Schematization and foundation
The cases considered above show how proportions can isolate systematic patterns while avoiding problematic or undermotivated morphotactic or morphosyntactic commitments. The transparency and representational neutrality of this format contributes to its role in pedagogical traditions. Yet, as discussed in Chapter 4.4, grammatical devices and principles drawn from pedagogical or reference grammars often carry along idealizations that have no status in a general model of morphology.11 The directional character of the preceding descriptions and proportions has this artifactual status, reflecting the same kind of pedagogical convention as the selection of exemplary lexemes or principal parts. As Aronoff (1994) observes, there is no formal basis for choosing either the supine or the past passive participle as basic: 
It is clear... that by judging from form alone, there is no way to choose either one of the supine or the perfect participle as underlying the other or to choose either one as underlying the future participle, since they are the same morphome, identical in all constant aspects of their sound forms... In fact, the grammar books are just about evenly divided as to whether they derive the future participle from the perfect participle or the supine. (Aronoff 1994:38f.)
The parity that Aronoff (1994) describes is, however, only inconvenient for the descriptive or pedagogical grammarian. From the standpoint of the learner or speaker it is clearly desirable that the forms of a conjugational series should be mutually informative and that deductions of other forms should be, as far as possible, independent of the order in which the forms are encountered.
From a speaker’s perspective, it is the generality of a pattern that is most important. Does the relation exhibited by a pair of forms characterize just a single lexeme, a whole inflection class or even a word class? The most systematic attempts to address this issue are developed in the context of analogy-based models, particularly in the work of Kurylowicz. Anderson’s (1992) summary of this tradition bears repeating at length here:
The most extensive attempt in the philological literature to provide a genuinely theoretical understanding of the operation of analogical change is that associated with Kurylowicz (1949,1964, and elsewhere), whose goal was to find principles that cover the cases in which analogy could (or could not) operate.
Kurylowicz’s theory is based on the model of proportional analogy, but he makes it abundantly clear that not all proportionals are well formed, even where they involve related terms. Thus, write : writer = receive : receiver is valid, but write : receive = writer : receiver is “nonsensical as between write and receive there is not only no grammatical relation but not even a lexical one” (Kurylowicz 1964:37). In other words, one side of a proportion must instantiate a relation between two (classes of) forms that is governed by some rule of grammar; the other side represents the extension of this rule to other forms. A proportion relates ‘basic’ forms to forms “founded” on them, and a relation of foundation a ^ b must exist in order for a : b = c : d to be admissible as a proportion.
Relations between founding and founded forms play much the same role in Kurylowicz’s system that rules play in a generative grammar, but there are also important differences. Most interestingly, if b is founded on a, this means that the existence of b presupposes the existence of a, rather than that b is constructed by starting with a and adding something. Thus, the stem of a paradigm is founded on the various fully inflected forms, rather than vice versa. Kurylowicz regards the grammar as a set of relations among full surface forms (much as de Saussure did: see Anderson 1985), rather than a set of rules specifyingthe construction of complex forms from simple components. (Anderson 1992:369)
A ‘complex system’ perspective is implicit in the conception that Anderson attributes to Kurylowicz. A relation of foundation between a and b that “means that the existence of b presupposes the existence of a” is an abstractive implication over a network of forms. But proposing that antecedents of a proportional analogy must stand in a “relation of foundation” or instantiate a “rule of grammar” just states a formal prerequisite for valid analogies, without specifying an effective procedure for validation. What is missing is a notion of ‘informativeness’ that measures how reliably one element identifies another. Yet informativeness is precisely what remains implicit in the classical model.
The same fundamental issue recurs in a different guise in the problem of selecting principal parts. Within the descriptive traditions that grew up around individual languages and language families, there was often a consensus regarding the number of classes in a language and the choice of diagnostic forms. But there was again no procedure for selecting diagnostic forms or for determining the informativeness of single forms or sets of forms. Although the problem was posed in a way that reflected irrelevant pedagogical biases, the difficulties that arose in principal part analyses reflected a genuine shortcoming in classical models, which had no means of measuring informativeness.
In both of these cases what is missing is a strategy for making explicit the “knowledge” that Matthews (1972) appeals to in the quotation on p. 106. Of all of the kinds of morphological knowledge, two varieties are particularly relevant to the validation of analogies and the selection of principal parts. The first kind of knowledge concerns ‘unconditioned’ dependencies between cells. Knowing that the present infinitive form of a Latin verb is X implies that the imperfect subjunctive is Xm. Similarly, knowing that a genitive singular of an Estonian noun is X entails that the nominative plural is Xd. The implication is not dependent on variation in the shape of X. A second kind of knowledge concerns conditioned dependencies. For example, feminine nouns in German can follow four of the plural patterns in Table 4.10. But feminine nouns ending in schwa almost always follow pattern P2 and have a plural in -n. Knowing the nominative singular form of a Russian noun is likewise reasonably informative about the accusative singular. But knowing that the nominative singular ends in -a identifies a second declension noun and sanctions an even more reliable deduction of an accusative singular in -u.
The ‘knowledge’ involved in these deductions concerns the reduction in uncertainty about one form based on acquaintance with another form. The reduction of uncertainty can, in turn, be modelled directly by information-theoretic notions. The conditional entropy between one cell and another measures how much uncertainty is left in the second if one knows the form of the first. Specific conditional entropy applies to conditioned dependencies. Mutual information between cells likewise gives a symmetrical measure of informativeness. Recasting proportional analogies in information-theoretic terms thus provides a means of calculating the soundness of a putative “relation of foundation”. This formal reconstruction of classical WP analyses also allows for the generalization and schematization of proportions. One can determine the reliability of deductions between cells, between cells with particular patterns of affixation, between cells with particular patterns of stem allomorphy, combinations of affixes and stems, etc. As in standard proportions, patterns are motivated by their predictive value and carry no larger commitment.
More generally, a description of a speaker’s knowledge of morphological patterns in terms of discriminative learning and information-theoretic measures can incorporate the dependencies exhibited by proportional analogies without endorsing the four-part format in which these patterns are conventionally expressed. A formalization of classical WP models in terms of notions of uncertainty and uncertainty reduction is outlined in Chapters 7 and 8. However, before considering this approach, it will be instructive to review the realizational tradition, which represents the first attempt to incorporate insights from classical models into a framework of morphological analysis.
-  As discussed in Chapters 7 and 8, the goal of describing a speaker’s knowledge of a language asa symbolic representation of the patterns in the language reflects an even more basic idealization, onethat is incorporated in the notion of a ‘formal’ grammar.
-  Aronoff (1994) appears to discount the possibility that a longer form, in this case the future activeparticiple, could be the basic member of the series. From an implicational perspective, there is noreason to make this assumption. Longer forms are more informative than shorter forms in cases, likethe nominative singular in Estonian, where the longer form retains bits of form that cannot be predictedfrom the shorter form.
-  Itkonen (2005) presents an extended discussion of analogy; some of the issues raised in this workare subsequently considered in the papers in Blevins and Blevins (2009). For computational approachesto analogical modelling, see discussions of AM (Skousen 1989, 1992) and TIMBL (Daelemans andVan den Bosch 2005).