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		10.25 Corollary For all ,

		10.26 Corollary For all , .

		We have the next corollary as an immediate consequence of Proposition 10.15.

		10.27 Corollary For each , .

		This last corollary yields Corollary 10.28, which is the language counterpart of Corollary 10.16. Corollary 10.28 says that if the bound on the number of errors allowed in both the additional information and the converged grammar are fixed, then requiring a scientist to converge to the same grammar for any upper bound is restrictive.

		10.28 Corollary For each and all ,

		It is tempting to be somewhat metaphoric about the above result and read into it an observation about human language learning—we yield to the temptation. Children tend to learn a language effortlessly, compared to adults in a class room setting. A possible explanation may be that the learning criteria used by. children allows them to converge to any grammar for the language, whereas adults are constrained to learn some unique grammar prescribed by the teacher!

		As in the case of functions, it is still open whether the above result holds for c = *. Similar notions of additional information for other language learning paradigms (TxtFex and TxtBc) are treated in the exercises.

		§10.3 Approximate Hypotheses as Additional Information

		Pursuant to our introductory discussion, we now explore a second conception of a priori information about the environment, and again allow this information to depend on which potential environment is ''actual." This time we provide the scientist with an approximately true theory about the language or function with which they are confronted. To model the approximate character of the information provided, we rely on the "density of agreement" concepts defined in Chapter 7.