Number fission from a formal and functional perspective

2.1 Inflectional properties

Diachronically, most Germanic modals are derived from the class of the so-called preterite-present verbs (PPs). This class comprises strong verbs with mainly cognitive semantics whose preterite (continuing the Indo-European perfect) has assumed a resultant state, i.e. present interpretation (see Birkmann 1987). The standard example is the verb wissen ‘know’: Its Old High German (OHG) equivalent ih weiʒ can be rendered as ‘I have seen and, therefore, I know’ (cf. Latin videō). Due to this temporal shift, the paradigm of the PPs becomes defective and the ensuing gaps are gradually compensated for. Formally, they stand between the two main inflectional classes, combining typical properties of both weak and strong verbs in an idiosyncratic manner (Dammel 2011: 140). While ablaut (mainly signaling number) constitutes a strong property (8), additive formation of the preterite and the past participle are typical weak properties (9).

Additionally, PPs show several features that cannot be found in either class: They exhibit ø-exponence in the 1/3pres.sg and, most importantly, phonologically non-conditioned umlaut in the pres.pl and the infinitive (10). These innovations are connected to the analogical extension of the i-umlaut in Middle High German (MHG) and Early New High German (ENHG) times. In terms of inflectional exponence, PPs showed the characteristic second person allomorph -t in the 2pres.sg that gradually came to be replaced by the ‘regular’ flexive -st (Fertig 2019). Conversely, they are regarded as paving the way for the overarching syncretism of the 1pl and the 3pl since it first only occurred in this verb class (Birkmann 1987: 131). In ENHG, new perfect forms after the paradigm of the weak inflectional class (and the ablaut stem of the pst.pl) emerge (Schmidt et al. 2013: 252).

2.2 Inflectional classes as point of reference

Inflectional classes and properties associated with them (e.g. specific class markers or allomorphs) are a classical topic of investigation in morphology since they constitute clear evidence for its autonomy. However, the PPs show several convergencies that are connected to different grammatical domains and sharpen their profile. Most prominently, this class became to be reduced to verbs with modal meanings, leading either to complete regularization or demise of verbs that do not match this profile. Of the 11 PPs of OHG (cf. Braune and Heidermanns 2018: 420–425), 6 still exist and all of them have modal meanings in a broader sense. This even applies to wissen ‘know’ in its ability reading, e.g. Er weiß sich zu helfen ‘He knows how to look after himself’. Some PPs became regularized (e.g. tugan ‘be useful’), others died out (e.g. eigan ‘own’). Conversely, there are also neophytes like wollen (originally stemming from a root verb), and, most prominently, brauchen ‘need’, which is becoming integrated into the modal/PP paradigm (see Maché 2019: 176–220 for a detailed discussion). Thus, the property that singles out PPs the most is that there is a tight link between their morphological irregularity and several syntactic and semantic properties. This connection has often been noted in the relevant literature. Plank (1981: 41), for instance, speaks of the “semantically natural class of modal verbs”; in the same vein, Wurzel (2001: 149) takes it as being indicative of an “extra-morphological motivation”. Simon and Wiese (2011) regard the overall development of PPs as a process of ‘entropy reduction’ in the sense of Ackermann and Malouf (2013), i.e. a stabilization of irregular behavior across different grammatical levels.

As discussed in Dammel and Schallert (2021), two relevant features of canonical inflectional classes (Corbett 2009) lend themselves as suitable point of reference for describing and analyzing the morphomic properties of PPs. On the one hand, their hybrid morphological irregularity can be related to distinctiveness (11a). I will argue in the following section that also umlaut and contraction add to this property. On the other hand, functional motivation of their properties (semantics) undermines independence (11b).

3.1 The empirical dimension

In order to arrive at a more precise picture of number contrasts with modals, I conducted a study on the basis of a wide array of grammatical descriptions.^[2] Such descriptions are abundant in traditional dialectology and have proven to be a very useful source for studying morphological systems (see Schmidt and Herrgen 2011: 88–95, 112–115 for a historical overview on this genre and Birkenes 2014 as well as Fischer 2018 for current studies based on it). In total, 308 respective descriptions were considered, yet only a smaller portion contained sufficient information on the relevant forms for the present study (see below).

The reason for choosing müssen ‘must’ and können ‘can’ was to pick one modal with no number distinction due to ablaut (as is the case with the first verb mentioned) and one that showed such a distinction. If umlaut came indeed to be used as a device for signaling (verbal) number, there should be more pressure on müssen because with können, the number distinction is already established via ablaut. The following factors are in the center of attention:

1. Spread of the umlaut to different paradigm cells. In addition to the pres.sg and the pres.pl also the infinitive and the ptcp were considered. In (12), the relevant forms of müssen ‘must’ are given.

As mentioned in the introduction, older studies like Sčur (1961) or Beckmann (1990) assume that umlaut spread along the implicational scale in (12a), in particular the pres.pl precedes the infinitive. According to the recent investigation by Dammel and Schallert (2021), which is based on a smaller sample of dialect grammars, the directionality is not as clear as previously assumed: While a larger group of dialects corresponds to this cline, there is still a non-negligible number that show the reversed pattern, i.e. umlaut in the infinitive but no umlaut in the pres.pl. In general, it is not so easy to correctly identify relevant forms because other phonological regularities can interact with umlauting. This was one of the reasons for not considering dürfen ‘may’ because ablaut/umlaut distinctions can interact with lowering processes in certain environments. Other cases, however, are quite clear-cut, e.g. when umlaut feeds de-rounding, which occurs quite often in German dialects (see the map in König et al. 2019: 148–149).

1. The second factor is contraction, i.e. elision of stem-final elements in the pres.sg versus pres.pl (13a). In this context, consonant mutations can occur as well, as is illustrated by (13b).

Contraction (and also consonant mutation) is the outcome of a complex interaction between phonological and morphological processes and shows sensitivity to factors like inflectional class or paradigmatic position (see Nübling 2000: Ch. 3 for a detailed discussion and Becker and Schallert 2021: 222–232 for case studies on MHG). Modals are among the core verb classes that show this phenomenon, with first reflexes already emerging in MHG times, e.g. went for wellent (3pl) ‘want’, sün for süllen ‘should’ (inf) or mün for mügen ‘may’, etc. (Paul et al. 2007: 282; Klein et al. 2018: 976, 985). Also in modern dialects, it is abundantly attested (Schirmunski 1962: 548–551).

Another aspect where fission is exploited or even deepened occurs in temporal marking. In the dialect of Mühlheim/Ruhr (Southern Low German) it also occurs with regard to temporal marking. As exhibited by the paradigm given in Table 1, the present forms use a stem ending in the cluster /r/ plus labiodental fricative (going back to Old Low German thurban ‘need’), whereas the past forms exhibit stem-final -s. These forms can be related to OLG durran ‘dare’ (cf. MHG türren) that ceased to exist as an independent verb. Thus, we have a case of heteroclisis, i.e. the inflectional paradigm of this verb is amalgamized of two different stems (Stump 2006).

Umlaut, truncations and consonant mutations can lead to stem allomorphy and even suppletion and thus have a highly irregularizing effect on the inflectional system as a whole, yet they are not randomly distributed. As a rule of thumb, they are more likely to occur in forms with higher token frequency (Nübling 2001a: 66). From a functional perspective, there is a trade-off between shortness (productive economy) and distinctiveness (receptive economy; Nübling 2001a: 69). What is unclear, however, is whether these strategies express a categorial contrast at some diachronic stage, i.e. plurality in the verbal domain. I will return to this question at the end of this section.

Let us now have a look at the different patterns we can observe in the dialect sample. In Table 2, the sample sizes for the different factors under investigation are given. Table 3 shows the spreading of umlaut in the different paradigm cells under consideration. Both müssen and können show a strong tendency to mark umlaut in the plural versus singular forms (99/135 cases, respectively). Strikingly, the converse pattern is not attested. However, there are also systems that either generalize umlaut to the whole pres (10/2) or do not show any traces of this phenomenon at all (31/17).

A closer look at other paradigm cells, as shown by Table 4, reveals that umlaut quite often extends beyond the pres.pl to the inf and the ptcp, once again with both müssen and können (60/68); a typical example is given in (14a). Cases where it spreads only to the inf are also very frequent (28/52). These proportions are followed by the number of systems that show no traces of umlaut at all (15/4), see (14b). Violations of the expected pattern like (14c) and (14d) do occur, yet they are comparatively rare (5/6) (X denotes ‘no umlaut’, U ‘umlaut’). Note that in many German dialects, de-rounding can disguise umlauted vowels, as is the case with (14a). In the present context, this complication is irrelevant since the respective vowels clearly contrast from their un-umlauted counterparts.

Turning to contraction patterns, as given in Table 5, the two modals show an opposite behavior. While müssen favors truncation in the plural (13/0), cf. (15a), können is prone to short forms in the singular^[3] (0/25), see (15b). This is against the background that uncontracted forms in both the singular and the plural are by large more common (139/124), while we rarely also observe short forms in the whole pres (3/1). With müssen, contraction ‘conspires’ with umlaut (11 cases), yet there are hardly any signs for such an interaction with können (2 cases).

Regarding other positions in the paradigm (Table 6), there is robust evidence for a mirror-image behavior of the two verbs considered (C denotes ‘contraction’, X ‘full form’). Typical examples are given in (16). Irrespective of contraction, the infinitive always patterns with the pres.pl (5/25). In many cases, however, no truncated forms are observed in any position (müssen: 120, können: 113).

Another system that is sufficiently attested in my data (7 cases) only occurs with müssen. In contrast to the mirror image patterns reported in Table 6, it involves contraction in the pres.pl but a full form of the pres.sg and the inf. This system is illustrated and analyzed in Section 4.1.

Let me summarize: Both verbs show a strong tendency to mark umlaut in the plural versus singular. What is more, umlaut can spread to the inf and the ptcp as well. Thus, the idea of umlaut as transcategorial number marking device has some diachronic plausibility in terms of its quantitative distribution (sg vs. pl). However, its spreading to other paradigm cells is clear evidence for a morphomic distribution because the inf and the ptcp are underspecified for number features. As regards contraction, we encounter a more complex situation: With können, it favors the sg, with müssen, it is the other way round, meaning that number fission is created from different directions. Both verbs also show extensive evidence for short forms of the inf, which is once again indicative of the morphomic nature of this phenomenon.

3.2 Overdifferentiation

One possibility somewhat related to Nübling’s approach is to take analogical umlaut in the context of modals as a case of overdifferentiation, i.e. “the situation where a particular lexeme (or group of lexemes) makes ‘too many’ distinctions, in comparison with the majority” (Corbett 2015: 157). A typical example would be the paucal in languages like Bezhta (a North Caucasian language), which only appears with a handful of nouns, e.g. sik ‘wineskin’ (sg) – sika (pauc) – sikla (pl) (Xalilov 1985: 137; see also Corbett 2015: 158). Analogically, number could be regarded as a feature value that is only present in a subclass of verbs, along the lines of the working definition in (17).

The problem with this approach is that number as a verbal inflectional feature value in German shows highly cumulative exponence, meaning that it cannot be easily teased apart from person or even tense features. This can be seen in (18) where some forms of the pres for strong (18a), weak (18b) and modal verbs (18c) are given. In all cases, person and number information overlaps on the affixal level. Apart from umlaut in the pres.pl of modals, some strong verbs show umlaut in the 3sg (and also the 2sg), i.e. with these verbs it is the singular that is marked by this non-concatenative strategy.^[4]

Note that the weak verbs, the class with the highest type frequency, show syncretism between the 3sg and the 2pl, cf. sie mach-t (3sg) versus ihr mach-t (2pl). Since this type of syncretism is crosslinguistically rare, we would expect it to be a natural target for reinforcing number distinctions.^[5] However, number is always unambiguously marked via agreement rules: Only pronominal DPs trigger person and number agreement (19a) whereas full DPs always come with 3sg or 3pl (19b)/(19c), depending on the controller’s number features (e.g. in coordinations or with pluralic subjects). Impersonal passives (20a) and clausal subjects (20b) necessitate the 3sg form, which is indicative of its default status (Wöllstein [Duden] 2022: 125–126; Himmelreich & Hartmann 2023, and others).

As an overarching tendency we can note that there is no functional need for (transcategorial) number marking in the verbal domain because number is always unambiguously marked – either in the verbal domain itself or via agreement regularities.^[6] Consider the minimal pair in (21): In (21a), verbal syncretism between 3sg:2pl is disambiguated by the subject pronouns while in (21b) it is verbal inflection that compensates for the syncretic pronoun, so to speak.

One reviewer rightly notes that what distinguishes Nübling’s concept of transcategorial number marking from overdifferentiation is the ‘depth’ of a split within a paradigm, so that “there is no need for an additional category that other verbs don’t have”. In this context, the paucal example is somewhat dubious because number as a category (and plural as the respective value) is not altogether missing in verbal inflection. There are only different means of how it is expressed, i.e. via additive morphs (albeit with fusion of number and person) or via stem alternations (umlaut). While I concede that overdifferentiation might be a misnomer for the situation we observe, the general question remains why number should only be reinforced in one particular verb class.

3.3 Where does umlaut with PPs come from?

Let us shortly address the question of the historical genesis of umlaut with PPs. Different previous explanations are discussed and reviewed by Nübling (2009: 212–219) so I will not refer to them in a lengthy manner. Instead, I focus on her notion of ‘transcategorial plural marking’ (see Section 1). The upshot is this: In diachronic terms, transcategoriality is a plausible explanation for the state of affairs in MHG. With the core modals (those PPs that still belong to this class in Modern German), umlaut in the pres.pl is already robustly attested, yet not very frequent (Klein et al. 2018: 891–892). In some cases, this alternation is also attested in the sg (müezen) or in the ind.pst (mügen; Klein et al. 2018: 906–907). Only in ENHG there is robust evidence for umlauted infinitives but the exact directionality and chronology of this spread is unclear (see Dammel and Schallert 2021: 209 for further references). Another neat aspect of this explanation is that it fits well with what we know about areal diffusion. In Upper German, particularly Alemannic, umlaut in the nominal domain is much more frequent than in Standard German and other varieties of German (cf. Nübling 2009: 220). Basically, transcategorial number marking is an explanation that resorts to proportional analogy (Paul 1920: Ch. 5), as captured by the proportional equation in (22). Picking up the basic distinction between intra- and inter-paradigmatic analogy (Nübling et al. 2017: 79), transcategorial plural marking belongs to the latter type if we assume a broader interpretation of ‘paradigm’ in the sense of different lexical categories and their inflectional properties.

In synchronic terms, however, transcategorial number marking fails to be a valid explanation for the occurrence of umlaut in the inflectional paradigm of PPs because it does not code for the feature ‘plurality’ in a consistent way. There is a substantial number of dialectal systems where it has also spread to the infinitive and (albeit less frequently) to the ptcp. Of course one could argue that there is also robust evidence for transcategoriality. In the sample of Dammel and Schallert (2021: 209), which covers all five core modals, there are dialectal systems that correspond exactly to the predicted distribution so there might be additional factors that explain the further spreading to other paradigm cells. However, there is a more general problem with functional explanations like this one: They do not conform to the standard model of scientific explanations, the Hempel–Oppenheim schema (Hempel and Oppenheim 1945). In this model, the explanandum (the respective phenomenon) is explained as the deductive consequence of the initial conditions and general or covering laws. In particular, functional explanations fail to specify the necessary/sufficient conditions (C) under which a certain morphological feature or feature value (F) is expressed (see Haider 2018: 85; Hempel 1959: 283–284).

1. If C _i is necessary for F (F ⇒ C), then there should be no C _j that also fulfills F.

2. If F is sufficient for C (C ⇒ F), then inference from F to C is logically invalid (abduction).

As discussed in Section 3.2, umlaut (or other strategies of deepening number fission with PPs) is not necessary because in the verbal inflectional paradigm, number is already – and unambiguously – coded by other morphological means. If it were a sufficient condition, we would wrongly conclude C ⇒ F, F ⊢ C because C ⇒ F does not imply F ⇒ C (‘affirming the consequent’). Actually, this sort of inference constitutes abduction, a concept introduced by Charles Peirce (1839–1914) to model thought processes that lead to new hypotheses (Peirce 1931: 188–189). While not logically valid, abduction has proven to be powerful tool for modeling the cognitive mechanisms responsible for language change (Andersen 2017) and even the classic analogy concept may involve this kind of inference.^[7]

Minnameier (2010) gives an interesting characterization of analogical reasoning. In the psychological literature on this phenomenon, it can be described as a process where schemata are transferred from a source (where the analogy is drawn) to a target domain (the domain where it is applied). This process comprises two subprocesses, matching and mapping: “[F]irst a target […] and a source […] have to be matched, then the relevant features of the source have to be mapped onto the target” (Minnameier 2010: 108). Crucially, analogical reasoning involves both an inductive and an abductive step, which can for the present purposes be roughly characterized as follows:^[8]

1. Step 1: There is a more or less superficial relation between source and target, i.e. plurality is marked both in the nominal and verbal domain by stem alternations (umlaut in the first, ablaut in the latter).

2. Step 2: Stem alternation via umlaut as a sign for ‘plurality’ is mapped on the target: As bridging context, the verbs OHG muos- ‘must’ and mag- ‘like’ act (cf. Nübling 2009: 210) because they have a uniform stem in the pres.^[9]

Analogy could also be responsible for umlaut in the infinitive of PPs even though its directionality relative to the pres.pl is unclear. Sčur (1961: 216) assumes that homophony with the 3pl.sbjv.pres in MHG emerged as the causing factor (cf. Birkmann 1987: 196). Crucially, also morphomic processes can be extended by analogy (Maiden 2005) so that an explanation along these lines seems feasible. However, I have to leave this matter open for further research. Besides their diachronic implications, there are plausible ideas how functional explanations can be made to work. They necessitate a detailed analysis of the containing system and its form-function interactions:

[…] functional ascriptions do require relativization to a ‘functional fact’ about a containing system, i.e., to the fact that a certain capacity of a containing system is approximately explained by appeal to a certain functional analysis. (Cummins 1975: 763)

Viewed from the angle of inflectional classes as the containing system, number fission is a bundle of irregularization strategies that has a stabilizing effect on the PPs and sharpens their profile (see Section 2.2). In particular, it strengthens distinctiveness as one feature of canonical inflectional classes.

4.1 Stem spaces

The tool for capturing stem allomorphy or (weak) suppletion is the concept of stem spaces (Bonami and Boyé 2006): Lexemes are associated with a vector of possibly different phonological representations; they belong to the lexeme’s lexical entry (24). Inflectional rules specify which coordinate in the vector is used as input. In the type logic of IBM, a morphological word is represented as an ordered list of lexemes that are stored in the attribute m-dtrs (‘morphological daughters’), cf. (23). Each lexeme is equipped with its stem space with n slots, as illustrated with a verbal category in (25) and (26), and inflectional rules specify which slot of the stem space is used (27).

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Let me now explore the different stem spaces we observe with modals. A caveat: I do not address the question how inflectional affixes are analyzed because this aspect necessitates a more detailed analysis of the respective dialectal systems. In many cases, it is difficult to arrive at a clear segmentation of stem plus affix, particularly with contracted forms. In a word-based account, this poses no general problem since bound morphs do not carry semantic information per se – structural descriptions license specific word forms.^[10] In the logic of IBM, concatenative processes can be modeled via the append relation (⊕) that modifies the phonological shape (phon-attribute) accordingly. This distinction in terms of modeling exponence is relevant because in several German dialects the suffixes of the inf and the pres.pl become dissociated while still targeting the same stem level.^[11] Conversely, syncretism between the two forms appears very early and can be regarded as a characteristic of the PPs in MHG times (Klein et al. 2018: 896). Note that there is a connection to the uniform pres.pl we observe in many Alemannic and Low German varieties because the respective allomorphs -en/-ed can be either based on the old form of the 1.pres.pl or the 3.pres.pl (Ebert et al. 1993: 248–249; König et al. 2019: 128). This means that with the -en class, syncretism can extend to the inf.

For PPs, we can assume the partial hierarchy of stem spaces in (28) that models stem selection for the pres.sg, pres.pl and the inf; for the pst and ptcp certain adaptions would be necessary, which I do not discuss in detail.

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The simplest case is presented by PP-lexemes that do not show any stem allomorphy; they represent the default specification regular in (28) and are described in (29), where only the content of one slot is listed (cf. Bonami and Boyé 2006: 371). The typical case would be sollen ‘should’ in Standard German. Assuming the analysis of umlaut described in Section 4.2, also müssen ‘must’ belongs to this class.

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The next stem type, contr-sg, subsumes PP-lexemes with fission due to ablaut and/or contraction. In this class, the form of the infinitive always shares the phonology of the pres.pl, yet in both cases the forms are uncontracted, cf. (30).

Nübling (2000: 9) regards the form of the infinitive as a distinguishing feature of the morphological class of short verbs. This cluster of verbs, among them have, come or give, developed highly irregular forms in most Germanic varieties due to factors like token frequency and/or analogical interactions with root verbs (i.e. verbs that do not have stem-forming elements historically). She does not regard the infinitive as the base form of the respective verbal paradigms but observes that there seems to be an implicational relationship between finite short forms in the pres and a contracted inf. Against this background, I assume that there is a designated slot in the stem space (bse ‘base’) that is reserved for the phonology of this form and that can be targeted by morphosyntactic rules like the substitute infinitive (IPP). This slot also constitutes the interface between the stem spaces of PPs and short verbs, which share many similarities, among them analogical umlaut in the pres.pl and other paradigm cells (see Nübling 2009: 220–223). Thus, this type has the stem space described in (31).

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The stem type no-contr-sg in (32) shows a full stem in the pres.sg and the inf but contraction in the pres.pl; it is illustrated with the example in (33).

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To model the mirror-image effect with können ‘can’ and müssen ‘must’ that is frequently attested in my sample, one additional stem type, is necessary. It is contr-inf, as described in (34) and illustrated by (35). It inherits the full form of the pres.sg and the contracted form of the pres.pl from no-contr-sg but shows a contracted form of the inf.

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One may wonder, as one reviewer does, why the stem vowels in the different stem representations do not match the respective umlauted forms in the dialect examples while contraction is represented directly. This becomes clear in the following section: Umlaut is regarded as a morphophonological rule that applies to the stem space with the highest index, whereas contraction is inherently a feature of stem spaces themselves (and the selection rules specified for them). The logic behind this approach is that contraction is a synchronically less transparent rule than umlaut. It is the consequence of a diachronic process that leads to weak suppletion and the different geometries of stem spaces we observe.

4.2 Umlaut

Let us now have a closer look at the analysis of umlaut. In modern Standard German, this alternation involves the back vowels /a/, /o/, /u/ and the diphthong /aʊ̯/ that contrast with the fronted vowels ä /ɛ/, ö /ø, œ/, ü /y, Y/, and äu /ɔɪ̯/. Haspelmath and Sims (2010: 214–217) discuss several properties that distinguish purely phonological (automatic, in their terminology) from morphophonological alternations. While the first class can be described in purely phonological terms, this is not possible in the latter case, and German umlaut belongs here. For expository reasons, I focus on some of the criteria; a more thorough discussion can be found in Wiese (1996).

Firstly, umlaut always applies in derived environments.^[12] While it can be observed in an impressive range of morphological environments, e.g. plural formation (36a), derivational processes like similatives (36b), subjunctive formation (36c), adjectival comparatives (36d), etc., there is always a base value of the respective category that does not show umlaut.

Secondly, umlaut is clearly morphologically or lexically conditioned, meaning that idiosyncratic exceptions occur (Wiese 1996: 122–123). Historically, it is first attested in OHG.^[13] In its first stage, it existed only as an allophonic process and was then integrated into the grammatical system in a stepwise process (Twaddell 1938). In MHG we find clear evidence of umlauted vowels while their historical triggers (i, j) were largely leveled to e in unstressed syllables (Salmons 2018: 206). During this period, and in particular in ENHG, umlaut spread via analogical extension to environments where it had never been phonologically triggered (see Sonderegger 1979: 297–319 for a detailed chronology).

An instructive example for the lexical/morphological dimension of umlaut is comparative morphology: It only affects a rather small range of (usually token-frequent) adjectives and only those with the stem vowels a, o, u (irrespective of their phonological correspondences), e.g. hart ‘hard’ – härter, groß ‘big’ – größer, dumm ‘stupid’ – dümmer; the diphthong /aʊ̯/ and polysyllabic stems are exempt, the only notable exception being gesund ‘healthy’ – gesünder (Augst 1971). Nowak (2017: 92) concludes that adjectives corresponding to the phonotactic scheme [(C)(C)aSC] are most prone to umlauting while deviations correlate with a lower rate of this alternation (C = ‘consonant’, S = ‘sonorant’). Note, in passing, that in contrast to automatic alternations, loans or borrowings are not systematically affected even if there are some examples like e.g. Popo ‘butt’ > Popö-chen (dim); Europa ‘Europe’ > europä-isch ‘European’ (Wiese 1996: 122).

The third criterion has different facets that are closely related so that I address them together; they involve phonological coherence/distance and the creation of new segments. Morphophonological alternations are not phonologically coherent, meaning that the alternating segments do not form a natural class. Describing umlaut only in terms of e.g. a fronting rule along the lines of [+ front] is not sufficient because a-sounds (/aː, a/) are also raised (/ɛː, ɛ/), and for the diphthong /aʊ̯/ additional rounding has to be assumed (Wiese 1996: 121–122). While umlauting in its first phase did indeed create new vocalic segments (fronted allophones), they later became phonematized. Thus, front vowels can also be found in stems and independently of morphological contrasts, e.g. fühl-en ‘feel’, Schlüssel ‘key’, schön ‘beautiful’, Hölle ‘hell’, etc.

For the present purposes, I assume the analysis of umlaut developed by Wiese (1996: 120–122), who treats it as a lexically-governed, phonological rule. Concretely, it can be regarded as a floating feature that is realized via a nonlinear linking rule, and the problems with phonological coherence can be overcome with a suitable feature geometry and redundancy rules. This approach straightforwardly reflects the ‘irregular’ and sometimes idiosyncratic character of this alternation. What about the morphological dimension, then? There is no denying that umlaut is employed as a morphological marker, but “the range of the categories involved comprises more or less the whole universe of morphological distinctions to be overtly marked in the German language” (Wiese 1996: 124). One might also add that this polyfunctionality even includes the morphomic level, i.e. signaling nothing more than inflectional class coherence (‘distinctiveness’ in Corbett’s 2009 terminology). Against this background, the morphology of umlaut has more to do with the organization of stem spaces than with feature signatures.^[14]

In IBM, umlauting can be implemented as follows: There is an additional slot in the stem space (ul-stem); as lexically-governed rule, it applies to the slot in the stem space with the highest index (37). If we revisit the three irregular stem spaces from Section 4.1, here repeated as (38)–(40), we see that this analysis captures the relevant facts correctly. For the infinitival stem (bse), we can assume that if there is an ul-stem targeted by the UL-rule, it can also be applied to this slot (but not vice versa).

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5.1 Main findings

In this paper, I showed that number fission with PPs is the effect of different irregularization strategies, most prominently umlaut and contraction. My observations point to the conclusion that these facets of irregularity are, synchronically speaking, morphomic because they also systematically co-occur in other paradigm cells: They serve no other purpose than signaling inflectional class coherence. A promising approach to capture these complex patterns of stem allomorphy is IBM and the concept of a stem space (Bonami and Boyé 2006), which allows to express implicational or directed generalizations in stem formation on a purely morphological level. In general terms, I favored a more form-based (or ‘formal’) account of the relevant facts. This does not mean, however, that I do not see any room for functional explanations. Which role they play in my account and how I think about them more generally is the topic of the concluding section.

5.2 General remarks: form versus function in morphology

Functional explanations have a long tradition in linguistics, particularly in the typological and diachronic branch. In the latter, for instance, the unidirectionality of grammaticalization processes or the question whether it constitutes a distinct process from reanalysis has lead to a lively and, I believe, fruitful debate (Campbell and Janda 2001; Norde 2009, and others). In the 1990s and early 2000s, an intensive discussion emerged about their general validity (Haspelmath 2000; Newmeyer 2003), yet without any negative effect on the popularity of such explanations. In Haider’s (2018) opinion, this must be seen against the background of many unwarranted and ill-founded developments in mainstream generative grammar (aka the Minimalist Program, cf. Chomsky 1995 et seq.) and the increasing appeal of usage-based models like e.g. Construction Grammar (Goldberg 1995 et seq.) as their most important proponents. For capturing and explaining crosslinguistic generalizations, usage-based theories “appeal to general cognitive constraints together with the functions of the constructions involved” (Goldberg 2003: 219). In some sense, formal and functional explanations have gained axiomatic status that circle around the three autonomy theses discussed in Newmeyer (1998: Ch. 2). The most important one is Chomsky’s (1965, pp. 3–4) (in)famous distinction between competence (“the speaker-hearer’s knowledge of his language”) and performance (“the actual use of language in concrete situations”). Most generative models accept this premise, while most usage-based models reject it (Diessel 2015). In my opinion, there are good reasons to stick with the competence-performance dichotomy, however. Several convincing arguments are presented in Newmeyer (1998: 55–77, Ch. 3) and Newmeyer (2003). Another hotly debated question is the independence of grammatical levels like syntax from semantics or, in the realm of morphology, the relative autonomy of morphological patterns (morphomes in Aronoff’s 1994 diction).

Quite paradoxically, those debates hardly touch on the daily business of linguists working in different fields, which I regard as a good thing. Our main goal is to better understand why grammatical systems are the way they are and, more specifically, how form relates to function (and vice versa). Nobody would claim that there is a strict isomorphism between these two levels, and in order to tackle possible interactions, very different grammatical frameworks have come up with useful tools. It is in the use of these devices where many convergences can be found, some more hidden, others more readily apparent. Typed feature structures (in the form of attribute-value matrices, i.e. AVMs) and type hierarchies, for instance, were developed in head-driven phrase structure grammar, a genuinely ‘generative’ framework, yet they were soon carried over to construction grammar because they are a simple and powerful means for modeling grammatical dependencies. Conversely, implicational rules (usually bundled as scales) are widely used outside of the realm of functional-typological approaches where they started their career, so to speak.

More specifically, there are several functional explanations in morphological analysis that have proven to be insightful. Why should we abstain from using them as tools in more ‘formal’ theories, then? One example is (diagrammatic) iconicity in the guise of Bybee’s (1985) relevance principle or structural iconism, as assumed in Naturalness Theory (Mayerthaler 1981: 25; Wurzel 2001: 22). As discussed by Newmeyer (1998: 114–118, 129–130), cognitive pressure for structure-concept iconicity is a plausible external (i.e. usage-based) force responsible for shaping grammatical structure. Another functional motivation we had a closer look at is ‘transcategorial number marking’ (Nübling 2009). It resorts to analogy as a cognitive factor underlying the extension of number fission, most prominently with umlaut. Even if there are problems with the synchronic validity of such an explanation, it offers a plausible scenario for the first step of this process. Viewed from the angle of inflectional classes as the “containing system” (Cummins 1975: 763), number fission is a bundle of irregularization strategies that has a stabilizing effect on the PPs and sharpens their profile with respect to the other classes (see Section 2.2). This morphomic stability can be successfully captured with the logic of typed feature structures and a suitable geometry of stem spaces.