Descriptive and entropic economy
The absolute adequacy of the PEP is again of less interest here than the factors that allow it to work as well as it does. In the initial formulation of the PEP, cited on p. 186 above, Carstairs (1983:127) describes it as “an absolute constraint on the organization of the inflexional resources for every word-class in every language” without specifying an explicit constraint that would have this effect. In subsequent elaborations of economy principles, Carstairs-McCarthy (1994:742) proposes what he terms the ‘No Blur Principle’ (NBP) “to refer to the corollary of the Principle of Contrast [(Clark 1987)] which flows from the proposal to treat inflection class membership as part of the ‘meaning’ of an affix”. His statement of the NBP principle is repeated below:
Within any set of competing inflectional affixal realizations for the same paradigmatic cell, no more than one can fail to identify inflection class unambiguously. (Carstairs-McCarthy 1994:742)
Although the NBP is expressed as a constraint on the meaning of individual realizations associated with a paradigm cell, rather than with the number of realizations, it will be violated by the same types of patterns that violate the PEP. In the Finnish example above, neither the partitive singular realization -A nor -tA can be said to “identify inflection class uniquely”. This simple pattern is deliberately chosen to isolate the interactions that challenge the PEP and NBP; more complex classes of counterexamples are discussed by Stump (2005b). Yet the structure of these patterns is again of far greater interest than their status as counterexamples to specific economy principles. This reflects the fact that the patterns will contribute to uncertainty given any objective measure of system complexity, whereas the theoretical import of particular cases will depend on the status of a range of auxiliary assumptions regarding the various ‘extensions’, ‘exemptions’ and ‘codicils’ proposed in the economy literature.
The difference between patterns and their theoretical import rests on a basic property of economy principles like the PEP or NBP. The PEP is stated as a “constraint on the organization of ...inflexional resources” (Carstairs 1983:127) and the NBP as an inflectional version of a “pragmatic principle assisting the acquisition of vocabulary” (Carstairs-McCarthy 1994:783). However, both principles are ultimately constraints on the form of morphological descriptions, rather than on the organization or acquisition of the systems themselves. It is the essential reference to inflection classes in these principles that determines their status as constraints on descriptions. By treating classes as “things in a language rather than merely part of our equipment for the analysis and description of the language” economy principles are guilty of “confusing one’s machinery of analysis with one’s object of analysis” (Hockett 1967:221). Like the ‘morphophonemes’ discussed by Hockett on p. 158 above, ‘inflection classes’ form part of a frame of descriptive reference for representing paradigmatic variation; their number and type depends on the goals of the description. This point is often stated explicitly in descriptions of individual languages, which do not, in general, assert that there is some fixed number of classes in a language, but instead assume that the number assigned depends on the level of precision to which the classes are described or the purposes for which they are defined. Thus Karlsson (2006:476) summarizes the variation in the number of declension classes in Finnish in the following terms:
There is no consensus on how many inflectional classes there are for nominals and verbs. Traditional Finnish lexicography as manifested in Nykysuomen sanakirja (Dictionary of modern Finnish, 1951-1961) postulates 82 inflectional classes for nominals, whereas at the other extreme, a generative description such as Wiik (1967) operates with none but a wealth of ordered (morpho)phonological rules. A surface-oriented morphological approach would recognize at least 10 nominal inflectional classes.
The existence of a range of class-size estimates does not in itself preclude the possibility of identifying a ‘theoretically correct’ estimate within that range. If one assumes that standard grammars incorporate a degree of redundancy that is useful for reference or pedagogical descriptions, then why could economy principles not be satisfied by compressing this redundant description into a theoretically concise analysis? A Russian grammar might, for purely pedagogical purposes, recognize a redundant fourth declension, but the system could be brought into conformance with the PEP by recognizing masculine and neuter subclasses of a general first declension. However, the appeal to ‘subclasses’ in this reanalysis exposes the fact that a class count of four is already highly idealized. A split between classes and subclasses may be motivated by descriptive or pedagogical goals, but within a grammar, this relocates rather than reduces the variation in the system. In the absence of practical goals, it is unclear what would motivate the division. Two paradigms that exhibit distinct inflectional patterns in each of their cells would be assigned to distinct classes in any classification. But how many cells must vary? Does one always suffice, or does partial overlap between two paradigms give rise to ‘subclasses’? Does it matter if variation involves suppletion or some other form of irregularity? How many items must follow an inflectional pattern in order to constitute a ‘class’? Presumably single suppletive items do not form classes. But what then is the item threshold that separates classes from residual suppletive patterns?
The issues raised by these types of questions cannot be resolved merely by stipulating an arbitrarily consistent policy for defining classes, subclasses, item- specific patterns, etc. The challenge for an approach that assigns some objective reality to notions like inflection classes lies in arriving at a principled, task- independent, basis for critical definitions. This challenge is scarcely addressed within the economy literature, which, for the most part, consists of case stud- ies. Studies start from an apparently uneconomical class system, often obtained from a traditional descriptive source, and then outline strategies for bringing the description into conformance with a given economy principle. Both the original and economical descriptions depend on the assumption that a distinction can be drawn between classes and subclasses. The fact that traditional sources may agree on the number of classes in a language does not validate this assumption, insofar as the consensus will again tend to reflect shared descriptive or pedagogical goals within a grammatical tradition.
Like ‘principal parts’ and ‘exemplary paradigms’, ‘inflection classes’ are components of a descriptive framework for exhibiting paradigmatic patterns and variation. Principles that make essential reference to these notions constrain the space of grammatical descriptions, much like the requirement of ‘scientific compactness’ proposed by Bloomfield (1933:238) on p. 70 above. As Finkel and Stump (2007, 2009) show, a morphological description can minimize its principal part inventory, or choose the same parts for each member of a word class, but in at least some cases must chose between minimization and uniformity. There are no comparable studies of exemplary paradigm selection, since it is more generally understood that nothing of consequence hinges on the choice of an exemplary item. Yet as with inflection class size, there is often a consensus regarding the choice of exemplary items within a grammatical tradition. Economy principles that constrain the size of inflection classes or regulate the relationship between realizations and classes fall squarely within the literature that deals with the metatheory of grammatical descriptions.
As with principal part classifications, the applicability of economy conditions will correlate with objective properties of a language. Information theory again provides a means of measuring these properties. According to the PEP, a fully economical system is one in which the number of realizations associated with the maximally allomorphic cell, i.e., the 'cell sum’, determines the number of inflection classes. A fully uneconomical system is one in which the product of all of the cell realizations, i.e., the 'cell product, determines class size. These limiting cases can be characterized directly in terms of paradigm (joint) entropy.24 Let us first generalize the definition in Figure 7.11 to apply to more than a pair of cells, as in Figure 7.13. Then the entropy of a paradigm with cells C1,..., Cn can be defined as the joint entropy H(C1,..., Cn).
Joint entropy values are bounded by the values of their component variables, as specified in Figure 7.14. The minimal value is defined by the variable with the greatest entropy, max[H(C1),...,H(Cn)]. The maximum value is bounded by the sum of their variables, H(C1) + ... + H(Cn). These boundary values define the limits identified by paradigm economy. In any system, there will be a maximally entropic cell (which need not be unique). If, as in Carstairs (1983) and Ackerman and Malouf (2013), the frequency of realizations is disregarded, the maximally entropic cells will correspond to the maximally allomorphic cells. In a fully economical system, the entropy of a paradigm will then correspond to the entropy of a maximally entropic cell. This cell must eliminate uncertainty about every other cell, since otherwise the system will exhibit independent variation that would make the system uneconomical. In a maximally uneconomical system, the entropy of a paradigm corresponds to the other boundary value, the sum of the entropies of its cells. In this case, cell variation is fully independent, and no cell is informative about any other cell.
Figure 7.13 Joint entropy of multiple cells
Figure 7.14 Boundary values for joint entropies
This notion of'paradigm entropy’ measures the cumulative uncertainty associated with the cells of a paradigm, in contrast to the notion of'paradigm cell entropy’ discussed in Section 7.2.2, which measures the average uncertainty associated with individual cells.
An entropy-based PEP could, like the class-based principle, posit that the entropy of a paradigm is “close to the logical minimum” (Carstairs 1983:127). There would, however, be at least three significant differences. The first and most obvious is that the entropies of cells and paradigms would be determined by the distribution of realizations over cells and would not require references to classes or other descriptive or theoretical constructs. The second is that the calculation of entropies from descriptions or corpora would permit an empirical investigation of how closely the entropy of inflectional systems approach the logical minimum. The third difference brings us back to the issue of the robustness of entropy-based measures, discussed in Section 7.2.1. Given a description of a system that specifies only inventories of inflectional realizations, an entropy ceiling can be estimated for the system. However, as more complete and accurate information about frequencies is added, the measures of system economy will show a corresponding increase in precision.
A general benefit of an entropy-based economy measure is that it provides the basis for a cross-linguistic comparison of inflectional economy. An initial study of this kind is presented in Ackerman and Malouf (2013), summarized in Section 7.2.2, which attempts to isolate the factors that contribute to inflectional economy and measure their influence on system complexity across a 10-language sample. This study represents a break from much of the previous literature on inflectional economy, which is taken up with debates about whether particular prima facie counterexamples are genuine violations of a proposed economy condition. The notion of economy that emerges from these debates tends to be binary: either a system is ‘economical’ because it can be made to conform to a given condition, or it is ‘uneconomical’ because it exhibits variation that violates the condition. Because these debates operate at the level of grammatical descriptions, languages with familiar and well-established descriptive traditions are somewhat overrepresented. The relative inflectional simplicity of familiar modern European languages also contributes to the fact that, as Ackerman and Malouf (2013) observe, many of the languages that figure in discussions of economy exhibit low entropy:
it is evident that the systems [Carstairs(-McCarthy)] examines all display low entropy: low entropy provides a unifying explanation that is affected by the specific factors such as those arising from cognitive principles. (Ackerman and Malouf 2013:446)
-  For some discussion of the status of the PEP, see Nyman (1986,1988); Carstairs (1988), as well asCarstairs-McCarthy (1991) and other papers in Plank (1991).
-  A parallel situation obtains in descriptions of Estonian (Blevins 2007, 2008a).
-  Though see Bochner (1993) for a ‘pattern-matching’ metric and Sagot and Walther (20Г3) for ametric based on minimum description length (Rissanen 1978).
-  Ackerman and Malouf (2015) subsequently propose that the NBP is best regarded neither as adesign feature of language nor as a part of morphological theory proper, but as a special case that fallsunder the Low Conditional Entropy Conjecture.