Implicational economy

Interpreting implicational relations in terms of uncertainty reduction offers a useful perspective on a range of other traditional issues. Perhaps the most direct implications are for the phenomena subsumed under the rubric of ‘economy of inflection’ (Plank 1991). Inflectional economy effects are of particular interest to classical WP models for two principal reasons. Since economy effects are defined with reference to the paradigm, they provide a measure of support for the theoretical relevance of paradigms. As Carstairs (1983:116) puts it, “inflexional paradigms do indeed need to be recognized as central in morphological theory, because it is only by reference to them that we can state an important fact about how inflexions behave”. Moreover, from a classical WP standpoint, the economical organization of paradigms provides a type of evidence for the general implicational structure of inflectional systems.

Paradigm economy

The contemporary interest in paradigm economy effects stems largely from the influential discussion in Carstairs (1983), which draws attention to these effects and attributes them to a Paradigm Economy Principle (PEP). Subsequent elaborations of this approach treat economy effects as morphological reflexes of more general synonymy avoidance principles (Carstairs-McCarthy 1991, 1994). However the general conception of economy that underlies this family of proposals is articulated most clearly in the initial formulation of the PEP. In contrast to a classical WP model, the PEP is not concerned with distinctive inflectional patterns in general but with ‘inflectional resources’, by which Carstairs (1983) means ‘affixal resources’:

Paradigm economy provides at least a partial answer to a question which, so far as I can discover, has not been asked before—a question about how, in any inflected language, the inflexional resources available in some word-class or part of speech are distributed among members of that word class. (Carstairs 1983:116)

The distribution of inflectional resources is characterized in terms of upper and lower bounds on the space of possible inflection classes, where these bounds are determined by the number of different affixal strategies available for realizing paradigm cells in a given word class. This can be illustrated by a schematic example. Consider a simple declensional system with two contrastive properties, number and case, two distinct number features, N1 and N2 and four case features, Q, C2, C3 and C4. Since each combination of number and case features defines a paradigm cell, these number and case features define the family of eight-cell paradigms summarized in Table 7.10.

The actual number of realizations for each cell are determined mainly by the size of the nominal lexicon, which is independent of the economy of the inflectional

Table 7.10 Schematic paradigm structure

N1

N2

Q

(N1A)

(N2A)

C2

(N1A)

(N2A)

C3

(N1A)

(N2A)

C4

(N1A)

(N2A)

system. Hence to describe inflectional economy one must first isolate patterns of inflectional exponence. For each cell (Ni, Cj), let ^(Ni, Cj) represent the set of patterns that realize the cell and ^(Nj, Cj) | the number of patterns in that set. There will be at least one inflectional pattern associated with every cell, since the existence of the pattern is a precondition for recognizing the cell in the first place. In every class system, there will also be a maximum number of patterns realized by any cell, though there need not be a unique ‘maximally allomorphic’ cell. Let us call this highest value the cell sum and represent it by H.15 In the singular German patterns in Table 4.9, the value of H is 3, since there are three patterns of exponence in the genitive. In the partial description ofRussian in in Table 75, the value of H is again 3, since the dative and instrumental singular cells both have three realizations.

The space of possible inflection classes can then be defined with reference to the allomorphic variation exhibited by individual cells. It is usually assumed that two inflection classes can be distinguished if they exhibit (morphologically conditioned) allomorphy in any cell. From this assumption, it follows that the cell sum H defines the minimal number of classes. This minimum just reflects the fact that there must be at least as many classes as there are realizations for the maximally allomorphic cells, since otherwise some class would have multiple realizations for those cells. At the other extreme, the largest space of classes corresponds to the cell product P, i.e., the product of all of the realizations of the individual cells. In the case of the the system in Table 7.10, the cell product is defined as ^(N1, C1) x ^(N1, C2),..., x|f(N2, C4).16 The cell product defines the largest class system, since it exhaustively enumerates all of the combinations of patterns exhibited by individual cells.

Continuing with the schematic system in Table 7.10, let D represent the system and Cd be the number of inflection classes in D. The assumptions adopted so far dictate only that the value of Cd must fall somewhere between the cell sum Hd and the cell product Pd. However, as Carstairs (1983) remarks, inflection class systems are not normally distributed within this space. Instead, they appear to cluster closely around the cell sum value:

when we apply the traditional notion of‘paradigm’, we find that the actual total of paradigms is at or close to the minimum logically possible with the inflexional resources involved, and nowhere approaches the logical maximum. (Carstairs 1983:127)

To account for this clustering, Carstairs (1983:127) proposes the PEP as “an absolute constraint on the organization of the inflexional resources for every word- class in every language” which has the effect of “keeping the total of paradigms for any word-class close to the logical minimum”. In the schematic terms adopted

15 This value corresponds to the Maximum Realizations count in in Table 7.9.

16 More generally, in an inflectional system with categories F, G..., H and corresponding features p, q,..., r the cell product will be

above, the PEP requires that for any word-class system W, the class size, Cw is equal to (‘or close to’) the cell sum Hw. In principle, the PEP would appear to impose a maximally restrictive constraint on class size, since it effectively requires that each inflectional system must be organized into the smallest possible set of classes. In practice, the restrictiveness of the PEP depends on how close is ‘close enough’ for compliance and, more fundamentally, on how classes are individuated. One immediate qualification is introduced by the decision to “disregard... stem alternations” (Carstairs 1983:120) and restrict the notion of‘inflectional resources’ relevant to the PEP to ‘affixal exponents’. This exclusion reduces the overall number of inflection classes in languages with distinctive, class-specific stem alternations, and contributes to the goal of bringing the count closer to the logical minimum.

Yet even as a constraint solely on the distribution of affixal resources, the PEP places an extremely tight constraint on the relationship between cells in a paradigm. In a system that conforms to the PEP, each maximally allomorphic cell partitions the system into classes, and every pair of maximally allomorphic cells partitions the system into the same classes. No cell in a paradigm can have realizations that vary independently of the realizations of any maximally allomor- phic cell. This means that for every realization of every maximally allomorphic cell there will be a unique realization in every other cell (including other maximally allomorphic cells). This entails that every maximally allomorphic cell is diagnostic of class. Moreover, since every class contains a maximally diagnostic cell, it follows that every class will be identifiable from the realization of a single cell. So a system that conforms to the PEP will realize the pedagogical ideal in which a single principal part suffices to identify class.

There are of course prima facie counterexamples to any principle that imposes this kind of tight organization on paradigm structure. As Carstairs acknowledges, traditional descriptions of German noun declensions do not appear to conform to the strictures of the PEP. Table 7.11 lists the exponents that Carstairs isolates from the traditional principal parts of these declensions.17 Since the smallest class space is defined by the maximally allomorphic cell, the five exponents that realize the non-dative plural in Table 7.11 define a minimum class size of five. This corresponds closely to the number of plural classes recognized in traditional sources such as Duden (2005) (provided again that stem alternations are disregarded in defining classes).

Table 7.11 Exponents of principal parts in German (Carstairs 1983:125)

Nom Sg

0

Gen Sg

0, -(e)s, -(e)n, -(e)ns

Nom/Acc/Gen Pl

0~-e, -"(e), -er~-"er, -s, -(e)n

Parentheses and dashes mark alternations treated as phonologically conditioned or stylistic, and exponents of the form ‘-"er’ indicate an umlauted stem vowel.

To conform to the PEP, the choice of plural ending must determine the nominative and genitive singular endings. The nominative is trivial, since it is always unmarked. However, the genitive appears to vary independently of the plural. This variation is exhibited in Table 7.12, which plots the co-occurrence of plural and genitive singular endings.18 These patterns are obtained from Table 4.11 by collapsing pairs of rows that differ solely in stem umlaut, and replacing the exemplary lexemes by plural and genitive endings.

At first glance, this pattern appears fully incompatible with the PEP, since every plural ending except-er in the second column corresponds to two or even three distinct genitive singular endings. Yet on closer inspection the system is in fact almost maximally economical. As Carstairs argues more generally, traditional ‘declensions’ (and ‘conjugations’) can be brought into closer conformance with the PEP by restricting attention to genuine inflectional variants, and grouping paradigms into common classes if they differ solely with respect to lexically-conditioned properties. In the specific case of German, Carstairs (1983:126) observes that “Gender is lexically determined for German nouns; and we can readily combine each of the Feminine-only ‘declensions’... with some non-Feminine ‘declension, just as we traditionally combine Latin Neuters and non-Neuters in Declension II”. Returning to the classes defined by the P1, P3 and P5 plural endings in Table 7.12, one can see that they exhibit a general contrast between masculines and neuters, which are marked by -s in the genitive singular, and feminines, which are uninflected in the singular. The distribution of umlauted vowels (suppressed in Table 7.12) is likewise a lexical property of nouns in classes P3 and P5.

The only real challenge to the PEP is posed by class P2. There is, first of all, some artificiality in collapsing what are arguably two synchronically distinct plural strategies. Plurals in -(e)n are the default for ‘native’ feminine nouns, as well as for feminine nouns formed with productive endings such as -ung and -heit, as discussed in Chapter 4.3.1. The formation of masculine and neuter plurals in -(e)n is much more restricted and cannot be regarded as productive in the modern language. Even within the non-productive subclass, neuters have unambiguous singular forms. So it is just the masculines that show inflectional variation, with weak nouns such as prinz following the weak singular pattern S2, and mixed nouns

18

Table 7.12 Co-occurrence of plural and genitive singular endings in German

Plu

S1

Masc

Neut

S2

Masc

S3

Fem

P1

-s

-s

-s

_

0

P2

-(e)n

-s

-s

-(e)n

0

P3

-e

-s

-s

_

0

P4

-er

-s

-s

_

_

P5

0

-s

-s

_

0

Though omitting -(e)ns, which occurs with a small and declining set of nouns.

Table 7.13 Economical class space in German

P1

P2

P3

P4

P5

Masc

S1

S1 S2

S1

S1

S1

Neut

S1

S1

S1

S1

S1

Fem

S3

S3

S3

S3

like staat following the strong singular pattern S1. The resulting class space is exhibited in Table 7.13.

Hence by consolidating gender-conditioned variation, one can arrive at a space of six classes, assuming just two P2 subclasses, or seven classes, if productive feminines in -(e)n are distinguished from the masculines and neuters. This exceeds, but is indisputably close to, the minimum of five determined by the patterns of plural exponence. One can in principle bring the system into conformance with the PEP by coercing plurals in -(e)n into a single class and making a “specific exemption” for mixed paradigms, as Carstairs (1983:127) suggests. This kind of exception is of course very different from the consolidation of lexically-conditioned variation. However it is also true that the evaluation of a general principle like the PEP should not hinge too directly on patterns that even traditional sources regard as mixed or hybridized.

It is arguably more instructive to diagnose the general characteristics of German that allow the PEP to work, or at least to work as well as it does. One pivotal property is the clean dissociation of inflectional and lexical variation. The patterns of plural affixation P1-P5 can reasonably be regarded as inflectional, while affixal variation between the singular patterns S1 and S3 can, as Carstairs argues, be attributed to lexical factors. Hence there is only one case, involving masculines of class P2, in which there is inflectional variation in both the plural and singular. Precisely this case is problematic for the PEP.

Although there may be other divisions of labour that promote economical paradigm organization, the separation of lexical and inflectional variation is less a feature of inflection class systems in general than a symptom of the near-complete loss of singular contrasts in German. Hence one useful test case for the PEP comes from declensional systems that are traditionally described in terms of multiple principal parts. Finnish declensions provide a familiar example. Traditional descriptions often list up to five principal parts (typically the nominative, genitive and partitive singular, along with the partitive plural and a plural ‘local’ case such as the inessive). Some of these forms identify stem alternations that are disregarded in determining compliance with the PEP, while others identify affixal patterns. As in the paradigm of Estonian pukk ‘trestle’ in Table 4.8, local case forms in Finnish have mostly invariant endings. Consequently, the grammatical case exponents in Table 7.14 exhibit nearly all of the inflectional variation that is relevant to the PEP.

Examination of Table 7.14 identifies the partitive singular as the most highly differentiated cell, with three distinct patterns -A, -tA and -ttA. The archiphoneme ‘A’ ranges over the vowels a and a, which show harmony with the final stem vowel, so that the three patterns in Table 7.14 have six surface realizations. The partitive

Table 7.14 Grammatical case exponents in Finnish (Karlsson 1999)

Sg

Plu

Nom

0

-t

Gen

-n

-den~-tten, -en

Part

-A, -tA, -ttA

-A, -tA

Table 7.15 Grammatical case forms in Finnish (Pihel and Pikamae 1999)

Sg

Plu

Part Sg

Part Plu

Gen Pl

A

A

asemaa

asemia

asemien

‘position’ (13) [15]

A

tA

perunaa

perunoita

perunoiden

‘potato’ (17) [3]

tA

A

lohta

lohia

lohien

‘salmon’ (33) [24]

tA

tA

leikkuuta

leikkuita

leikkuiden

‘haircut’ (25) [18]

ttA

tA

huonetta

huoneita

huoneiden

‘room’ (78) [4]

A

A~tA

karitsaa

karitsoja

karitsoita

karitsojen

karitsoiden

‘lamb’ (15) [6]

A~tA

A

lahtea

lahta

lahtia

lahtiin

‘bay’ (34) [12]

plural is realized by the first two endings, -A and -tA, but these endings are usually described as varying independently of the partitive singular endings. The genitive plural is realized by three endings, but two of these, -den and -tten are regarded as variants.[1] For the grammatical cells of the Finnish declensional system, the cell sum H is then 3, and the cell product P is 12 (3 x 2 x 2). The actual variation exhibited by grammatical case forms (according to standard descriptions) is set out in table 7.15.[2]

The first two columns in Table 7.15 plot the co-occurrence of partitive singular and plural exponents, followed in the next two columns by forms of exemplary items. We can with no loss of generality restrict attention to word types with unique partitive singular and plural affixes, exhibited in the first five rows in Table 7.15. These word types define the five classes C1-C5 in Table 7.16. A class size of five is closer to the maximum of six defined by the partitive exponents than to the minimum of three dictated by the PEP. But it is still far from the cell product of 12, due to the fact that genitive plural endings do not add any new classes. Instead, variation in the genitive plural largely conforms to the expectations of the PEP in aligning with the partitive plural, with -A implying -en and -tA predicting -den (Karlsson 1999:92f.).

Table 7.16 Classes defined by unique partitive exponents

C1

C2

C3

C4

C5

Part Sg

A

A

tA

tA

tA

Part Pl

A

tA

A

tA

ttA

Gen Pl

en

den

en

den

den

The challenge presented by Finnish is that the partitive plural is itself not predictable from the partitive singular. The four logical possibilities defined by the exponents -A, and -tA are illustrated in the first four rows in Table 7.15: singular -A co-occurs with each of the plural endings -A and -tA and the same is true of singular -tA. Since there is no grammatical gender in Finnish, it is not possible to collapse the classes in Table 7.16 into ‘macroparadigms’ whose internal variation is conditioned by gender differences, as in German. Nor are there any other evident non-inflectional properties that condition the choice of partitive singular realizations. Rather, Finnish simply appears to exhibit an unremarkable pattern of cross-variation. Two of the three realizations of the most highly variable cell, the partitive singular, may each co-occur with one of two partitive plural realizations. The interaction of these choices defines a class space that exceeds the theoretical minimum of 3 determined by the cell sum. This of course does not expand the class space to anything approaching the cell product. Significantly, the variation exhibited by the partitive singular plays no direct role in restricting further expansion of the class space, which is constrained by the coocurrence of partitive and genitive plural realizations.

  • [1] “The ending -den can always be replaced by the ending -tten” (Karlsson 1999:93).
  • [2] The numbers in parentheses identify the word type numbers from the authoritative Nykysuomensanakirja ‘Dictionary of modern Finnish’ (Hakkinen 1990), which are used in Pihel and Pikamae (1999),and the number in square brackets indicates how many of the 8 5 word types exhibit that pattern ofpartitive exponence.
 
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