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fact be addressed in Chapter 12. Let. us here consider a related issue, bearing on the infinitary nature of scientific success as we have defined it. |
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Texts are infinitely long, and convergence takes forever. These features of identification will be preserved in most of the paradigms discussed in this book. However, language acquisition is a finite affair, so our theory might seem from the outset to have little bearing on linguistic development. Similar remarks apply to scientific discovery. |
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We offer two replies to the foregoing objection. First, although convergence is an infinite process, the onset of convergence occurs only finitely far into an identified text. What is termed "language acquisition" may be taken to be the acquisition of a grammar that is accurate and stable in the face of new inputs from the linguistic environment; such a state is reached at the onset of convergence, not the end. |
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This first reply notwithstanding, convergence involves grammatical stability over infinitely many inputs, and such ideal behavior may seem removed from the reality of linguistic development. We therefore reply, secondly, that our theory is best interpreted as relevant to the design of a language acquisition system, not to the resources (either spatial or temporal) made available to the system that implements that design. Analogously, a computer implementing a standard multiplication algorithm is limited to a finite class of calculations, whereas the algorithm itself is designed to determine products of arbitrary size. In this light, consider the language acquisition mechanism  built into the three-year-old child. However mortal the child,  is timeless and eternal, forever three years old in design. Various questions can be raised about  , for example: What class of languages does  identify? If comparative grammar is cast as the study of the design of the human language faculty — as abstracted from various features of its implementation — then such questions are central to linguistic theory, just as analogous questions are central to the study of scientific competence. (For more discussion, see Matthews and Demopoulos [129].) |
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§3.6 Characterization of Identifiable  |
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The present section provides a necessary and sufficient condition for the identifiability of a collection of languages. Its proof rests on a preliminary result that will appear often in our work. |
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First, we introduce some important notation. |
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