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		not simply one text. A weaker version of the exact identification paradigm is discussed in Exercise 4-21.

		Since each scientist exactly identifies only one collection of languages, TxtExExact can contain no more collections than there are computable scientists to exactly identify them. Hence, the cardinality of TxtExExact is at most countably infinite. On the other hand, every singleton collection {Wi} of languages is exactly identifiable. So TxtExExact is countably infinite. In contrast, TxtEx is uncountable. This is because TxtEx contains collections with infinitely many members (, for example) and each of the uncountably many subsets of such a collection is also in TxtEx. We conclude:

		4.49 Proposition (TxtEx - TxtExExact) is uncountable.

		Proposition 4.49 shows that the criterion of exact identification places more constraint on theories of human language than does identification tout court. Should we require of such a theory that it embrace an exactly identifiable collection? Although the remarks at the beginning of this section suggest an affirmative response, the matter is clouded by the following consideration.

		Human languages are not only learnable, they are also highly expressive in the sense that many thoughts can be communicated within any one of them. Let us therefore stipulate that a language be counted as human just in case it is both learnable and highly expressive. Now consider the impoverished language consisting of the single expression "Go!" with its usual meaning. The Go-language may well be learnable by children through casual exposure. If so, then not every learnable language is human, and hence the human languages are a proper subset of the class of learnable languages. This entails that a theory of human language can be legitimately evaluated against the standard of identifiability, but not against the standard of exact identifiability.

		It may be possible to disarm the foregoing objection to exact learning. There is evidence that children exposed to inexpressive languages (e.g., pidgins) as well as children denied access to any ambient language (e.g., deaf children in certain circumstances) invent linguistic devices of considerable complexity and communicative potential (see Feldman, Goldin-Meadow, and Gleitman [57], Sankoff and Brown [166], Bickerton [17], and Gleitman and Newport [79]). These findings suggest that children may not be capable of learning profoundly inexpressive languages. If this is true, then the human languages coincide exactly with the learnable languages, and exact identifiability is the appropriate standard for the evaluation of theories of comparative grammar.

		Finally, suppose that certain inexpressive languages turn out to be learnable after all. In this case it is possible that comparative grammar can be investigated more successfully