|
|
|
|
|
|
3.29 Corollary  is not identifiable. |
|
|
|
|
|
|
|
|
Proof: Let  . The condition expressed by 3.26 fails for N, because for each finite  , there is  such that  ' and  . This L' has the form N - {x} for some  . |
|
|
|
|
|
|
|
|
To gain insight into the phenomenon of nonidentifiability, consider a scientist F that identifies N, and let us see why F does not identify  . By Corollary 3.25 let s be a locking sequence for F on N. Let T be the text:  . Then F converges on T to an index for N, whereas T is for  . Since no index for N is an index for a finite set, F does not identify T and hence does not identify the finite set content( s ). |
|
|
|
|
|
|
|
|
Corollary 3.28 yields some lattice-theoretic properties of the class of identifiable subsets of e . |
|
|
|
|
|
|
|
|
(a) Although  is identifiable, none of its proper extensions are identifiable. |
|
|
|
|
|
|
|
|
(b) Although both  and {N} are identifiable, their union is not identifiable. |
|
|
|
|
|
|
|
|
Corollary 3.30b may be expressed this way: the class of identifiable subsets of e is not closed under finite union. This aspect of identifiability is exploited in Chapter 9. |
|
|
|
|
|
|
|
|
§3.7 Some Alternative Paradigms |
|
|
|
|
|
|
|
|
If normal linguistic development is correctly construed as a species of identification, then Corollaries 3.28 and 3.29 yield nonvacuous constraints on theories of comparative grammar. No such theory, for example, could admit as natural some infinite and all finite languages. The force of such constraints is attenuated, however, by the fact that identification is far from adequate as a representation of normal linguistic development. Children's linguistic environments, for example, are probably not arbitrary texts for the target language: on the one hand, texts do not allow for the grammatical omissions and ungrammatical intrusions that likely characterize real environments; on the other hand, many texts constitute bizarre orderings of sentences, orderings that are unlikely to participate in normal language acquisition. In addition, the identification paradigm provides no information about the special character of the child's system for language acquisition. To claim that this system implements some function from SEQ to the set of (indexes for) programs is to say essentially nothing at all. Even the criterion of |
|
|
|
|
