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(b) only gn is total; and |
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(c) Mi fails to identify gn's canonical text. |
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On the other hand, if the construction proceeds through infinitely many stages, then: |
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(a) infinitely many functions f, g0, g1, . . . are specified; |
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(c) Mi fails to identify f's canonical text. |
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So in either case, Mi fails to identify the one, total function specified. Finally, observe that for all of the functions h specified in the construction (whether partial or total), h(0) = i. |
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§4.4 Parameterized Scientists |
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Real scientists are sensitive not only to evidence from their laboratories but also to general information about the character of the scientific problem they confront. In particular, the hypothesis produced by scientists in response to given data depends in part on the background knowledge they obtain from teachers and colleagues. Such knowledge rules out certain logically possible realities and embraces others. In the paradigms discussed so far, it is not possible to communicate with scientists in this sense. They thus have "one-track minds," being adapted to a single class of possibilities but no others. This feature of scientists is appropriate for modeling language acquisition — since parents have no means of informing infants about the class of human languages — but limits the fidelity of our results to adult, scientific discovery. The present section introduces a model in which scientists are given a description of a class of theoretical possibilities and must then adapt their conjectures accordingly. The ideas in this section are due to Jantke [98], and explored further by Osherson, Stob, and Weinstein [141]. |
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§4.4.1 Communicating with Scientists |
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Numbers will be used to inform scientists of the class of theoretical possibilities they face. The following definition shows how numbers are used for this purpose. |
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(a) Any mapping of N into the power set of e is a descriptor for languages. |
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(b) Any mapping of N into the power set of R is a descriptor for functions. |
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