Against Gradience John J. McCarthy Date of this version: April 1, 2002 Because thou art lukewarm, neither hot nor cold, I will vomit thee out of my mouth. Revelation 3:16 Abstract In Optimality Theory, a constraint can assign multiple violation-marks in two ways. Any constraint can assign several marks if there are several violating structures in the form under evaluation. Furthermore, some constraints are claimed to be gradient: they can assign multiple marks even when there is just a single instance of the non- conforming structure. In this paper, I argue that gradience is not a property of OT constraints. I examine the full range of gradient constraints that have been proposed and show that none is necessary, some are insufficient, and some are actually harmful. The archetypic gradient constraint is ALIGN, and I point out various inadequacies of gradient ALIGN, arguing instead for a quantized, categorical alternative. 1. Introduction In Optimality Theory (Prince and Smolensky 1993), a constraint can assign multiple violation-marks to a candidate. This happens in two situations. First, there can be several places where the constraint is violated in a single candidate, as when ONSET assigns two marks to the candidate a.pa.i. Second, some constraints are evaluated gradiently, measuring the extent of a candidate’s deviance from some norm. The best-known gradient constraints come from the alignment family (McCarthy and Prince 1993a); for instance, the constraint ALIGN(Ft, Wd, R) assigns three violation-marks to [(pá.ta).ka.ti.ma], one mark for each syllable that separates the right foot-edge “)“ from the right word-edge “]“. Both sources of violation-marks are treated as equivalent in OT, so they are added together when assessing a candidate. Again, alignment supplies the best-known example. When ALIGN(Ft, Wd, R) evaluates a word with several unaligned feet, each unaligned foot is a separate locus of violation, and each locus is assessed gradiently. The violation-marks accumulated from these two sources are treated homogeneously, as (1) shows. 2 (1) Evaluation by ALIGN(Ft, Wd, R)1 Ft-1 Ft-2 Ft-3 ALIGN(Ft, Wd, R) a. [(´σσ)1 (´σσ)2 (´σσ)3 σ] σσσσσ σσσ σ ********* b. [(´σσ)1 (´σσ)2 σ (´σσ)3] σσσσσ σσσ Ø ******** c. [(´σσ)1 σ (´σσ)2 (´σσ)3] σσσσσ σσ Ø ******* d. [σ (´σσ)1 (´σσ)2 (´σσ)3] σσσσ σσ Ø ****** e. [σσσσσ (´σσ)3]Ø ØØ° In (1a), for instance, the three feet are misaligned by five, three, and one syllables, respectively. So this candidate receives eight violation-marks from ALIGN(Ft, Wd, R). For the purposes of this article, it will be useful to make explicit the ideas that underlie (1). The hypotheses in (2) state outright the assumptions implicit in Prince and Smolensky (1993). (2) Multiple Violations in OT a. Locus hypothesis. A violation-mark is assigned for each instance or locus of violation in a candidate. When presented with the right candidate, then, any OT constraint can assign multiple violation-marks. b. Gradience hypothesis.2 Some constraints, by virtue of their formulation, assess violations gradiently. These constraints can assign multiple violation-marks even when there is just a single locus of violation. c. Homogeneity hypothesis. Multiple violations of a constraint from either source are added together in evaluating a candidate. No distinction is made between multiple violation-marks derived from the Locus hypothesis and those derived from the Gradience hypothesis. In my view, the Locus hypothesis is an important insight that should be maintained if possible. It allows candidates with, say, different numbers of feet to be compared very straightforwardly. Imaginable alternatives involve a great deal more finagling, such as comparing candidates by matching the feet in one with the feet in the other. The Gradience hypothesis, though, is less solid. In this article, I will argue that the Gradience hypothesis is both unnecessary and even problematic. And, of course, if gradience is eliminated from the theory, then the Homogeneity hypothesis is superfluous. 1This approach, which is found in McCarthy and Prince (1993a), ultimately derives from a suggestion made by Robert Kirchner. 2 The term gradience is not always used in the sense I assume here. For example, Hume (1998) describes LINEARITY (“no metathesis”) as a gradient constraint because it assigns two violation-marks to the mapping /C1V2C3V4C5/ 6 C1V2C5C3V4. This is not true gradience, in my view. Rather, it is a case of multiple loci of violation: the input asserts various linear precedence relations among the segments, and the output contradicts two of those relations, C3 > C5 and V4 > C5. 3 As an alternative to gradience, I propose that categorical constraints may be distinguished by extent of violation. Instead of gradient ALIGN(Ft, Wd, R), for example, there is a family of constraints, one for each type of prosodic constituent that can intervene between a foot-edge and a word-edge. (3) Quantized ALIGN(Ft, Wd, R) a. ALIGN-BY-σ(Ft, Wd, R) No syllable stands between the right-edge of Ft and the right-edge of Wd. b. ALIGN-BY-FT(Ft, Wd, R) No foot stands between the right-edge of Ft and the right-edge of Wd. The candidates [σσσ(σσ)σ], [σσ(σσ)σσ], and [σ(σσ)σσσ] are each assigned one violation-mark by ALIGN-BY-σ(Ft, Wd, R) — this constraint is categorical, so it treats all three candidates alike, distinguishing them only from the non-violator [σσσσ(σσ)]. The candidate [σσ(σσ)(σσ)] is assigned one violation-mark by ALIGN-BY-σ and another by ALIGN-BY-FT(Ft, Wd, R). The candidate [σ(σσ)(σσ)σ] receives two violation-marks from ALIGN-BY-σ — one for each foot under the Locus hypothesis, which is not in dispute — and another from ALIGN-BY-FT. It is apparent from the preceding paragraph that quantized alignment constraints do not make the same distinctions as gradient alignment. Two examples emphasize the differences. First, [σσσ(σσ)σ] and [σσ(σσ)σσ] are equally harmonic under quantized alignment but distinct under gradient alignment. Second, [σσσ(σσ)σ] and [σσ(σσ)(σσ)] violate the same gradient alignment constraint to different extents, but quantized alignment distinguishes them categorically because only [σσ(σσ)(σσ)] violates the quantized constraint ALIGN-BY-Ft(Ft, Wd, R). Quantized alignment, then, is both more and less exact than gradient alignment. A principal goal of this article is to show that these differences weigh in favor of quantized alignment when the full range of evidence is considered. §2 of this paper briefly reviews some of the gradient constraints that have been proposed in the literature. Except for alignment, all have an obvious and probably necessary reformulation in categorical terms. Alignment constraints, as I have suggested, can also be reformulated categorically, and §3 shows that they should be, based on evidence from infixation. §4 then turns to the role of alignment in stress. Following Kager (2001), I argue that gradient ALIGN(Ft, Wd) constraints overgenerate and can be replaced by constraints on stress lapses. I also show that other applications of gradient alignment, such as locating main stress, can be reanalyzed using quantized alignment constraints. Gradient alignment constraints have also been invoked as a way to produce autosegmental spreading effects, so this is the topic of §5. As we will see, this use of gradient alignment is by no means a settled matter, and plausible alternatives exist. Finally, §6 sums up the results and discusses some of the remaining issues. Before we begin, there is a final remark to be made about the logic of the argument pursued here. The Gradience hypothesis is not an essential element of classic OT. (“... [T]he division of constraints into those which are binary and those which are not, a division which we have adopted earlier in this section, is not in fact as theoretically fundamental as it may at this point appear.” (Prince and Smolensky 1993: 81)) If it can be shown that gradient constraints are unnecessary, then that is sufficient reason to reject them. In this article, I will show that they are indeed unnecessary, since reasonable non-gradient alternatives can be developed. The argument, however, goes beyond 4 this essential step to show that gradient constraints both underpredict and overpredict the range of observed behavior in certain situations. The argument, then, is two-pronged: supply an alternative to gradience and highlight the descriptive superiority of that alternative. 2. Making Gradient Constraints Categorical This section reviews some of the uses to which gradient constraints have been put in the OT literature. The section goes on to show how these constraints can — and in some cases must — be reformulated categorically. A taxonomy of gradience is useful to help organize the discussion. (4) Types of Gradience in the OT literature a. Horizontal gradience. Assign violation-marks in proportion to distance in the segmental string. Example: ALIGN(Ft, Wd, R), ALIGN(Prefix, Wd, L). b. Vertical gradience. Assign violation-marks in proportion to levels in a hierarchy. Example (paraphrasing Spaelti 1994): WEAKEDGE . assign a violation-mark for each prosodic category whose right-periphery is non-empty. E.g., [((dog)σ)Ft]Wd receives three marks. c. Collective gradience. Assign violation-marks in proportion to the cardinality of a set. Example (Padgett 1995a): CONSTRAINT(Class) . assign one violation-mark for each member of the feature-class Class that does not satisfy CONSTRAINT. E.g., aõgMba receives one mark from ASSIM(Place) and angMba receives two marks. d. Scalar gradience. Assign violation-marks in proportion to the length of a linguistic scale. Example (Prince and Smolensky 1993: 16): HNUC / “A higher sonority nucleus is more harmonic than one of lower sonority.” I.e., assign a nucleus one violation-mark for each degree of sonority less than a.