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7.18 Definition Suppose  and  . |
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(a) The asymptotic uniform agreement between a and b on intervals of  (written: auam( a , b )) is  . |
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(b) The asymptotic uniform disagreement between a and b on intervals of  (written: audm( a , b )) is 1 - auam ( a , b ). |
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(c) An M Huexa,m-identifies f (written:  ) if and only if  and  . |
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(d)  . |
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(e)  . |
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If  , then  for all intervals I of length m or more. Note that there is no notion of an M Huexa-identifying a particular f. If  , then for some m we have that for every interval I of length m or greater and for every  ,  . The basic properties of the Huex criteria are easy to establish. |
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a) For all a,  with a < b,  . |
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(b)  . |
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(c)  . |
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(d)  . |
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Proof: For parts (a), (b), (c), and (d), adapt the proofs of Proposition 7.7, Proposition 7.12, Corollary 7.11, and Proposition 7.10, respectively. |
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The most interesting property of the Huexa (for a < 1) criteria is that they fail to contain the Ex* criterion. That is, |
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7.20 Proposition  . |
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Proving the proposition requires taking a closer look at the Exn criteria. This is accomplished in the propositions that follow, from which Proposition 7.20 is obtained as an immediate corollary. |
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A basic problem with the Exn criterion for large n is that it permits a scientist to converge to an explanation that is correct on all but n points, but these n points many include all the experiments for which one would like correct predictions. Proposition 7.21 |
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