from consistently uttering (1) and (2a) together.
The lesson is probably something like this: Sentences like "He believes p" and "She believes p" (where p is any proposition), are descriptions of some part of the world - namely the beliefs of other people. However, the first person equivalent, "I believe p" seems to function not as a description of me, of the fact that I believe p; rather, it functions simply as an affirmation of p itself. If I'm committed to affirming "p", then I'm committed to affirming "I believe that p", and vice versa. The same is not true of the third person cases. We might say that there is an asymmetry between the logical form of first- and third-person attributions of belief.
Attributions of knowledge do not follow exactly this pattern. Consider: "I know that it's raining, even though it isn't" is a contradiction, but so is "John knows that it's raining, even though it isn't." The same asymmetry does not recur, because saying that someone knows p, unlike saying merely that someone believes p, is an endorsement of p itself in both the first and third person cases: you do not call any belief knowledge unless you hold it to be a true belief. Comparable paradoxes, however, can be generated for knowledge attributions.