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7
Inference of Approximations |
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Lakatos [117] claimed that most scientific theories are "born refuted." That is, the theories are put forth in the face of prima facie anomalies. To back this claim Lakatos sketched the history of a number of successful scientific theories that were nonetheless awash in an "ocean of anomalies." These theories, which are approximate explanations for the phenomenon in question, succeed in part because they were judged to be close enough approximations to be useful. Approximations arise in language learning also. To see this, have a conversation with a three-year-old child who, we will assume, speaks English. Now, the child's grammar almost certainly does not match ''standard" English grammar. For example, irregular verbs may be regularized, adjectives and adverbs may appear in peculiar places in sentences, and whole grammatical categories may be missing. Yet despite this, you can probably carry out a successful conversation. So there are quite reasonable informal criteria under which a learner of a language can be judged to have learned a sufficiently good approximation to the target language, even though this approximation may have vast differences from the target. |
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We previously considered inference of approximations in Chapter 6 where the Exa and TxtExa ( ) criteria were introduced and studied. It is questionable whether Lakatos' "oceans of anomalies" are properly reflected in, say, the setting of the Ex* criterion which demands that final explanations be correct on a co-finite set. In defense of Ex* one might argue that a clever coding of experiments could fence the anomalies into a finite set. (For instance, if a theory was known to be anomalous in the X-ray region of the EM-spectrum, then one could simply code all experiments on the X-ray region as a single "don't care" experiment.) But the cleverness seems only to mask Lakatos' insight. More is accomplished by attempting to frame success criteria that permit infinitely many anomalies in final explanations or grammars. Such is the topic of the present chapter. |
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Criteria permitting infinitely many anomalies in final explanations or grammars usually make some restriction on the allowed distribution of these anomalies. (Otherwise, the criteria are typically degenerate—see Exercise 7-1.) Most of this chapter concerns criteria that require that the "density" of anomalies be no more than a fixed amount, where the notion of "density" is formalized in various ways. We begin with a short |
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