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Journal of Semantics 24: 307330doi:10.1093/jos/ffm005Advance Access publication July 16, 2007

The Processing Consequences of the Imperfective

Paradox

GIOSUE BAGGIO

F.C. Donders Centre for Cognitive Neuroimaging, Radboud University Nijmegen

MICHIEL VAN LAMBALGEN

Department of Philosophy, University of Amsterdam

Abstract

In this paper we present a semantic analysis of the imperfective paradox based on theEvent Calculus (van Lambalgen & Hamm 2004), a planning formalism characterizinga class of models which can be computed by connectionist networks. We report theresults of a questionnaire that support the semantic theory and suggest that differentaspectual classes of VPs in the progressive give rise to different entailment patterns.Further, a processing model is outlined, combining the semantic analysis with the

psycholinguistic principle of immediacy in the framework of recurrent networks. Themodel is used to derive predictions concerning the electrophysiological correlatesof the computations described by the Event Calculus.

1 INTRODUCTION

Recently, a number of studies have brought experimental data to bearon semantic theories (see, for instance, the work on quantifiers byGeurts & van der Slik 2005 and McMillan et al. 2005, 2006). However,formal semantics and psycholinguistics have reached their most impor-tant results independently, even when the issues at stake could havebeen addressed more completely by joining efforts. One example is thestudy of discourse-based inference in psycholinguistics, where formalnotions of truth, entailment and veridicality have often been neglected(see Cook et al. 2001 and Frank et al. 2003 among others).

The assumption behind this paper is that there exists a relativelyunexplored territory in which the two disciplines can interact produc-

tively. Here we consider a small portion of this territory: the imperfectiveparadox and its processing consequences. In the remainder of this sectionwe provide a minimal methodological background against which semantictheories can be combined with processing models and we introduce thesemantic paradox which we shall work with. In section 2 we present an

The Author 2007. Published by Oxford University Press. All rights reserved.For Permissions, please email: [email protected].

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analysis of the paradox based on the Event Calculus (van Lambalgen &Hamm 2004), a planning formalism characterizing a class of models whichcan be computed by connectionist networks. In section 3 we report the

results of a questionnaire that support the semantic theory and suggestthat different aspectual classes of VPs in the past progressive give rise todifferent entailment patterns. In section 4 a processing model is outlined,combining the formal semantic analysis with the psycholinguisticprinciple of immediacy in the framework of recurrent networks. Themodel is used to derive predictions concerning the electrophysiologicalcorrelates of the computations described by the Event Calculus.

1.1 Methodological preambleFrom a semantic analysis of a given linguistic structure it is usually notpossible to derive predictions concerning the complexity and timecourse of the processes involved in producing or comprehendingutterances in which the structure occurs. Although there may be casesin which processing hypotheses can be formulated on the basis of thesemantic theory alone (Geurts & van der Slik 2005; McMillan et al.2005), this approach is unlikely to work in general. The reason is that

processing largely depends upon the particular algorithms and neuralmechanisms that, in a physical system such as the human brain, com-pute the linguistic structures posited by the theoryor functionallyequivalent structures. Because semantics typically does not describealgorithms and neural mechanisms (nor perhaps it should), there ap-pears to be no direct way to relate semantic theories to what is observedin language-processing experiments.

The solution is to adopt a theoretical framework that allows formalsemantic analyses to be explicitly related to processing algorithmsinspired by the available psycholinguistic evidence and ultimately tomechanistic models of the kind investigated in neuroscience. Marrs(1982) scheme for the analysis of cognitive systems seems an appro-priate choice in this regard.

Marr argued that information-processing systems should be un-derstood at three nearly independent levels of analysis. The first levelcontains a description of the computations to be performed by thesystem and more precisely a characterization of the goals that have to be

attained in order to solve the information-processing problem. Forexample, a sentence S can be seen as introducing a specific goal,namely, the construction of a cognitive model making S true. In thesemantic analysis proposed below, the tense and aspect of VPs like thepast progressive are treated as instructions to update the current

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discourse model so as to achieve that goal. We regard it as the task ofsemantics to describe information-processing goals and updateinstructions. The actual processing steps are described at the in-

termediate level, where constraint satisfaction algorithms, implementedin artificial neural networks (Marr & Poggio 1976), and processingprinciples such as immediacy and incrementality are combined. Marrcompletes the picture with a third level, at which the neurobiologicalarchitecture is described. We discuss issues relevant for the computa-tional level in section 2 and for the algorithmic level in section 4.

1.2 The imperfective paradox

Vendler (1957) famously classified VPs as states (know, love etc.),activities (write, run etc.), accomplishments (write a letter, runa mile etc.) and achievements (finish, reach etc.). We are concernedwith the inferences licensed by sentences containing activities andaccomplishments in the past progressive. The following exampleinvolves the accomplishment write a letter:

(1) The girl was writing a letter when her friend spilled coffee on thetablecloth.

From (1) the reader would typically conclude that, barring unforeseencircumstances, the girl attained the desired goal and would thus assentto the statement The girl has written a letter. Such inference is basedon the assumption that spilling coffee on the tablecloth is usually neutralwith respect to the writing activity, that is it is not a typical immediatecause leading to its termination. It is possible to imagine scenarios inwhich writing is temporarily interrupted or even terminated by theaccident. Nonetheless, as the data reported in section 3 will demon-

strate, failing to explicitly mention a disabling condition in the dis-course is sufficient for leading the reader to assume that there was nosuch obstacle to attaining the intended goal.

Inferences to a goal state are non-monotonic, that is they can besuppressed if the discourse describes an event which terminates therelevant activity:

(2) The girl was writing a letter when her friend spilled coffee on thepaper.

Assuming that writing was intended to occur on the same paper sheetson which coffee was spilled, the accident is sufficient for terminatingthe activity and it is therefore a disabling condition for obtaininga complete letter. Accordingly, on the basis of (2) the reader would bemore likely to assent to The girl has written no letter.

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One important observation is that suppression can obtain only withaccomplishments and not with activities (Rothstein 2004). Since a sen-tence containing an activity in the past progressive, such as writing

letters, does not involve a canonical goal, it is interpreted as entailingthat The girl has written one or more letters regardless of the con-sequences of the second event on the writing activity:

(3) The girl was writing letters when her friend spilled coffee on thetablecloth.

(4) The girl was writing letters when her friend spilled coffee on thepaper.

Upon reflection, there is something paradoxical about examples (1)and (2) which is not found in (3) and (4). Although it belongs to themeaning of the accomplishment writing a letter that the writingactivity is directed towards the goal state of a complete letter, the actualoccurrence of that goal state can be denied without contradiction. Buthow can a seemingly essential component of the meaning be deniedwithout affecting the meaning itself? This is known as the imperfectiveparadox. The semantic literature is replete with attempted solutions ofthe paradox, ranging from explaining the problem away (Michaelis

2001) to various invocations of possible worlds (Dowty 1979; Landman1992; de Swart 1998). It is impossible to review all proposed solutionshere. Instead, we will focus on some representative claims. Possibleworlds analyses are based on the idea that

. . . the progressive picks out a stage of a process/event which, if itdoes not continue in the real world, has a reasonable chance ofcontinuing in some other possible world. (de Swart 1998)

They differ however in the (largely informal) descriptions of thepossible worlds used. For example, Dowty (1979) would claim that thefollowing are equivalent:

a. The girl is writing a letter is true in the actual world;b. The girl will have written a letter is true in all so-called inertia

worlds, worlds which are identical with the present world untilnow, but then continue in a way most compatible with thehistory of the world until now.

These approaches are intensional in the formal sense of using possibleworlds. In fact, most authors (but not all) would agree that the progres-sive creates an intensional context: even though the accident in (2) mayhave terminated the writing activity at a stage in which it was unclearwhether the girl was writing a letter or, say, a memo, still only one of




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(5) The girl was writing a letter.(6) The girl was writing a memo.

is true of the situation described by (2). Explicitly denying thatthe progressive creates an intensional context, Michaelis (2001)argues that1

. . . the Progressive sentence She is drawing a circle denotes a statewhich is a subpart not of the accomplishment type Shedrawa circle but of the activity type which is entailed by the causalrepresentation of the accomplishment type. Since this activity canbe identified with the preparatory activity that circle drawingentails, circle drawing can in principle be distinguished fromsquare drawing etc. within the narrow window afforded by theProgressive construal [and] does not require access to culminationpoints either in this world or a possible world.

We find this questionable, for without access to a persons intention itmay be very hard to tell initially whether she is drawing a circle or asquare, writing a letter or a memo. But that persons intention in per-forming an activity is characterized precisely by the associated con-sequent state, even though the latter cannot yet be inferred from theavailable data.

Here the Event Calculus will come to our rescue because thenotion of goal or intention is built into the semantics from the start.In particular, the meaning of a VP is represented by a scenario whichdescribes a plan for reaching the goal. However, unlike approachessuch as Parsons (1990) where one quantifies existentially over events,the scenario is a universal theory and does not posit the occurrenceof the intended consequences. Although the plan is appropriate for

that purpose, attaining the goal is guaranteed only in a minimal model(in which no unforeseen obstacles occur) of the scenario plus theaxioms of the Event Calculus. The meaning of an accomplishment(as embodied in the scenario) involves a culminating event type,which therefore must exist; but no existential claims are made con-cerning the corresponding event token, which, as in example (2),may also fail to occur. Type and token are handled by differentmechanisms in the Event Calculus. These notions form the basis ofour semantic analysis of the imperfective paradox, to which we nowturn.

1 Replace drawing a circle with writing a letter and drawing a square with writing a memoin Michaelis examples to see the connection with our examples.




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2 SEMANTIC ANALYSIS

The reference to goal states in the preceding section suggests thata semantic analysis of the progressive can be based on a planningformalism which is able to talk about goals and actions and whichincludes a theory of causality together with a principle of inertia(Hamm & van Lambalgen 2003). Such a formalism is presented in vanLambalgen & Hamm (2004). It consists of an Event Calculus which hasfound applications in robotics, reformulated using the computationalmachinery of Constraint Logic Programming.

The reader may wonder why planning can provide a source ofinspiration to linguistics. The reason can be found in the nature of

planning computations, which typically proceed as follows. First a goalis specified, which may be an action to be performed at a particularlocation (e.g. pick up outgoing mail in an office building). Next a planis computed, that is a sequence of actions to get to the requiredlocation, derived by backward chaining from the goal to obtaina sequence of subgoals, the last one of which can be executed in theagents initial position and state. Planning requires a situation model(including a map of the building, a causal theory of the agents actions,

a specification of values of variables such as door open/closed, theagents initial position and state and a record of its past and currentactions), a repertoire of possible actions (follow wall and go throughdoor) and observations (door open/closed). While the agent executesthe plan, it also registers its observations and actions in the situationmodel; knowledge of its actions may be important for the agent toestimate its current position. Particularly relevant for our purposes isthat a plan might have to be re-computed in mid-course when theinitial situation model is updated due to new observations (for instance,

a closed door which was expected to be open on the basis of the initialmodel or a wrong estimate of the agents current position). Later on weshall see how this re-computation relates to the imperfective paradox.

This short description should be sufficient for enabling the reader tosee the connection with language processing. The comprehender startswith an initial discourse model, in which an incoming sentence orclause must be integrated. Suppose the main verb of the sentence isnon-stative, for instance, an activity. If the sentence is in one of the

simple tenses, it is unpacked in the relevant action and its participants,and the discourse model is updated accordingly. This is the analogue ofupdating the situation model with representations of individuals andactions in planning. In more complex cases, such as sentences involvingaccomplishments like (1) and (2), the VP is taken to express the




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existence of a goal-directed plan. If, on the contrary, the main verb ofthe sentence is stative, the sentence can be viewed as analogous to anobservation report, and the discourse model is updated with a property.

2.1 The Event Calculus

The Event Calculus is a planning formalism which allows one to talkabout actions, goals and causal relations in the world. Its main functionis to return a plan given a goal, the initial state and causal relations.Formally, the Event Calculus is a many-sorted predicate logic. It hastwo different sorts for events, viewed either perfectively or im-perfectively. The former are called event types and are symbolized by

e, e#

,. . .

, e0,. . .

. The latter are called fluents (Newtons name for time-dependent variables) and are symbolized by f, f#, . . ., f0, . . .. One maythink of event types as action types, such as, for example break orignite; fluents can be thought of as time-dependent properties, such asbeing broken or writing; the time parameter in fluents is implicit,but they can have further parameters (for instance, for the subject ofwriting). The real distinction between event types and fluents comesfrom the different roles they play in the axioms of the Event Calculus.The universe also contains sorts for individuals (the girl), real numbers

interpreted as instants of time, and various other real quantities (such asposition, velocity and degree of some quality).

The primitive predicates comprise the minimum necessary to talkabout two forms of causality, instantaneous (as in a collision) and con-tinuous (as when a force is acting):

(7) a. Initially(f): fluent f holds at the beginning of the discourseb. Happens(e, t): event type e has a token at time tc. Initiates(e, f, t): the effect of event type e at time t is the

initiation of fd. Terminates(e, f, t): the effect of event type e at time t is the

termination of fe. Trajectory(f1, t, f2, d): if f1 holds from t until t + d, then at t +

d f2 holdsf. Releases(e, f, t): event type e releases the continuously-varying

fluent f2

g. HoldsAt(f, t): fluent f holds at at time t

2 The predicate Releases is used to reconcile the two notions of causality: while instantaneouschange leads to one form of inertia, where properties do not change their value between theoccurrences of two events, continuous change requires that variable quantities may change valuewithout concomitant occurrences of events. The solution of this conflict is to exempt, by means ofthe predicate Releases, those properties which we want to vary continuously from the inertia of thefirst form of causation.




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While (7ad) are the predicates required by instantaneous change,(7ef) are used to model continuous change.

Three further notions will be used in the semantic analysis presented

below. The first is that of scenario.3

Whereas the axiom system ECprovides a macro-theory of causality, scenarios provide micro-theoriesstating the specific causal relations obtaining in a given situation.Scenarios can be used to describe the causal environment of actions andevents at the level of granularity expressed in natural language, such as,for instance, writing a letter or drawing a circle. Scenarios can be takento represent, in a tenseless fashion, the meaning of VPs. For instance,the scenario for the accomplishment writing a letter specifies that thewriting activity is causally related to the amount of letter produced, thatthe termination of the activity is temporally contiguous to thecompletion of the letter and so on.

The second notion is that ofintegrity constraint. As we said above, weregard a sentence as introducing an information-processing goal (MakeS true) to be achieved by updating the current discourse model orconstructing a new one. Integrity constraints regiment such updates.They can take the form of either obligations ?u succeeds, requestingan update of the discourse model satisfying u, or prohibitions ?u fails,

blocking updates of the discourse model satisfying u. While scenariosdescribe the meaning of VPs in a tenseless fashion, integrity constraintsspecify the contribution of tense, aspect and temporal adjuncts to thesemantics of VPs.

The third notion is that of minimal model. The axioms of the EventCalculus constitute a correct theory of causality if and only if thefollowing two conditions are satisfied:

1. The discourse model contains only those occurrences of events

forced to be there by the discourse and the axioms;2. The interpretation of the primitive predicates is as small as is

consistent with the discourse and the axioms.

These two requirements define minimal models. They imply that nounforeseen events are allowed to happen and that all causal influencesare as expected. The choice to work with minimal models instead of allmodels leads to non-monotonicity in discourse interpretation: addinga new sentence or clause to the discourse may invalidate a conclusion

derived from the initial model. In van Lambalgen & Hamm (2004) it isargued that it is precisely the possibility to retract previously inferred

3 See the related notions of frame in Minsky (1977) and script in van Dijk (1980). For a moreexplicit connection with planning, see Schank & Abelson (1977).




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The contribution of the past progressive is represented by theintegrity constraint

(9) ?HoldsAt(write, R)^

R < S succeeds

which forces an update of the discourse model such that the activityfluent write holds at the reference time R located in the past of themoment of speech S (Reichenbach 1947). As we noted above aboutexamples (1)(4), different inferences can be drawn from activities andaccomplishments in the past progressive. In the Event Calculus, thisfollows from the fact that updating the initial model according to (9)leads to different models depending on whether the scenario representsan activity or an accomplishment. Proposition 1 provides informationon the inferences licensed by a discourse containing an accomplishmentin the past progressive.

Proposition 1. Let D be a discourse consisting of scenario (8) givenabove. Suppose D is extended to D# so that the query ?HoldsAt(write, R) ^ R < S succeeds in D#. Suppose also limt/Ngt> c.Then there is a unique minimal model of D# and in this modelthere is a time t > R for which HoldsAt(letter(c),t). By virtue of thestipulation Letter(letter(c),t), there will be a letter at time t.6

If write holds at R, as required by (9), it must either hold initially orhave been initiated. The latter requires an event start which initiatedthe writing activity. Since the starting event is not provided bydiscourse, we must assume that write holds initially. The clause (8g)states that the writing activity will increase the stage of completion ofthe letter. As time tends to infinity, the latter will be at least equal to c(the final completion stage). We have stipulated that a letter whose stageof completion is c is a finished letter. Hence, there must be a time atwhich the letter is considered finished. The writing activity will stop assoon as a complete letter is obtained, as is guaranteed by (8de). Noticethat this holds for accomplishments but not for activities: if the VP isthe activity writing letters, (8c) and (8e) are not part of the scenarioand writing will continue also after one letter has been completedinfact, it will continue indefinitely, if no terminating event is described.

We have just seen how the Event Calculus deals with the inter-pretation of VPs in the past progressive. We will now discuss the con-

tribution of subordinate when clauses to the meaning of (1)(4).Because spilling coffee on the tablecloth usually does not terminate thewriting activity, the dynamics of the scenario will lead to a complete

6 See van Lambalgen & Hamm (2004) for a more rigorous statement and a computational proof.




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letter in (1) and to an indefinite number of letters in (3). The argumentmirrors the one given above as a gloss of Proposition 1. The situation isdifferent for examples (2) and (4). Spilling coffee on the paper is

typically sufficient for terminating the writing activity. This bit of worldknowledge can be expressed by the following addition to scenario (8):

(10) Terminates(spill, write, t)

But what is more important here is the integrity constraint introducedby the when clause

(11) ?Happens(spill, R) ^ R < S succeeds

which requires the accident to occur at R, while the writing activitywas taking place. Since during the writing process there is no completeletter (yet), spilling coffee on the paper will terminate the activity beforethe letter is finished. Therefore, there will be no complete letter in thefinal discourse structure. As for (4), the theory is consistent both withthe situation in which the writing activity was terminated before evena single letter had been completed and with the case in which one ormore letters were finished when the accident happened. The proposedsemantics predicts that readers would assent to The girl has written

a letter in (1), to The girl has written no letter in (2) and to The girlhas written one or more letters in (3). This is consistent with receivedsemantic wisdom about entailments of activities and accomplishmentsin the progressive. Behavioral data supporting this claim are presentedin the next section.

To summarize, the attainment of the consequent state can bederived in a minimal model of a discourse containing an accomplish-ment in the progressive. However, the computation of discoursemodels is non-monotonic: if the minimal model is extended witha sentence or a clause describing an event which terminates the relevantactivity (what we called a disabling condition), the derivation of thegoal state is blocked. Non-monotonicity affords a neat solution of theimperfective paradox, as there is no contradiction in assuming thatthe representation of the goal state is both essential to the meaning ofthe progressive VP (as an event type in the scenario) and suppressibleon the basis of additional discourse information (as an event token inthe minimal model).

3 BEHAVIOURAL DATA

In the preceding sections we have argued that accomplishments andactivities in the past progressive behave differently as regards entailment.




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An accomplishment such as writing a letter implies that The girl haswritten a letter provided that no obstacles are described in discourse.Failing to mention a disabling condition is sufficient for inferring that

there is no such obstacle to attaining the goal. If, however, a disablingcondition is introduced, the accomplishment will imply that The girlhas written no letter. An activity such as writing letters implies thatThe girl has written one or more letters regardless of further discourseinformation. Since disabling conditions affect the possibility ofattaining a predefined goal, and such predefined goals are part of themeaning of accomplishments but not of activities, accomplishmentswill be sensitive to the presence of a disabling condition in thediscourse context, whereas activities will not. We administered anentailment questionnaire aimed at testing this claim.

3.1 Subjects

Thirty-two native speakers of Dutch (mean age 22.5, 27 female)completed the cloze probability test (see below) and 36 (mean age 22.5,28 female) the entailment questionnaire. Participants were selectedfrom the database of the F.C. Donders Centre for CognitiveNeuroimaging in Nijmegen. The two sets of subjects were disjoint.

3.2 Materials

The set of Dutch materials used in the tests included 160 items. Eachitem comprised two context sentences (C) providing a neutral settingfor the events described by critical sentences,7 four critical sentences(S1)(S4) and two probe pairs (P1)(P2):

(C) De deur van de woonkamer was gesloten. Binnen speelde de

radio klassieke muziek.The door of the living-room wasPST closedPRT. Inside playedPST theradio classical music.The door of the living room was closedPST PRT. Inside the radioplayedPST classical music.

(S1) Het meisje was brieven aan het schrijven toen haar vriendin koffieop het tafelkleed morste.The girl wasPST letters on the to-writeINF when her friend coffeeon the tablecloth spilledPST.The girl was writingPST PRG letters when her friend spilledPST coffeeon the tablecloth.

7 That is, no disabling condition for the event described by the progressive VP was introduced bycontext items.




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(S2) Het meisje was brieven aan het schrijven toen haar vriendin koffieop het papier morste.The girl wasPST letters on the to-writeINF when her friend coffee

on the paper spilledPST.The girl was writingPST PRG letters when her friend spilledPST coffeeon the paper.

(S3) Het meisje was een brief aan het schrijven toen haar vriendinkoffie op het tafelkleed morste.The girl wasPST a letter on the to-writeINF when her friend coffeeon the tablecloth spilledPST.The girl was writingP ST P RG a letter when her friend spilledPSTcoffee on the tablecloth.

(S4) Het meisje was een brief aan het schrijven toen haar vriendinkoffie op het papier morste.The girl wasPST a letter on the to-writeINF when her friend coffeeon the paper spilledPST.The girl was writingP ST P RG a letter when her friend spilledPSTcoffee on the paper.

(P1) Het meisje heeft een of meer brieven geschreven.The girl hasPRS one or more letters writtenPRT.

The girl has writtenPRS PRF one or more letters.Het meisje heeft geen brief geschreven.The girl hasPRS no letter writtenPRT.The girl has writtenPRS PRF no letter.

(P2) Het meisje heeft een brief geschreven.The girl hasPRS a letter writtenPRT.The girl has writtenPRS PRF a letter.Het meisje heeft geen brief geschreven.The girl hasPRS no letter writtenPRT.The girl has writtenPRS PRF no letter.

Critical sentences were constructed manipulating the aspectual classof the VP in the past progressive (activity or accomplishment) and thecausal type of the condition introduced by the subordinate when clause(neutral or disabling with respect to the event described in the progressiveclause). Activities and accomplishments differed only in the object NP:an indefinite (een brief) was used for accomplishments, a bare plural

(brieven) was used for activities. Disabling and neutral conditionsdiffered only in the prepositional or object NP, for instance, papier forthe former and tafelkleed for the latter. The distinction between neutraland disabling conditions was made according to the authors judgment.The probes (P1) were used with activities, (P2) with accomplishments.




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The following linguistic properties of critical sentences werenormed. The mean length and the raw frequency of the differingwords in the NP of subordinate when clauses were matched using the

CELEX corpus (Baayen et al. 1996). To determine the cloze prob-abilities of the verbs in the subordinate clauses, context items followedby a critical sentence with the final word blanked were presented tosubjects. Participants were asked to fill in the blank with the first wordthat came to their mind. Four versions (40 items per condition) ran-domized and balanced across conditions were constructed. Mean clozeprobabilities are not different across conditions (T-tests, all comparisonsP> 0.05). This was established for each version as well as for the entireset of experimental items. Therefore, the same materials and testversions were used in the entailment questionnaire.

3.3 Procedure

Copies of the questionnaire were distributed to all subjects in thedatabase meeting the following criteria: they had to be native speakersof Dutch with an age between 18 and 50 and with no history ofneurological, psychiatric or cognitive disorders. The first 36 subjects

who returned a completed questionnaire were included in theanalysis.

The first page of each booklet contained the test instructions.Participants were informed that the questionnaire consisted of 160 shorttexts and that each comprised three sentences (the two contextsentences and the critical sentence) and was followed by a pair ofprobes. Subjects were instructed to read each sentence carefully andselect the probe which they deemed correct on the basis of theirexpectations (verwachtingen) about the continuation of the text(over het verlog van de tekst). Participants were asked to respond asquickly and accurately as possible and to write a brief comment ontheir answer in the blank space following the probes.

The reference to expectations in the test instructions requires someexplanation. Notice that we are not interested in what subjectssemantically know given a progressive clausepresumably that thewriting activity was taking place some time in the past and that onlya part of the letter was then completed. For this would amount to

asking subjects what is true at the reference time, which is captured by theintegrity constraint (9) above and requires no inference based on therelevant models. Rather, we are interested in what subjects infer aboutthe outcome of an action described using the progressive. That is, wewant to know how subjects reason about goals and, more precisely,




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what is the projected course of events after the reference time. In order toaccomplish this, we constructed the probe pairs (P1) and (P2) using theDutch present perfect (which focuses on the present consequences of

a past event), we avoided probes of the form The girl has written a partof the letter and we asked subjects to decide which of the two probes(positive or negative) matched their expectations about the continu-ation of the narrative. Participants written explanations provide noevidence that (P1) and (P2) were insufficient for giving accurateresponses such that, for instance, a probe of the form It is unclearwhether the girl has written a letter was necessary. Rather, subjectscomments suggest that, if the discourse implied that only a part of theletter was completed, as in (S4), then the negative probe in (P2) had tobe selected.

3.4 Data analysis

Subject-based and item-based statistical analyses were carried out. Forthe former, we used a repeated-measures analysis of variance (ANOVA)model with Subject as the random effect, Aspectual Class (Activity orAccomplishment) and Condition Type (Neutral or Disabling) as fixed

effects and the mean number of negative responses (i.e. of negativeprobes chosen as responses) as the dependent variable. To generalizeover both subjects and test items,8 we employed a parallel repeated-measures ANOVA model in which Test Item (as defined above, i.e. asa set of context, critical and probe sentences) was the random effect,Aspectual Class and Condition Type were the fixed effects and themean number of negative respondents (subjects giving a negativeresponse) was the dependent variable. Univariate F-tests were

computed in both cases.

3.5 Results

The subject-based analysis revealed significant effects of Aspectual Classand Condition Type and a significant interaction between the twofactors (see Table 1). Neutral activities (S1) had the lowest mean ofnegative responses (M 2.72, SD 3.22), followed by disabledactivities (S2) (M 8.06, SD 7.05), neutral accomplishments (S3)(M 10.03, SD 9.23) and disabled accomplishments (S4) (M 25.14, SD 8.02). Except for neutral activities (S1), the distribution of

8 See Clark (1973) for the main motivation and Wike & Church (1976) for a critical discussion.




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the data in the different conditions appears rather similar, as indicatedby standard deviations (SDs) and box height and whisker length inFigure 1a. Figure 1b shows that the pattern of responses appearsas predicted by the theory: accomplishments are more sensitive thanactivities to the presence of a disabling condition in the discoursecontext. A similar pattern of effects was revealed by the item-based

analysis (Table 1). Neutral activities (S1) had the lowest mean ofnegative respondents (M 2.35, SD 4.13), followed by disabledactivities (S2) (M 7.13, SD 7.24), neutral accomplishments (S3)(M 9.54, SD 7.09) and disabled accomplishments (S4) (M 22.62, SD 9.04).

Figure 1 Results of the questionnaire. (a) Boxplot for the subject-based analysis. Solid lines

within the boxes represent the median, box height is equal to the inter-quartile range, whiskers

indicate adjacent values and empty circles denote outliers. The maximum of potential negative

judgments is 40. (b) Interaction plot for the subject-based analysis. Trace endpoints represent

the mean number of negative responses computed across participants in each experimental

condition. The maximum of potential negative judgments is 40.

Subject-based analysis Item-based analysis

Aspectual Class F(1,35) 64.763 F(1,159) 619.240P < 0.001 P < 0.001

Condition Type F(1,35) 112.560 F(1,159) 237.270P < 0.001 P < 0.001

Aspectual Class 3

Condition Type

F(1,35) 61.832 F(1,159) 100.210P < 0.001 P < 0.001

Table 1 Summary of ANOVA statistics for the questionnaire data




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4 PROCESSING MODEL

In the preceding sections we have proposed an analysis ofthe imperfective paradox based on the Event Calculus and wehave presented some behavioural data supporting the theory. Recallthat we chose to design the semantic theory as a computational analysisin Marrs sense. We are now ready to move on to the intermediate levelof analysis and consider the processing steps subserving thecomputation of discourse models.

The processing model is based on the combination of the EventCalculus and Constraint Logic Programming used in van Lambalgen &Hamm (2004). The algorithms presented there specify in an abstract

manner the computational steps involved in satisfying integrityconstraints or, more precisely, they spell out the derivation (the proof)of a given statement within the theory. In order to derive predictionsabout the complexity and the time course of the relevant computations,it is necessary to add an explicit processing component to the formalmachinery of the Event Calculus: the principle of immediacy.

4.1 The principle of immediacy

In a paper discussing language-processing models and their neuralimplementation, Hagoort (2006) proposed six general architecturalprinciples for comprehension beyond the word level. One of these,also known as the principle of immediacy, is particularly relevant inthis context. Immediacy has often been debated in the psycholinguisticliterature (Marslen-Wilson 1989; Jackendoff 2002), where it is viewedas a hypothesis about the time course of the access and integration oflexical meanings in a sentence structure and is contrasted with syntax-first models (Frazier 1987; Friederici 2002). Immediacy is defined byHagoort (2006) as

. . . the general processing principle of unification. Semanticunification does not wait until all relevant syntactic information(such as word class information) is available, but starts immediatelywith what it derives on the basis of the bottom-up input and theleft context. The corollary of immediacy is incrementality: outputrepresentations are built up from left to right in close temporalcontiguity to the input signal.

This general statement cannot be directly related to our Event Calculusanalysis. The reason is that it refers only to lexical integration and not tothe computation of denotations and discourse models. Hagoortsnotion of immediacy is sufficient for deriving processing predictions




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concerning semantic composition. However, to be relevant fordescribing the construction of discourse models, the principle ofimmediacy must be thus reformulated:

Proposition 2. The algorithms updating a minimal model so as tosatisfy an integrity constraint are executed as soon as the integrityconstraint is given as input.

The proposed definition of immediacy is a general hypothesis about thetime course of the construction of discourse representations: it statesthat minimal models are computed as soon as update instructions, inthe form of integrity constraints, are fed into the system by relevantlexical and morphosyntactic material. Propositions 1 and 2 form thecore of our processing model.

4.2 The re-computation hypothesis

Let us now consider the processing steps leading to the construction ofdiscourse models for (S3) and (S4). The first update instruction, in theform of integrity constraint (9), is introduced by the progressive VP.Since at that stage the information provided by discourse amounts to

the context (C) and the main (progressive) clause of (S3) or (S4), noobstacle for the writing activity is described. Proposition 2 implies that(9) is satisfied as soon as the progressive VP is processed and Proposition1 guarantees that a minimal model is computed such that there is a timeat which the goal state holds. The upshot is that, when an accomplish-ment in the progressive is processed, a minimal model is immediatelycomputed in which the goal state holds.

Considering the subordinate clause, we must distinguish twoprocessing steps. The first is carried out by adding (10) to the scenario(8), updating the discourse model according to (11) and deriving theexistence of a time later than the initiation of the writing activity atwhich writing was terminated. The second step computes the sup-pression of the goal state inference by deriving the further statementthat there is no time at which a complete letter was obtained. These aretwo distinct operations: for activities that are terminated by a disablingcondition, such as (S2), only the first step is carried out when thesubordinate is processed because there is no goal state to be cancelled;

for disabled accomplishments, such as (S4), both steps are implemented.Recall that the satisfaction of the goal state is derived in a minimalmodel of the progressive VP. As a consequence, the hypothesis licensedby the theory is that, when an accomplishment in the progressive isfollowed by a subordinate clause describing a disabling condition,




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the initial minimal model is re-computed to the effect that, in the newdiscourse structure, there is no time at which the goal state holds.

Re-computation is thus the main processing consequence of the

imperfective paradox. Predictions concerning the complexity and thetime course of semantic computations for sentences like (S1)(S4) canbe derived from our processing model. The theory implies that thesubordinate clause in (S4) involves the re-computation of the minimalmodel computed while processing the progressive clause, while in (S3)the initial model is simply extended (we return to the differencebetween re-computation and extension in 4.3). Furthermore, theactivity cases (S1) and (S2) will also involve a monotonic extension ofthe initial model, such that the termination of the writing activity iscomputed for (S2), but no cancellation of the goal state (sincea canonical goal is not involved in activities). In short, the subordinateclause in (S4) will be more complex to process compared to (S1)(S3),as no re-computation is triggered in the latter conditions. As regardstime courses, the Event Calculus requires the causal and temporalinformation carried by the verb morste to activate the additionalscenario clause (10), satisfy (11) and derive the failed occurrence of theconsequent state. Therefore, re-computation is expected to surface

only at the final word in (S4).

4.3 Re-computation and perceptron learning

The reader might wonder why computing a model that is incompatiblewith a previous structure (re-computation) would be different from,and in particular more complex than, computing a minimal model thatmonotonically extends the initial one. The answer can be found in thebehaviour of the neural networks that compute minimal models. Dueto the syntactic restrictions inherent in Logic Programming(Doets 1994), the models characterized by the Event Calculus can beviewed as stable states of the associated neural networks. It has beendemonstrated that recurrent neural networks are sufficient and suitablefor computing minimal models for propositional logic programs(Stenning & van Lambalgen 2005, 2007). It has also been shown that,for any propositional logic program, there exists a three-layerfeedforward network of binary threshold units that computes the

semantic operators on which the construction of minimal models isbased (Hitzleret al. 2004). The language underlying the Event Calculusis not propositional but is a many-sorted predicate logic (see section2.1), with matters being complicated further by the use of integrityconstraints. Recent research suggests, however, that recurrent networks




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can also approximate the semantic operators for first-order logicprograms and their fixed points (Hitzler et al. 2004). Moreover, asshown in Stenning & van Lambalgen (2007), integrity constraints can

be modelled via a form of back-propagation called perceptronlearning (Rosenblatt 1962; Rojas 1996).9

In this framework, any computation on a given minimal model, suchas adding a logic program clause (a scenario clause), will somehow bringthe network from its initial stable state to another stable state,corresponding to the new minimal model. Nevertheless, if the neuralrepresentation proposed in Stenning & van Lambalgen (2007) isapproximately correct, there is a large difference in neural activitybetween, on the one hand, a monotonic extension of a minimal model,and a non-monotonic re-computation of a minimal model on theother. Consider first the case of a monotonic extension as envisaged byour processing model. Retrieving a clause such as (10) from semanticmemory, assuming that spill denotes spilling coffee on the tablecloth,will result in the activation of a number of units (neurons) which werepreviously silent; but the activation state of the remaining units,including those representing the goal state (the complete letter), willremain the same. However, retrieving a different clause from semantic

memory, for instance (10), where spill now denotes spilling coffee onthe paper, will result not only in the activation of neurons which weresilent but also in the readjustment of the activation patterns of unitswhich were previously active. For instance, the neurons representingthe consequent state (the complete letter) will no longer be active. Thischange in activation is achieved in the neural network by applications ofperceptron learning. The difference between monotonic extension andnon-monotonic re-computation can thus be found in the occurrence ofthe readjustment of connection strengths driven by perceptron learning.

These considerations imply that non-monotonic re-computation in(S4) has more drastic consequences for neural processing as comparedto the monotonic extension of a minimal model in (S3). We must nowshow that a processing model based on monotonic semantics does notpredict a similar effect. In a strictly monotonic progression ofstructures, the goal state is necessarily not represented, for otherwise,

9 Multilayer feedforward networks (Hornik et al. 1989) and recurrent networks (Schafer &Zimmermann 2006) can approximate any Borel-measurable function to any desired degree ofaccuracy. Therefore, despite earlier claims advanced by Levelt (1990), connectionist networks canapproximate arbitrary recursive functions as well, suggesting that they are suited for modellinglinguistic processes. The fact that their computational power is restricted to that of finite-statemachines (McCulloch & Pitts 1943), given that the human brain is itself a finite-state machine, addsto their plausibility in psycholinguistics and cognitive science more generally (van der Velde 1993;Petersson 2005).




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the model may have to be re-computed, which is not allowed by themonotonic logic underlying the theory. In the Event Calculus frame-work this means that the predicates and axioms for continuous change

are not used, at most those for instantaneous change. Furthermore,because the progression of models is monotonic, one never actuallycomputes minimal models.10 Under the assumption of monotonicitylittle semantic computation is going on. In particular (S3) and (S4) willboth lead to monotonic extensions of the initial model and bothextensions will be computed at the same time, that is when the finalverb is processed: in (S3) the update will lead to a model in which(given the results of our entailment questionnaire) the consequent stateis represented as being attained, in (S4) it will lead to a model in whichthe consequent state does not hold. Because there is no principledsemantic or processing reason to assume that one final model would bemore complex to compute than the other, it follows that a strictlymonotonic semantics predicts no difference between the conditions.

The re-computation hypothesis can be tested in a dedicatedelectrophysiological study. Event-related potentials (derived fromelectroencephalography (EEG) signals) or fields (derived frommagnetoencephalography signals) and power changes in specific

frequency bands provide direct insights into the complexity and thetime course of neural processing, as opposed to reading times and eyemovements, which are indirect and cumulative measures of processingload (Luck 2005). An EEG study using the materials of ourquestionnaire as stimuli would be able to determine whether theevent-related potentials (ERP) correlate of model re-computation isthe N400the negative shift peaking around 400 ms after the onset ofsemantically anomalous words (Kutas & Hillyard 1980), words withlower cloze probabilities (Kutas & Hillyard 1984; Hagoort & Brown1994) and words which provide information conflicting with thediscourse context (van Berkum et al. 1999, 2003) or world knowledge(Hagoort et al. 2004). Since the computations underlying thesephenomena pertain more to the domain of semantic composition thanto the construction of discourse models, re-computation might evokea different ERP effect. Preliminary data show that the final verb in (S4)elicits a sustained anterior negativity (SAN; thus different from thetransient, centro-parietal N400) which is not observed in conditions

(S1)(S3). The amplitude of the anterior negative shift in (S4) iscorrelated with the number of negative responses in the probe selection

10 Formally, one never computes the completion of the scenario plus the EC axioms. For thenotion of completion see van Lambalgen & Hamm (2004).




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task, which is what should be expected if the SAN effect was a brainsignature of re-computation: assuming that re-computation evokesa SAN in each (S4) trial in which a negative response is given, the

larger the number of negative responses (the larger the number of trialsin which re-computation took place), the larger the amplitude of theERP component revealed by averaging. Current experimental work inour laboratory is aimed at corroborating these findings.

Acknowledgements

This paper benefited from many discussions with Travis Choma, Bart Geurts, Peter

Hagoort and Karl Magnus Petersson. We thank Ira Noveck and two anonymousreviewers for their comments. We are grateful to The Netherlands Organization forScientific Research for support under grant NWOCOG/04-19.

GIOSUE BAGGIO

F.C. Donders Centre for Cognitive Neuroimaging

Radboud University Nijmegen

P.O. Box 9101

NL-6500 HB NijmegenThe Netherlands

e-mail: [email protected]

MICHIEL VAN LAMBALGEN

Department of Philosophy

University of Amsterdam

Nieuwe Doelenstraat 15

NL-1012 CP Amsterdam

The Netherlands

e-mail: [email protected]

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