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		So, if T is 3.3 then for , and Fperiods(T[5]) =o.

		Two other examples of scientists can be entered here.

		3.9 For all :

		(a) Ffinite( s ) = the least index for the language content( s );

		(b) Ffive ( s  = 5.

		Scientist Ffinite behaves as if its current evidential state includes all the numbers it will ever see. Consequently, it conjectures a grammar for the finite language made up of just the elements seen to date. Being parsimonious, Ffinite selects the smallest possible grammar. Scientist Ffive has fixed ideas about the incoming language; indeed, its behavior does not even depend on the data presented. We shall have occasion to refer to these two functions later.

		Our conception of scientist is broad in some respects. The everywhere undefined function is a scientist, as are functions that seem to make non-optimal use of data. For example, if our scientist keeps poor records of past experiments, her hypotheses might depend only on the last datum examined, i.e., only on T(n) rather than on T[n + 1]. Or she might change her conjecture upon encountering #, even though # provides no information about the underlying language. Such scientists might be termed "subrational," and will be central to paradigms that attempt to model children (for example, in Section 3.8 below). In addition, as Chapter 5 will reveal, our intuitions about scientific rationality are not always trustworthy in the context of precise models of empirical inquiry. So it is wise at this stage not to eliminate any potential scientists from consideration.

		Our conception is broad, as well, in allowing both computable and noncomputable functions to be scientists. For those convinced that human intellectual capacities are computer simulable, noncomputable scientists might seem of scant interest. Indeed, most of the paradigms to be discussed in this book restrict attention to the computable case. Nonetheless, our encounter with noncomputable scientists is useful for two reasons. First, despite considerable speculation it is still unclear whether noncomputable processes exist in nature (for discussion see Webb [191] and Suppes [187]). In particular, it is simply not known whether the human brain is suitably modeled as a computational agent. Worse yet, the answers to such questions depend on apparently arbitrary conventions about what to count as inputs and outputs to naturally occurring processes (see Osherson [137]). A second reason to study noncomputable scientists is that many results in the theory of inductive inference do not depend upon computability assumptions; rather, they are information theoretic in character (examples arise in Section 3.6 below). Consideration