1. Probabilistically Ranked Constraints: Derivation of the Gradient Grammaticality of Implicit Objects Tamara Nicol Medina Institute for Research in Cognitive Science (IRCS), University of Pennsylvania Proposal The grammaticality of an indefinite implicit object is gradient across verbs, varying in accordance with the Semantic Selectivity of the verb, Telicity, and Perfectivity. The gradient grammaticality can be derived using probabilistically ranked constraints within the framework of Optimality Theory (Prince and Smolensky, 1993/2004). The Indefinite Implicit Object Construction In English, indefinite object omission is preferred with certain verbs but not others: 1. John ate (something). 2. John found *(something). Verb Semantic Selectivity The grammaticality of an indefinite implicit object will be lower for verbs with weaker semantic selectional preferences.  Selectional Preference Strength (SPS) (Resnik, 1996): Model calculates relative entropy between a “baseline” distribution of the semantic argument classes of the direct objects in a corpus, and the distribution of the argument classes of direct objects given a particular verb. SPS will be greater, the greater the difference between the two probability distributions.  Verbs with higher SPS (stronger semantic selectional preferences) were more likely to occur without an overt object (Resnik, 1996). 3. Highly Selective Verb: John read (something). 4. Low Selective Verb: John wanted * (something). Telicity Since a direct object often specifies what constitutes the endpoint of a telic event, the grammaticality of an indefinite implicit object will be lower for telic verbs than for atelic verbs.  Refers to the existence of a natural end or result of an event: a telic event entails an endpoint, while an atelic verb does not.  Indefinite object omission is preferred with atelic (5) rather than telic verbs (6) (Tenny (1994), van Hout (1996), Olsen and Resnik (1997), and others). 5. Atelic: John sang (something). 6. Telic: John brought * (something). Perfectivity Since perfective aspect targets the endpoint of an event, the grammaticality of an indefinite implicit object will be lower given perfective than imperfective aspect.  Denotes the point along the temporal constituency of an event from the speaker’s perspective (Comrie, 1976; Olsen, 1997) : perfective indicates an event at its endpoint, while imperfective construes an event in progression.  Informal grammaticality judgments suggest that indefinite implicit objects are preferred with imperfective aspect (7) than with perfective aspect (8). 7. Imperfective: John was writing (something). 8. Perfective: John had written ? (something). Gradient Grammaticality Participants: 15 monolingual native speakers of English at Johns Hopkins University. Stimuli and Design:  140 sentences containing 30 two-argument verbs for which: − SPS had been calculated by Resnik (1996) using the Brown corpus of American English (Francis & Kučera, 1982). − Telicity was assessed: 14 telic, 16 atelic. − Each verb was used once in a sentence with perfective aspect and once in a sentence with imperfective aspect. Test Sentences: Implicit objects. Michael was bringing. / Michael had brought. Control Sentences: Overt objects. Sarah was bringing a gift. / Sarah had brought a gift.  20 sentences containing 10 two-argument verbs (5 telic, 5 atelic) Filler Sentences: With and without overt objects. Emma was sleeping. / Emma had slept. Andrew was sleeping a blanket. / Andrew had slept a blanket. Procedure: Rate the goodness of sentences on a scale of 1 (very bad) to 5 (very good). Results: Grammaticality judgments are gradient across verbs for the Test Sentences (shown here collapsing Perfective/Imperfective). A multiple linear regression was found to be significant (F = 9.68, p < 0.05), with each of the three factors making small but significant contributions: SPS accounted for 12% of the variance, Telicity accounted for 7% and Perfectivity accounted for 6%. An Optimality-Theoretic Analysis, cont. Grammaticality of an Implicit Object: Increases gradiently as a function of SPS, differently for each of the four aspectual types of inputs. Correlations between Model and Judgments: Telic PerfectiveTelic Imperfective r = 0.84, p < 0.05r = 0.88, p < 0.05 Atelic PerfectiveAtelic Imperfective r = 0.26, p > 0.05r = -0.09, p > 0.05 An Optimality-Theoretic Analysis Constraints: * I NT A RG (* O VERT I NTERNAL A RGUMENT ) The output must not contain an overt internal argument (direct object). F AITH A RG (F AITHFULNESS T O A RGUMENT S TRUCTURE ) All arguments present in the input must be present in the output. T ELIC E ND (T ELIC E NDPOINT ) The endpoint of a Telic event must be bounded by the presence of an overt internal argument in the output. P ERF C ODA (P ERFECTIVE C ODA ) The coda (endpoint) targeted by perfective aspect must be indicated by the presence of an overt internal argument in the output. An Optimality-Theoretic Analysis, cont. Input: catch (x,y), x = Jack, y = unspecified SPS = 2.47, [+ Past], [+ Telic], [+ Perfective] Probabilistic Ranking of Constraints: Follows partial ranking (Reynolds, 1994; Nagy & Reynolds, 1997; Anttila, 1997; Legendre et al., 2002; Davidson & Goldrick, 2003) and stochastic ranking (Boersma, 1997, 1998, 2004; Hayes & MacEachern, 1998; Boersma & Hayes, 2001) approaches.  * I NT A RG is the only constraint which is violated by the overt object output candidate; thus, what matters is the relative ranking of this constraint to each of the other three: p(* I NT A RG » F AITH A RG ), p(* I NT A RG » T ELIC E ND ), and p(* I NT A RG » P ERF C ODA ).  If F AITH A RG, T ELIC E ND, and P ERF C ODA are unranked with respect to each other, then there are eight possible orderings of the constraints.  Within this probability space of the eight partial orderings, the probability of each individual partial ordering, e.g., p(* I » F), is equal to the joint probabilities of the independent pairwise orderings that comprise it.  The probability (= grammaticality) of an implicit object is equal to the sum of the probabilities of each of the constraint orderings that give rise to the implicit object output candidate for an input according to its aspectual features: p(implicit) Telic Perfective = p(*I » {F, T, P}) p(implicit) Telic Imperfective = p(*I » {F, T, P}) + p(P » *I » {F, T}) p(implicit) Atelic Perfective = p(*I » {F, T, P}) + p(T » *I » {F, P}) p(implicit) Atelic Imperfective = p(*I » {F, T, P}) + p(T » *I » {F, P}) + p(P » *I » {F, T}) + p({T, P} » *I » F)  The probability of * INT ARG ranked above each of the other three constraints is defined by separate linear functions: p(* I NT A RG » F AITH A RG ) = p(* I NT A RG » T ELIC E ND ) = p(* I NT A RG » P ERF C ODA ) = * I NT A RG F AITH A RG T ELIC E ND P ERF C ODA Jack had caught.  Jack had caught something.  References Anttila, A. 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Aspectual Features in the Input Telic Perfective Telic Imperfective Atelic Perfective Atelic Imperfective *I » {F,T,P}implicit P » *I » {F,T}implicit T » *I » {F,P}implicit {T,P} » *I » Fimplicit F » *I » {T,P} {F,T} » *I » P {F,P} » *I » T {F,P,T} » *I Acknowledgements This research was conducted at Johns Hopkins University under the guidance of Géraldine Legendre, Paul Smolensky, Barbara Landau, and Philip Resnik, as part of the PhD dissertation, and was supported by an Integrative Graduate Education & Research Training (IGERT) Fellowship.