Optimal Gaps in Optimal Paradigms

Abstract This article develops a treatment of phonologically motivated gaps in paradigms within Optimal Paradigms Theory (McCarthy 2001). The gaps are instances of absolute ungrammaticality, i.e. situations in which there is no output for a given input. In the approach developed here, gaps arise when phonological markedness constraints outrank constraints requiring the expression of a particular morphological category. In this way, incomplete paradigms can be selected as optimal, since the incomplete paradigm will incur fewer markedness violations. The optimal gaps approach is illustrated with Norwegian imperatives and Hungarian CCik verbs. The analysis is compared with earlier treatments of absolute ungrammaticality, such as the null parse of Prince and Smolensky (1993). Key words: gaps, ineffability, Optimality Theory, Optimal Paradigms, phonology; Hungarian, Norwegian. * Various fragments of earlier versions of the material in this paper have been presented at the University of Göteborg (Scandinavian Workshop on Optimality Theory, November 2002), the University of Durham, University College London, Katholieke Universiteit Nijmegen, Rijksuniversiteit Leiden, Université de Toulouse-Le Mirail (Des représentations aux contraintes, July 2003) and the University of Tromsø (CASTL Kick-off Conference, October 2003), Eötvös Loránd University and the Hungarian Academy of Sciences. Thanks to those audiences, and also for helpful discussion to Ricardo Bermúdez-Otero, Sylvia Blaho, Patrik Bye, Dafna Graf, Ove Lorentz, John McCarthy, Bruce Morén, Péter Rebrus, and Miklós Törkenczy. An anonymous reviewer has also made several useful suggestions, for which I am grateful. Responsibility for error remains mine. Catalan Journal of Linguistics 4, 2005 155-170 Optimal Gaps in Optimal Paradigms* Curt Rice University of Tromsø Center for Advanced Study in Theoretical Linguistics (CASTL, http://castl.uit.no) NO-9037 Tromsø (Norway) [email protected] Table of Contents 1. Introduction 2. Phonologically motivated gaps 3. Optimal Paradigms Theory 4. Mardedness vs. category expression 5. Optimal gaps 6. Conclusions References Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 155 1. Introduction When a morphological operation yields a form which is phonologically infelicitous, that cell in the paradigm experiences one of three fates. Either it is filled with a phonologically «repaired» form, or it is filled with a substitution from elsewhere in the paradigm, or the cell remains empty such that there is a gap —which in turn may ultimately be filled by periphrasis or circumlocution. We refer to these three strategies for resolving phonological infelicity respectively as phonological neutralization, morphological neutralization and syntactic neutralization. A theory of grammar must allow language specific grammars which yield any of these possible resolutions in the face of such a situation. This paper focuses on gaps of the third type —syntactic neutralization, i.e. gaps which are seemingly left unfilled by the morphology and whose target is left unrepaired by the phonology, such that the only way to express the intended content of the cell is to turn to the syntax. A proposal for the representation of such cases is developed within the framework of Optimality Theory (Prince and Smolensky 1993) and specifically within Optimal Paradigms Theory (OP) (McCarthy 2001, 2002). In OP, candidates are paradigms, and there is a set of correspondence constraints —OP faithfulness constraints— exerting a leveling pressure on intraparadigmatic relations. The members of the candidate paradigms are also evaluated against the other constraints of the grammar, including markedness constraints. A relatively high ranked markedness constraint can yield a winning candidate paradigm with an «optimal gap». Several examples of this are given below. One particular advantage of this approach is that a grammar may respond to phonological infelicity differently, depending on whether it occurs in the context of a paradigm or not. For example, an ill-formed consonant cluster may go unrepaired in a paradigm, resulting in a gap, while it does get repaired in the context of a borrowed place name. A second advantage of the present model is drawn in comparison to the two existing models of this kind of ungrammaticality, namely MPARSE (Prince and Smolensky 1993; McCarthy 2002) and CONTROL (Orgun and Sprouse 1999). The analysis of gaps as part of a model of (un)grammaticality offers yet another domain in which the notion of paradigm is useful, supplementing McCarthy’s (2001) appeal to processes such as attraction to the unmarked. The context for this proposal includes work on ineffability, absolute ungrammaticality, and paradigm structures, many recent instances of which appear in Downing et al. (2005). In §2, two brief examples of phonologically motivated gaps are presented. Optimal Paradigms Theory is presented in §3, followed by my proposal for the representation of optimal gaps in §5. §4 highlights a technical problem in Optimal Paradigms Theory, and proposes a solution to that problem which results in a theory that optimizes gaps only in the desired situations. This slightly revised version of OP Theory is then applied to two more examples of paradigmatic gaps, namely the case of Norwegian imperatives, in §5.1, and the case of Hungarian CCik verbs, in §5.2. The paper ends with a discussion of the implications of the present development. 156 CatJL 4, 2005 Curt Rice Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 156 2. Phonologically motivated gaps The notion of a gap in a paradigm invokes the notion of a «complete» paradigm, a point which will be of crucial importance in §5 below. A familiar metaphor for this line of discussion refers to «cells» in a paradigm, where each cell represents the expression of a particular morphological category. A language in which there are six possible expressions of the present tense of a verb (3 persons, 2 numbers) has six cells for that part of the paradigm. When one of the cells is left empty, there is a gap in the paradigm. There are various reasons why cells might be empty; of interest here are cases in which cells are empty because the form which is expected to fill that cell is phonologically ill-formed. Subsections 2.1 and 2.2 each present an example of such a gap. 2.1. Turkish suffixation One of the familiar examples from the literature on phonologically motivated paradigm gaps involves suffixation in Turkish (Itô and Hankamer 1989; Orgun and Sprouse 1999; Raffelsiefen 2004). The output of suffixation in Turkish must meet a phonological requirement on its shape. In particular, a suffixed form must be minimally disyllabic. When this requirement is not met, suffixation fails, and the paradigm is incomplete. The particular danger for a gap in Turkish arises from attempts to combine monosyllabic vowel final stems with monoconsonantal suffixes. Example (1a) shows the affixation of a the genitive suffix -mto a CVC stem. In this case, epenthesis applies to create a well-formed disyllabic word. In (1b), however, epenthesis does not apply, since there is no immediate phonotactic violation such as an unsyllabifiable cluster (for details, see the references above). The «intended» CVC output is phonotactically well-formed. However, it fails to meet the minimal size requirement and therefore is ill-formed. The cell remains empty. (1) Disyllabic minimality for bimorphemic Turkish words a. (i) soly‘musical note G’ (ii) soly-üm ‘my G’ b. (i) do‘musical note C’ (ii) *do-m ‘my C’ 2.2. Swedish neuter adjectives The phonology of Swedish can block the normal process of neuter inflection (Eliasson 1975; Iverson 1981). Swedish adjectives agree in gender with the noun they modify; when the noun is masculine, it has no ending, while the neuter variant is inflected with a -t, as seen in (2). Optimal Gaps in Optimal Paradigms CatJL 4, 2005 157 Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 157 (2) Well-formed neuter marking in Swedish a. en rysk pojke (masc.) ‘a Russian boy’ b. et rysk-t barn (neut.) ‘a Russian child’ However, when an adjective has a short vowel followed by the voiced coronal stop [d], it can only be used in these constructions with masculine or feminine nouns, but not with neuter ones. Attaching the neuter suffix yields an ungrammatical construction, as seen in (3). (3) Ill-formed neuter marking in Swedish a. en rädd pojke (masc.) ‘a scared boy’ b. *et rädd-t barn (neut.) ‘a scared child’ The ungrammatical form in (3b) is not repaired by voicing assimilation or epenthesis or any other imaginable strategy. Rather, the neuter form of the adjective is unutterable, such that the paradigm has a cell which cannot be filled synthetically. The only way to express the concept is with a phrase, such as et barn som er rädd ‘a child who is afraid’.1The ungrammaticality of *rädd-t in (3b) is due to the infelicity of [dt] as a coda cluster and to a grammar in which repair by devoicing the [d] is not possible, possibly due to high ranking requirements preserving the integrity of the stem.2 2.3. Analyzing gaps The data given in this section from Turkish and Swedish —along with that from Norwegian and Hungarian analyzed in §5 below— present a challenge to Optimality Theory. In OT, the core strategy is not to look for an output which is perfect, but rather to find the candidate which is best, given a particular grammar. The architecture of OT is such that some candidate will always be best; an input is always mapped onto some output, such that there is no parallel to a crashed derivation. Yet, paradigms with gaps would seem to be situations in which no candidate is good enough. The literature includes at least two proposals for analyzing the situations under consideration here, the «null parse» of Prince and Smolensky (1993) and the theory of control proposed in Orgun and Sprouse (1999). Very briefly, the approach of 158 CatJL 4, 2005 Curt Rice 1. Predicative adjectives in such a construction should in principle agree in gender as well, cf. et barn som er ryskt ‘a child who is Russian’, but this gender agreement requirement is weaker for the predicative construction than it is for the attributive one. Hence, et barn som er rädd is marginally acceptable while *et räddt barn is unambiguously ungrammatical. 2. In fact, there are very few words having this shape —and therefore meeting this problem— in Swedish. Iverson (1981) reports only two, rädd ‘scared, afraid’ and fladd ‘flat (water)’. The same examples hold in Norwegian, where denominal examples can be constructed, e.g. nedsnedd ‘snowed in’, lit. down-snowed. There are also unutterable in the neuter. Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 158 Prince and Smolensky (1993) includes a candidate in each tableau which effectively is unpronounceable, which they call the null parse. That candidate violates certain well-formedness requirements, and will thereby be ruled out whenever the relevant constraint(s) are sufficiently high ranked. In the rare case that the relevant constraint(s) are relatively low ranked, however, this candidate will win. Since it is unpronounceable, this is essentially a model which maps an input onto no (surface) output, i.e. it is a model of absolute ungrammaticality. McCarthy (2002) expands on the formalism slightly, stipulating that the null parse (which he calls the «null output») violates exactly one constraint, MPARSE, by stipulation. Orgun and Sprouse (1999) reject the null parse approach and propose instead a new evaluation component of the grammar —a supplement to EVAL— which they call CONTROL. CONTROL is a domain in which constraints are «hard», i.e. their violation is fatal. The optimal output from EVAL is passed along to CONTROL; if it also satisfies the constraints in CONTROL, the form surfaces. If it does not satisfy the constraints in CONTROL, then it fails. For a recent critique of this approach, cf. Raffelsiefen (2004). The optimal gaps approach developed below is an alternative to these two, and superior to them because it allows elimination of the stipulative violation of MPARSE and because it does not require positing a new component of the grammar, cf. CONTROL theory. 3. Optimal Paradigms Theory Optimal Paradigms Theory invokes the notion of the paradigm to define a domain for correspondence relations (McCarthy 2001). In OP Theory, candidates are paradigms, such that the candidates which are in competition with one another will present different possible paradigms for some stem. The formal strategy for assessing intraparadigmatic correspondence is the OP-correspondence constraint, i.e. correspondence constraints limited to examining the stem portion of each member of the paradigm. In his paper, McCarthy (2001: 5) summarizes the principles of the theory as in (4). (4) Core properties of Optimal Paradigms Theory a. Candidates consist of entire inflectional paradigms. b. Markedness and input-output (IO) faithfulness constraints evaluate all members of the candidate paradigm. The violation marks incurred by each paradigm member are added to those incurred by all the others. c. The stem (shared lexeme) in each paradigm member is in a correspondence relation ℜOP with the stem in every other paradigm member. There is no distinctive base —rather, every member of a paradigm is a base of sorts with respect to every other member. d. There is a set of output-output faithfulness constraints on the ℜOP correspondence relation. Optimal Gaps in Optimal Paradigms CatJL 4, 2005 159 Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 159 This model is intended to formalize a strategy for analyzing leveling effects in paradigms, and in particular cases in which the attractor is not derivationally simpler. McCarthy notes that the phonologically least marked candidate stem can triggering leveling effects throughout the paradigm. To illustrate the theory and set the stage for our enhancements, part of McCarthy’s discussion of the Classical Arabic templates is briefly reviewed here. The Classical Arabic verbal template ends in CVC]. Both CVC] and CVCC] are impossible template shapes for verbs, although noun templates can end in either of these. Verbs and nouns share the following two properties. (5) Constraints on Classical Arabic verbal stems a. Trimoraic syllables are ungrammatical: *µµµ] σ b. The stem portion of each member of the paradigm should be identical w.r.t. vowel length, which can be captured, e.g., with OP-MAX(V)-µ. The differences between nouns and verbs fall out from the fact that there are C-initial verbal suffixes but no C-initial nominal suffixes. Since verbal suffixes may begin with a consonant, then a verbal stem ending in VC or VCC could only be syllabified in ways violating the prohibition on trimoraic syllables: VC.C... or VCC.C... Since the suffixes for nouns all begin with a vowel, stems may end in VC or VCC and still be acceptably syllabified under suffixation: V.CV... or VC.CV... A fragment of an OP tableau is given in (6), showing a hypothetical verbal input CVCVC, and candidate paradigms which crucially include one member with a V suffix and one with a CV suffix. (The stem portions standing in correspondence appear underlined in the tableaux below.) (6) Attraction to the unmarked Explicating the tableau in (6), the paradigm member with the -CV suffix is of particular importance (cf. McCarthy 2001: 14). In the candidate paradigm in (a), this form has a penultimate syllable with both a long vowel and a coda consonant, hence it is trimoraic, and violates the markedness constraint *µµµ] σ .The same form in candidate (b) shows closed syllable shortening since the vowel loses its length. Note that there is no such shortening in the form with the -V suffix, since suffixa160 CatJL 4, 2005 Curt Rice CVCVC*µµµ] σ OP-MAX(V)µIO-MAX(V)µ a. CVCV.CV, CVCVC.CV *! b. CVCV.CV, CVCVC.CV *! * ☞c. CVCV.CV, * CVCVC.CV * Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 160 tion there does not close the penultimate syllable. Hence, candidate (b) shows an alternation within the paradigm and thereby incurs a violation of OP-MAX(V)µ. Candidate (c) picks up on the best properties of candidates (a) and (b). First of all, the highly ranked markedness constraint is respected. Secondly, there is no intraparadigmatic alternation. In other words, candidate (c) shows leveling effects, such that the vowel length is eliminated from the entire paradigm. By now we can see that an input with an infelicitous shape (one which cannot bear a consonant initial suffix and be acceptably syllabified) will be mapped onto a paradigm with a better stem shape. Indeed, inputs with such shapes will never be posited as lexical representations for a word. The reason for this is as follows. The winning candidate in the hypothetical example (6) will also be the winning candidate for the input CVCVC. Given a competition between CVCVC and CVCVC to be the lexical entry for the winning candidate (c), the principles of lexical optimization will select CVCVC, such that there will never be a reason to assume a verbal stem ending in CVC]. Were there no C-initial suffixes, violation of the markedness constraint would not arise, and the winning candidate paradigm will preserve vowel length through the paradigm, as is the case for the nominal paradigms. Hence, there is a gap in the lexicon. There will be no verbs which have a lexical entry with final CVC. This follows from the interaction of markedness with the pressure for paradigm uniformity. This pressure is expressed through faithfulness constraints which hold among members of the paradigm, and which crucially dominate faithfulness constraints on other relations, especially the input-output relation. In this way, OP Theory gives unmarked members of paradigms status as attractors. The selection of a candidate paradigm with a gap as optimal will follow from the interaction of markedness and faithfulness, as is usual in OT. Markedness constraints (e.g. *µµµ] σ ,*CCC, SONSEQ) will interact with faithfulness constraints which are focused on two different correspondence relations. Some assess intraparadigmatic faithfulness (e.g. OP-MAX-µ, OP-IDEN(VOI)) while others focus on other faithfulness relations, such as the input-output relationship (e.g. IO-MAX-µ, IO-IDENT(VOI)). Reranking of the two categories of faithfulness constraints under the domination of markedness constraints yields the following two situations. When MARKEDNESS » IO-FAITH » OP-FAITH, the markedness constraints will assert themselves only on those members of the paradigm which violate the markedness constraint. On the other hand, when MARKEDNESS » OP-FAITH » IO-FAITH, then the effects of the markedness constraints are spread throughout the paradigm, giving leveling effects. 4. Markedness vs. category expression As noted above, the standard OT strategy for dealing with gaps is to include a candidate with the null parse. The importance of bearing this in mind in all OT analyses is emphasized in the following quotes. Optimal Gaps in Optimal Paradigms CatJL 4, 2005 161 Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 161 The Null Parse is a possible candidate which must always be considered (Prince and Smolensky 1993: 193). No matter what the input, the [null output] is among the candidates emitted by Gen (McCarthy 2002: 280). The requirement that we consider the null parse or the null output should be reflected in Optimal Paradigms Theory as well. Consider what this requirement would mean for the tableau in (6). We would have to include a candidate in which the paradigm is blatantly defective. For example, we would leave out all paradigm members with C-initial suffixes, keeping in mind that it is precisely those members which —when repaired— become the unmarked attractors. The defective paradigm —candidate (d) in (7)— will preserve V-length and respect OP faithfulness, and will thereby be optimal. Extending on the proposal in McCarthy (2002), the symbol is used as a place-holder indicating a gap in a paradigm. (7) A defective candidate paradigm EVAL assigns no violations to candidate (d) for IO-MAX(V)µeither. The defective paradigm in candidate (d) has only one member, and therefore only one opportunity to violate the IO-FAITHFULNESS constraint. The sole member of the paradigm in (d) in fact does not show the loss of a mora and therefore incurs no violation of this constraint. The symbol has no ontological status and participates in no correspondence relations, and hence provokes no violation of any faithfulness constraints. Indeed, does not incur any markedness violations either, such that it only violates MPARSE, and only then by stipulation. The proposal under development, as noted, will allow us to eliminate this stipulation from the theory. This illustration brings out a flaw in Optimal Paradigms Theory. EVA L assigns a violation of a markedness constraint to a paradigm for every violation of the constraint found within the paradigm. In (6) and (7), this means the stem ending in VC will incur a violation of µµµ] σ for every member of the paradigm having a Cinitial suffix. The motivation to have fewer violations effectively rewards paradigms with gaps. Although this basic notion will be important in the model of gaps advocated here, it is not entirely unproblematic. 162 CatJL 4, 2005 Curt Rice CVCVC*µµµ] σ OP-MAX(V)µIO-MAX(V)µ a. CVCV.CV, CVCVC.CV *! b. CVCV.CV, CVCVC.CV *! * c. CVCV.CV, *! CVCVC.CV * ☞d. CVCV.CV,  Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 162 Consider a tableau like (7), but with an input in which there is no vowel length, as in (8). If we also remove the vowel length from the candidate paradigms, then candidates (a-c) in (7) become identical, and they are therefore reduced to one candidate, candidate (a), in (8). This candidate is compared to one with a gap, as in candidate (d) in (7). (8) Two perfect candidates The given constraints do not distinguish the two candidate paradigms in (8). Neither of them have forms with trimoraic syllables, and there are no vowel length alternations between input and output or within the paradigm, hence no violation of the two faithfulness constraints. But, of course, the tableau is incomplete. It includes a host of low ranked markedness constraints, which will play a role in choosing between the candidates. Such constraints include prohibitions on essentially all structure. Given that a candidate paradigm is punished for every violation (e.g. of *[LABIAL] or any other such *STRUC constraint), a smaller paradigm will inevitably be preferred to a larger one. In this way, gaps will always be rewarded. To illustrate this, consider tableau (9), which includes a low ranked constraint *C as a stand-in for some relevant *STRUC constraint. (9) The emergence of the unmarked Taking this line of reasoning to its absurd extreme, the evaluation of paradigms by constraints referring to the markedness or faithfulness of phonological properties of the members of the paradigms will reward the paradigm with the most gaps. Indeed, a paradigm with gaps in every cell —the «null paradigm»— will be optimal, as seen in (10). Optimal Gaps in Optimal Paradigms CatJL 4, 2005 163 CVCVC *µµµ] σ OP-MAX(V)µIO-MAX(V)µ ☞a. CVCV.CV, CVCVC.CV ☞b. CVCV.CV,  CVCVC *µµµ] σ OP-MAX(V)µIO-MAX(V)µ*C a. CVCV.CV, *** CVCVC.CV *!*** ☞b. CVCV.CV, ***  Cat.Jour.Ling. 4 001-252 7/2/06 11:45 Página 163 References Clements, George N. (1990). «The role of the sonority cycle in core syllabification». In: Kingston, John; Beckman, Mary E. (eds.). Papers in Laboratory Phonology 1. Cambridge: Cambridge University Press, pp. 283-333. Downing, Laura J.; Hall, T. Alan; Raffelsiefen, Renate (eds.) (2005). Paradigms in Phonological Theory. Oxford: Oxford University Press. Eliasson, Stig (1975). «On the issue of directionality». In: Dahlstedt, Karl-Hampus (ed.). The Nordic Languages and Modern Linguistics / 2. Stockholm: Almqvist & Wiksell, pp. 421-445 Hetzron, Robert (1975). «Where the grammar fails». Language 51: 859-872. Itô, Junko; Hankamer, Jorge (1989). «Notes on monosyllabism in Turkish». In: Itô, Junko; Runner, Jeff (eds.). 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