artstein, ron - 2005 - quantificational arguments in temporal adjunct clauses

8/12/2019 Artstein, Ron - 2005 - Quantificational Arguments in Temporal Adjunct Clauses

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Quantificational Arguments in Temporal Adjunct ClausesAuthor(s): Ron ArtsteinSource: Linguistics and Philosophy, Vol. 28, No. 5 (Oct., 2005), pp. 541-597Published by: SpringerStable URL: http://www.jstor.org/stable/25001970.

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LinguisticsandPhilosophy(2005)28:541-597 ? Springer2005DOI 10.1007/s10988-005-6921-6

RON ARTSTEIN

QUANTIFICATIONAL ARGUMENTS IN TEMPORALADJUNCT CLAUSES

ABSTRACT. Quantificational arguments can take scope outside of temporaladjunct clauses, in an apparent violation of locality restrictions: the sentence fewsecretaries cried after each executive resigned allows the quantificational NP eachexecutive to take scope above few secretaries. I show how this scope relation is theresult of local operations: the adjunct clause is a temporal generalized quantifierwhich takes scope over the main clause (Pratt and Francez, Linguistic and Philosophy 24(2), 187-222. [2001]), and within the adjunct clause, the quantificationalargument takes scope above the implicit determiner which forms the temporalgeneralized quantifier. The paper explores various relations among quantificationalarguments across clause boundaries, including temporal clauses that are modifiedinternally by a temporal adverbial and temporal clauses with embedded sententialcomplements.

1. INTRODUCTIONTemporal clauses provide apparent counterexamples to the generalization that adjunct clauses form boundaries for quantifier scope. Thesentences in (1) have readings where the quantificational argument ofthe embedded clause takes wide scope with respect to the matrixsubject: the sentences are true if each resignation or termination isassociated with different crying secretaries.

Before( {Few secretary cried whenewsecretaries) \ after Jeach executive resigned.the board fired each executive.

Such readings are generally not available with non-temporal adjunctclauses: the sentences in (2) do not allow a wide-scope interpretationof the embedded argument.

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(2) A secretary cried althoughFew secretaries I aubecause{ each executive resigned.the board fired each executive.

The ability of a quantificational argument to take scope outside of atemporal adjunct clause therefore appears to be related to thetemporality of the clause. In this paper I show how this falls naturallyfrom a semantics that treats temporal adjunct clauses as temporalgeneralized quantifiers (Pratt and Francez 2001). Temporal adjunctclauses become temporal generalized quantifiers through the application of a temporal determiner, akin to the explicit determiner inPPs such as before/during/after each meeting. The determiner applieswithin the adjunct clause, so a quantificational argument can takescope over it; in turn, the entire temporal generalized quantifier cantake scope above a quantifier in the matrix clause, giving the quantificational argument scope over thematrix clause as well. Throughthis mechanism the scope of a quantificational argument transcendsthe clause that contains itwithout violating locality restrictions.A further illustration of the scopal properties of quantificationalarguments in temporal adjunct clauses is the ability of a quantifierinside such a clause to bind a pronoun outside it.The sentences in (3)all have such readings, which can be roughly paraphrased as "foreach boy, before/when/after he goes to sleep, I give him a kiss".

(3) (Before'When each boy goes to sleep, I give him a kiss.After

While superficially similar to "donkey" sentences like ifa farmer ownsa donkey he beats it, the above sentences must be instances of truevariable binding because universal quantifiers do not license"donkey" readings; the sentences in (4) do not have readings that canbe paraphrased as "for each boy, if/although/because he goes tosleep, I give him a kiss".

(4) IfAlthough each boy goes to sleep, I give him a kiss.Because



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 543Evidence that the contrast between (3) and (4) stems from thetemporality of the subordinate clause comes from looking atatemporal when-clauses such as (5) (Carlson 1979; Farkas andSugioka 1983).

(5) When a bear has blue eyes she is intelligent.The atemporal when-clause in (5) allows a "donkey" interpretation ofits subject a bear, but a universal quantifier in this position cannotbind a pronoun outside of its clause - sentence (6) is incoherent; whentheword when has a temporal interpretation, a universally quantifiedargument can bind a pronoun outside the adjunct clause (7).

(6) *When each bear has blue eyes she is intelligent.(7) When each bear is hungry she growls.

My discussion will concentrate on examples with scope ambiguitieslike (1) above, and ignore pronoun-binding sentences like (3).Whilethe ability to bind a pronoun in themain clause is one of the clearestillustrations of the scope-taking ability of quantifiers in temporaladjunct clauses, the question of how a pronoun gets bound isorthogonal to the problem of quantifier scope: once our semanticsgets the scope right, the binding of pronouns should follow from anytheory of pronoun interpretation.The ability of quantificational arguments to take scope outside thetemporal adjunct clauses that contain them is not a peculiar trait ofEnglish; however, it is not universal. Hebrew behaves like English,and in the examples below, the quantificational arguments in thetemporal adjunct clause may receive wide scope.

(8) lifnei semazkira bax-ta before thatsecretary cried-sg ksemazkirot sfurot bax-u whensecretaties few cried-pl axrei seafter that

kol menahel hitpatereach manager resignedha-direktoryon piterkol menahelthe-board fired each manager



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German, on the other hand, does not allow quantificationalarguments to take scope outside temporal adjunct clauses. In the endof Section 3.2 I suggest that the difference can be attributed tomorphology: while English and Hebrew employ an implicit temporaldeterminer, inGerman the temporal determiner constitutes part ofthemeaning of the complementizer, and therefore a quantificationalargument cannot take scope above it while remaining within theboundaries of the adjunct clause.Due to the recent interest in temporal quantification (Pratt andFrancez 2001; von Stechow 2002; Francez and Steedman to appear), Ifeel that it is appropriate to add a note that places this work inrelation to the others. Pratt and Francez introduce temporal contextvariables and temporal generalized quantifiers, and use themmainlyfor an account of temporal "cascades" - the modification of asentence bymultiple temporal adverbials - as inBill cried during every

meeting on Tuesday. The theory is extended by von Stechow toinclude temporal relations brought about by tense and aspect, andreformulated into a framework of transparent LF. Francez andSteedman extend the analysis to deal with locative prepositions andwith varying orders of temporal and locativemodifiers, reformulatingit in the framework of Combinatory Categorial Grammar.The present study is a detailed investigation of relations betweenquantificational arguments across the boundaries of temporal adjunctclauses. Temporal adjunct clauses are treated in all of the aboveworks, but quantifiers within such clauses only receive a cursory

mention in one of them (Francez and Steedman, to appear, Section5.1). Inmy investigation I largely ignorematters of tense and aspect;while I agreewith von Stechow's claim that "an adequate treatment oftemporal adverbs is only possible on the basis of an elaborate theoryof tense and aspect" (2002, p. 756), I find tense and aspect orthogonalto the problems of quantification discussed in this paper, so I set thesequestions aside in the interest of keeping the investigation focused. Ialso ignore Francez and Steedman's extension to locatives. Locativeclauses allow quantifier scope interactions with themain clause justlike temporal clauses (thanks toAnita Mittwoch for this observation).

(9) A tree grows where each car had crashed.Avoiding locative clauses in this paper is done only in the interest ofkeeping the semantics simple; it appears that an extension along thelines of Francez and Steedman (to appear) is possible. We should



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 545note that locative clauses are comparatively rare - unlike their temporal counterparts, ocativeprepositionslikeover,under,behindetc.do not take clausal complements. This property of locative prepositions appears to be valid crosslinguistically, and it does not receive anexplanation; the mystery, however, is independent of the ability ofquantifier scope to cross the boundaries of both temporal andlocative clauses.

The paper is organized as follows: Section 2 develops the basictheory of temporal generalized quantifiers, and is mostly areinterpretation of ideas presented in the three works mentionedabove. The heart of the paper lies in Section 3,which examines scopeinteractions across the boundaries of temporal clauses. The maininsight on how a quantificational argument escapes the scope of itsclause is developed in Sections 3.1-3.2; Section 3.3 looks at restrictions on the resulting readings, and Section 3.4 extends the treatmentto temporal clauses which have temporal modifiers within them.Section 4 compares the theory developed in this paper to the otherworks on the topic, and offers two mild arguments in favor of thecurrent formalism. Section 5 offers a summary and conclusion.

2. THE BASIC THEORY OF TEMPORAL MODIFICATION

I develop a theory of temporal modification, based on the proposalsof Pratt and Francez (1997, 2001), von Stechow (2002) and Francezand Steedman (to appear). These works develop similar semanticsystems using significantly different formalisms for the syntax-semantics interface: Pratt and Francez leave the interface unspecified,von Stechow uses a syntax with movement operations and abstractlogical forms, and Francez and Steedman use a highly lexicalized

Combinatory Categorial Grammar. The contribution of the presentpaper is mostly on the side of the semantics: I show how thesemantics of temporal modification allows a quantifier to escape fromwhat would otherwise be a scope island. As far as the syntax isconcerned, I choose themiddle way: I assume a fairly close matchbetween surface form and semantic interpretation, but I willintroduce certain semantic operations that take place at variousnodes of the surface representation which are not triggered bymorphology or syntax. This results in a system that overgeneratesmeanings, and in order to restrict the system Iwill posit a number ofconstraints on semantic derivations. Such restrictions would fit in the



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syntactic component of von Stechow's framework and in the lexiconof Francez and Steedman's framework. At the end of the paper, inSection 4, I will show how my formulation is better at handling twoparticular structures, namely non-persistent temporal predicates andlong-distanceemporaldependencies.

2.1. PreliminariesI use a two-sorted translation language with lambda abstraction(cf. Gallin 1975), which represents times explicitly: e is the type ofindividuals and i of time intervals. To keep the language simple, I willtreatwhat are ordinarily thought of as event predicates as if they arepredicates of time intervals; this will allow us to eliminate events fromour ontology and to simplify expressions of the formAi.3e[i = (e) A pred(e)] to themore readable form Ai.pred(i). Verbsand temporal nouns have temporal arguments, akin to event arguments in event semantics. The basic meaning of a verb like cry is arelation etween ndividualsnd timeintervalsAx;i.cry(x)(i) (typeeit),and a noun likemeeting in a temporal context is a property of timeintervals Ai.meeting(i) (type it) - those time intervals during which a

meeting takes place.Representations are enriched by temporal context variables, whichare variables of type i that stand for time frames for the evaluation ofsentences (cf. Pratt and Francez 2001). These can occur as free

variables, and as such they are treated as indexicals whose meaning isdetermined by an assignment function. The existence of temporalindexicals has been recognized at least since Partee (1973), who notesthat a sentence like I didn't turn off the stove makes reference to "adefinite interval whose identity is generally clear from extralinguistic context" (pp. 602-603). The most common use of free temporalcontext variables in this article will be to denote the overall temporalcontext of evaluation. Temporal indexing applies to both nouns andverbs (cf. En9 1986), so both will allow temporal context variables intheir representations.

Temporal modification means subordinating the temporal contextof a particular linguistic expression to that of another expression; thisis captured in our representation by binding the temporal contextvariable of the temporally subordinate expression. This treatment oftemporal indices is formally similar to the treatment of world indicesinGroenendijk and Stokhof (1982), where free world variables get



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 547

bound in forming intensions of sentences, but binding of an index hasa different meaning in our semantics - it does not result in anintension, but rather in the creation of a temporal property (seeexample (24) in Section 2.3).A few typographical conventions: constants like cry are set inboldface, and variables are set in italics. I use lower case letters forindividual variables and upper case letters for property variables; thefollowing letters (with primes, if necessary) stand for variables of themost commonly used types.

type e: x, y type et: P, Qtype i: i,j type it: I, JThe types of higher-order variables will be noted explicitly withsuperscripts when needed, e.g. (it)t. A distinguished temporalcontext variable stands for the overall temporal context ofevaluation, and it ismarked with a hat in order to make it visuallysalient: 1.

2.2. Basic Sentence MeaningsNatural language predicates are indexed to time intervals. Anintransitive verb like cry denotes a relation between individuals andtime intervals (type eit); the denotation we get for a sentence like Billcried is a property of time intervals - the set of times at which Billcried (I ignore tense for themoment, postponing the discussion toSection 2.5).

(10) cry ^-Lxxi.cry(x)(i)(11) Bill cried , Ai.cry(bill)(i)

The sentence Bill cried is true if a time that satisfies (11) exists withinthe general time frame in which the sentence is evaluated. We willcapture these truth conditions by altering the representation in (11),subjecting it to two operations that apply in succession: contextualization(12a) introduces the temporal context variable i which stands for thecontext of evaluation, and existential closure (13a) closes off the eventtime variable by turning the lambda operator into an existentialquantifier (the latter is akin to the existential closure of an eventargument in event semantics).



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(12)a. Contextualization operation (type (it)it): AI i.i c i A I(i)b. Bill cried + A.i c i A cry(bill)(i)(13)a. Existentialclosureoperation(type(it)t): L.3i[I(i)]b. Bill cried - 31ii c i A cry(bill)(i)]

The final representation (13b) is the translation of the sentence Billcried; it is true with respect to amodel M and an evaluation contexti if there exists an interval i included in i such that Bill cried at i inM.

An alternative to the operations (12a) and (13a)would be to definetruth directly on the representation (11). Since context of evaluationis not represented explicitly in (11) we would need an indirectmechanism for specifying the truth conditions, perhaps along thelines of the following definition of truthwith respect to one index ofevaluation trueuel) in terms of truth with respect to two indices ofevaluation (Dowty 1982, p. 33, example 24).(14) [1pJ= 1'I is true atf) iffthere is some i such that

[1 J- 1

But such a definition will not do for our purposes. Dowty's definitionis part of an account of tense, aspect, and non-quantificationaltemporal modifiers, and thus it can allow one index to always receiveimplicit existential quantification at the sentence level. This paper isconcerned with quantificational temporal modifiers; these operate onthe temporal context of linguistic expressions, and in order tomakecontext accessible to various quantifiers itmust be represented in thetranslation explicitly.The idea that temporal modifiers operate on the context (or"reference time") of modified expressions rather than on their eventtime appears at least as early as Dowty (1982). The reasons arediscussed in detail inPratt and Francez (2001, pp. 200-206), so here Iwill just repeat the crux of the argument. Modifying the event time, asdone for instance inDowty (1979) or Stump (1985), works fine withnon-quantificational temporal modifiers, as in the following examplefrom Stump (1985, p. 103).

(15) Yesterday John saw Mary in the morning, after Billarrived.



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 549The temporal modifiers can apply in turn to a temporal propertyformed by abstracting over the event time of the verb; this results in aseries of conjoined temporal predicates that do not stand in a scoperelation to one another.

But when temporalmodifiers are quantificational, they do stand inscope relations: in the following example, themodifier during mostconferences takes scope above after each meeting.

(16) John sawMary after eachmeeting during most conferences.In order to allow scoped relations, each modifier has to apply to adistinct temporal variable, but the verb (or any other temporalpredicate) only has one event time. Temporal modifiers must therefore operate on something other than the event time of the expressiontheymodify, and this is the context. Since each modifier introduces itsown context, the number of modifiers is in principle unlimited andleads to the phenomenon of "cascading" (Pratt and Francez 2001, seealso Section 2.8 below). The fact that the scope of temporal modifiersdoes not necessarily follow their surface order is treated by Francezand Steedman (to appear) using the power of Combinatory

Categorial Grammar, and will not be discussed here.The inclusion relation between the event time and context issimilar to the relation proposed by Kamp and Reyle (1993) betweenan eventuality and the discourse referent which plays the role of its"location time". Temporal inclusion of an event time in its context isconsidered by von Stechow (2002, Sections 10 and 11) to be themeaning of perfective aspect. In order to avoid the complicationsraised by aspectual class, I will always choose examples withpredicates for which inclusion is the appropriate relation.Contextualization must precede existential closure because it has toapply at a point where the time variable of the predicate is accessible.

2.3. Temporal Generalized QuantifiersTemporal modifiers like the PP after each meeting are temporalgeneralized quantifiers of type (it)t. I take common nouns likemeetingto denote predicates of times rather than predicates of events whenplaced in a temporal context, e.g., as the complement of a temporalpreposition; the temporal noun meeting is a predicate of type it.

(17) meeting - Ai.meeting(i)



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A predicate likemeeting is itself evaluated with respect to the overalltemporal context, so the temporal context variable i is introduced bythe contextualization operation.

(18) meeting ^ i.i c iA meeting(i)The determiner each in a temporal NP like eachmeeting is similar tothe familiar nominal determiner each, except that itdenotes a relationbetween predicates of times rather than predicates of individuals - itis of type (it)(it)trather than type (et)(et)t.Each combineswith thecontextualized translation of the noun meeting in the obvious way.

(19) each - I2J.VilI(i) -> J(i)](20) each meeting - %J.Vi [(ic 'Ameeting(i)) -* J(i)]

Since the common noun meeting was taken to be a predicate of times,it should come as no surprise that the expression eachmeeting, whenplaced in a temporal context, denotes a temporal generalizedquantifier, that is a predicate of predicates of times.The temporal preposition after translates as a function of type((it)t)(it)t,mapping temporal generalized quantifiers to temporalgeneralized quantifiers; this higher-order function isdefined using theprimitive temporal function AiAj.afterij) from interval pairs tointervals (type iii; the same function is called "time-from" in Prattand Francez (2001), and Fafterin Francez and Steedman (to appear)).

(21) after (j) is the interval spanning from the end of j to theend of i, if j c i;undefined otherwise.The proposition that an interval follows j (within a context i) can beexpressed by the formula C afteri(j). Iwill discuss the rationale forthis treatment of preposition meanings in Section 2.6, after we haveseen how thesemeanings are put into use by the semantics.As for the higher-order translation of the preposition after, wenote that the temporal generalized quantifier eachmeeting (20)mapsany temporal predicate J to the statement that J holds of eachinterval which is ameeting; the expression after eachmeeting should

map a temporal predicate J to the statement that J holds after eachinterval which is a meeting. Therefore the preposition after shoulddenote a function which takes a temporal generalized quantifier of



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 551the form LJ.4 and "injects" the temporal function after into thecomplement(s) of J in 4.

(22) after -,A (it)t.)J.T(ii.J (afters(i)))The preposition after uses the overall temporal context of evaluationi as the context argument of the temporal primitive after; this will beused in the account of temporal cascades in Section 2.8.The temporal PP after each meeting gets itsmeaning by straightforward application of the preposition after (22) to the temporalgeneralized quantifier each meeting (20). The result is a temporalgeneralizedquantifier.

(23) after each meetingLJ.Vi[( i Ameeting(i)) -> J (afters(i))]

The PP after eachmeeting modifies the sentence Bill cried (I assumefor themoment that the PP attaches as an adjunct to the entire mainclause, this will be revised in Section 2.7). The temporal generalizedquantifier (23) needs to apply to a temporal property; we form such aproperty by abstracting over the free temporal context variable i inthe existentially closed meaning of themain clause (13b). This has theeffect of subordinating the temporal context of themain clause tothat of the temporal modifier.

(24) Bill cried after each meeting)lJ.Vi[(i Ameeting(i))-> J(after1(i))](2i.3r [i'c IAcry(bill)(0')])= Vi[(i c iAmeeting(i)) --3i'[i c afters(i) A cry(bill)(i')]]

Abstracting over the temporal context variable in order to allowtemporal modification is essentially the same as Dowty's rule forcombining a time adverbial with a sentence (Dowty 1982, p. 35, ruleS43). We can view this abstraction as part of the rule of temporalmodification, or as an independent operation that necessarilyprecedes it; an independent application of an abstraction operationwill be needed for the computation of aggregate readings in Section3.4.1. Shortly below we will also see that abstraction over thetemporal context variable has to apply to types other than t (makingitmore general thanDowty's rule), and we will revise the abstractionrule accordingly in Section 2.7.2, example (54).



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2.4. Temporal ClausesTemporal clauses like after Sue arrived also denote temporal generalized quantifiers. Their meanings are derived in a similar way totemporal PPs, through application of the contextualization operationfollowed by a determiner meaning. The rawmeaning of a clause likeSue arrived is a property of times (25), to which a temporal contextvariable is added by the contextualization operation (26) in a mannercompletely analogous to that of the clause Bill cried (11)-(12b).

(25) Sue arrived A2i.arrive(sue)(i)(26) Sue arrived - {i.ic IA arrive(sue)(i)

However, unlike a matrix clause whose temporal argument isexistentially closed at this stage, a temporal clause needs to have adeterminer applied to it in order to turn it into a temporal generalizedquantifier. There is no overt determiner, so an implicit existentialdeterminer meaning (27) is applied, followed by the preposition after(22).

(27) Implicitexistentialdeterminer: IA2J.3i[I(i) J(i)](28) Sue arrived , J.3i[i c 'A arrive(sue)(i) A J(i)](29) after Sue arrived 1J.3i[ic iA arrive(sue)(i) A J(after((i))]

There are two reasons not to subsume the determiner meaning underthe semantics of after. The obvious reason is to keep the semantics ofafter with a clausal complement identical to that of after with anNPcomplement: after applies to NPs with determiners (that is totemporal generalized quantifiers), so its clausal complements shouldalso denote temporal generalized quantifiers, and this requires theapplication of an independent determiner. Another reason to assumethat the determiner is not part of themeaning of after is that certainoperations may intervene between after and the determiner meaning -for example, a quantificational argument in the temporal clauseshould be allowed to take scope above the determiner in order to giveit scope above thematrix clause; such constructions will be discussedin detail in Section 3. The existential determiner is therefore anindependent semantic operation, triggered by the temporalcomplementizer but applying below it.



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 553The temporal generalized quantifier (29) combines with the

meaning of thematrix clause and yields the correct results.(30) Bill cried after Sue arrived -JJ.31]i iA arrive(sue)(i) J(afteri(i))]

(AI3' [i'C IA cry(bill)(0')])= 31tic ~A arrive(sue)(i) A 3i'[i' c afterr(i) A cry(bill)(')]]Sentence (30) is true if there exists an interval (within the overall frame)inwhich Sue arrived, followed by an interval at which Bill cried.

Looking at the representation (30), we can already seewhere ouranalysis is headed: the temporal clause in (30) takes semantic scopeover the entire sentence, and this will be used in Section 3 to explainhow a nominal quantifier in a temporal clause can take scope above aquantifier in thematrix clause. Before we get there, we turn to twoissues that were glossed over in the preceding discussion, namelytense and temporal prepositions (Sections 2.5 and 2.6, respectively);the remainder of Section 2 then continues to develop temporalgeneralized quantifier theory, which is the basis for the semantics thatfollows.

2.5. TenseAn anonymous reviewer notes that if tense is taken to apply abovethe temporal modifier in (24), it will modify the context of thetemporal PP rather than the time of the verb. This is undesirable,because such an analysis would predict that in a sentence like Billcried before themeeting themeeting must be in the past, whereas thesentence can be uttered felicitously if the meeting is still going on.The conclusion from this example is that tense should apply belowthe temporal modifier, so that itmodifies the verb itself.Iwill demonstrate how this treatment of tense works by derivingagain the meaning of Bill cried after each meeting, this time interpreting the tensemorphology on the verb. Unlike temporal adverbsand PPs which modify a sentence's context of evaluation, tenseshould modify the event time directly. The reason is that adverbialmodification identifies the context of themodified sentence with theevent time(s) of the temporal modifier. In the sentence Bill criedtoday, the context of Bill cried is the event time of today; only theevent time of Bill's crying should be in the past, not its context. We



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therefore spell out the meaning of the past tense morpheme as amodifier of temporalproperties,type(it)it.(31) PAST /.J.Aitpast(i) A J(i)]

Past tensemust apply before existential closure, when the event timeis still accessible; its ordering with respect to contextualization is notcrucial.(32) Contextualized: Bill cried --* i.ic i A past(i) A cry(bill)(i)(33) Existentiallyclosed:Bill cried

3i[i c i A past(i) A cry(bill)(i)]The temporal PP after eachmeeting (23)works in the same fashion asin the previous section, applying to the temporal property formed byabstracting over the free temporal context variable i in theexistentially closed tensed sentence Bill cried (33).

(34) Bill cried after each meetingiJ.Vtii c IAmeeting (i)) -> J afteri (i))](2i.3i' [i'c A past(') A cry(bill)(i')])= VI(i c 1Ameeting(i)) ->3i'[i'c after (i)A past(') A cry(bill)(i')]]

The final representation has the desired result: each event-time i'of Billcrying is past, but tense does not directly modify the contexts of thecrying or themeeting times.Of course, themeetings in (34)must alsobe in the past if the sentence is true; this is an entailment that comesfrom the condition that each meeting precedes a past crying time.The above representation also shows us how tense can restrict theoverall context of evaluation. In an attempt to explain why in asentence like John called everyMonday the domain of quantificationappears to be restricted to past Mondays, von Stechow (2002, p. 759)suggests that tense applies above the temporal modifier PP. Thissuggestion is subject to the criticism noted at the beginning of thissection - the sentence can be uttered felicitously on aMonday andinclude the day of utterance in the domain of quantification. Itappears that von Stechow himself is not too happy with his solution,and he hints that a better approach would follow a proposal he citesfrom class notes by Irene Heim: tense should modify the verb, and



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a. AFTER(i, j,f) if and only if c afteri (i).b. afterV(j)= max(i2J.ArFER(i, j, ')),where max(J) is themaximal interval i such that J(i) is true, if such aninterval exists.

The reason I prefer the iii-type notation is as follows. We have seenabove in Section 2.3 that a formula like the one in (35a) belongsbelow the scope of the quantifier introduced by a temporal NP likeevery meeting. Syntactically, however, the quantified NP is acomplement of the preposition after. The problem is addressed byvon Stechow (2002) through a syntactic movement rule that bringsthe temporal NP above the preposition at Logical Form, but a theorythat is faithful to surface constituent structure requires a semanticmechanism to bring the temporal relation below the scope of thequantifier. This is achieved in this paper through a higher-ordertranslation of temporal prepositions, which builds on lower-typetemporal primitives (essentially the samemechanism is employed byPratt and Francez (2001) and Francez and Steedman (to appear)).Such a mechanism is easiest to write if we express the temporalprimitive as a function which yields an interval of type i; expressingthe primitive as a relation would make the translations more difficultto read.

Not all temporal primitives can be equally formulated as relations and functions. Whatever functional meaning we give to theprimitive throughout, the representation f C throughout,(j) does notexpress the appropriate relation, namely that the interval f laststhroughout the duration of j (thanks to Ariel Cohen for pointingthis out). The problem is with the temporal inclusion relation C,introduced by the contextualization operation. There are two solutions to this problem, both based on the observation that thepreposition throughout may only modify predicates of certain aspectual classes. One solution is offered by Pratt and Francez (1997):throughout is subjected to a lexical restriction which allows it to onlyapply to universally quantified temporal predicates; the other solution follows the suggestion of von Stechow (2002) to replace theinclusion relation C by modeling various verb aspects with distinctcontextualization operations. We do not need to go deeper intoeither of the solutions in this paper, because the issue does not seemto affect quantifier scope.



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2.6.2. Refinements oprepositionmeaningsThe semantic primitive after is not fine-grained enough. Forexample,theformuladerivedin (24)forBill criedaftereachmeetingis satisfied if Bill cried once, after the lastmeeting. One could arguethat this is correct on a very literal interpretation of the sentence,but the sentence normally implies a tighter connection between the

meeting times and the crying times. One way to capture this isthrough additional parameterization of the primitive after. Prattand Francez (2001, p. 199) note that the word before often has a

meaning of "just before" or "a short time before"; the same holdsfor after. Such a meaning can be expressed in our semantics by aprimitive function RiUAj.afteri,(j), which maps a context i, acontextually determined length of time e and an interval j to theinterval that begins at the end of j, has a maximal length E, andextends maximally to the end of i (a preposition meaning that takesa length argument is needed anyway for PPs that include an explicitdegree expression, e.g., at most twominutes after each meeting). Buta single, fixed length emay not be sufficient for Bill cried after eachmeeting, because the sentence normally implies that for each meeting, Bill cried after it and before the next meeting. We shouldtherefore take e as a function whose range is lengths of time andwhose domain includes various contextual factors, among them theevent time j (or even the event itself). As noted by Kamp and Reyle(1993, p. 628), it is very difficult to give a precise characterization ofall the factors that constrain the distance between the event times ofthe modified clause and the temporal modifier; I will not attemptsuch a characterization here.

An alternative to the parameterization of the function after is toenforce a connection between the times of the modifier and themodified clause through a matching mechanism along the lines ofRothstein (1995). Rothstein notices that sentences like every timeJohn rings the bell, Mary opens the door have a "matching effect",whereby each ringing event must be matched by a distinct openingevent. I am not clear as towhether such an effect exists in sentenceswith temporal prepositions. Take the sentence after each bell rings,Mary opens thedoor, or the sentence after each chime,Mary opens thedoor: if two bells ring or chime simultaneously, is one door openingsufficient tomake the sentence true? I am not surewhat the answer is.If there is amatching effect in these cases, it should be possible to addevents to our semantics and incorporate amatching function like that



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of Rothstein (1995). But even if amatching mechanism is applied, itwill still not enforce the temporal interleaving of crying times andmeeting times inBill cried after eachmeeting, so inserting contextualparameters to the function after seems inevitable.Both the question of temporal primitive parameterization and thequestion of a matching effect are orthogonal to the problem ofquantifier interaction across temporal adjunct clauses. We willtherefore continue with the samemeanings for primitives we definedin Section 2.3.2.6.3. Prepositions of temporal identityTemporal prepositions other than after have translations similar to(22), with the appropriate temporal function replacing after. Buttemporal prepositions which denote identity of time, such as during,on and in,have a curious property: when such a preposition takes anNP complement, themeaning of the resulting PP temporal generalized quantifier turns out to be identical to the original meaning of thecomplement NP temporal generalized quantifier. This follows fromtwo assumptions about natural language NP meanings: one is thattemporal determiners only apply to contextualized NPs, and theother is that natural language determiners are conservative (Keenanand Stavi 1986). Here is theproof. We startwith the definitions of thetemporal primitive during and the preposition during.

(36) during) is the interval j itself, ifj c i;undefined otherwise.(37) The preposition during translates as

AT(it)t.AJ.T(,i. J(during(i0)).Let CN be a temporal common noun with meaning i.Af(i) of typeit, and letDet be a conservative temporal determiner with meaning

7IAJ.D(I)(J) of type (it)(it)t. Conservativity is defined as follows.(38) D(I)(J) if and only ifD(l)(i.I(i) A J(i))

A quantificational NP Det CN is formed by applying the determinermeaning D to the contextualized common noun meaning Ai.iC i AN(i).

(39) ,J.D(i.i c IA A(i))(J)The preposition during applies to the above to give the PP duringDetCN.



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(40) )J.D(Ai.i c 1A A(i))(i.J(during(i)))By conservativity, this is equivalent to the following.

(41) i J.D (Ai.ic iAfA(i))(i.i c iA A(i) A J(during(i)))We can now simplify the expression in the second argument of >:bythe definition in (36), the expression duringi(i) is equivalent to i if iCi. Since duringi(i) in the above expression is part of an expressionthat's conjoined with iC i,we can safely rewrite it as i.

(42) lJ.T(Ai.i C i A A(i))(.i C A /(i) A J(i))Applying the conservativity hypothesis again, we arrive at arepresentation that is equivalent to themeaning of theNP Det CN(39).

(43) )J.D(Ai.i c i A /())(i.J(i))This concludes the proof thatwhen the preposition during applies to atemporal generalized quantifier formed from a conservativedeterminer and a contextualized temporal property, the result isidentical to applying the identity function on temporal generalizedquantifiers (AT.T). By hypothesis, this is the case with all naturallanguage temporal generalized quantifiers. From now on, in order tokeep things simple, Iwill treat during and similar prepositions as ifthey do indeed denote identity. It is important to remember that thisis not an arbitrary stipulation, but rather a result derived from the

meanings of primitive temporal functions.Of course, the preposition during is not redundant: it serves toindicate that the following NP is to be interpreted temporally. In thetheory developed here, the semantic type of common nouns dependson their syntactic position: during forces the head noun of itscomplement to be of type it rather than et. In a theory where commonnouns likemeeting invariably denote predicates of events (Pratt andFrancez 2001; Francez and Steedman to appear), the prepositionduring (along with all other temporal prepositions) is a type-changer,transforming event generalized quantifiers into temporal generalizedquantifiers.



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exactly three executives(47) Mary scolded few executives duringeachsome executive meeting.However, this paper concentrates on the semantics of temporalexpressions, so I will put aside the question of precise syntacticrepresentation. Various mechanisms can be used to account forscope, among themQuantifying-in (Montague 1973), QuantifierStorage (Cooper 1983) and Quantifier Raising (May 1985); to myknowledge, the data discussed in this paper do not present newarguments for choosing among them.All the scope mechanisms allowtwo different orders of application of the nominal and temporalgeneralized quantifiers to the verb; the same is true of the structures(45a) and (45b), and this is sufficient for our purpose.2.7.1. Wide scope temporaleneralizedquantifiersWe start with the wide scope reading for the temporal generalizedquantifier in the sentence some executive cried during each meeting.The meaning of the unmodified clause, some executive cried, is theresult of applying the subject as well as contextualization andexistential closure to the verb cry, which is a function of type eit.

(10) cry xAxJi.cry(x)(i)In Section 2.2 we saw how contextualization and existential closureapply to the entire sentence; contextualization and existential closurecan also apply to the verb directly by using a mechanism that givesthe subject scope above these operations, for instance Quantifying-in,Quantifier Storage or Quantifier Raising (thanks to Idan Landau forpointing out the importance of using a scope mechanism for thispurpose). In the sentence some executive cried, the relative scope ofthe subject and existential closure doesn't matter because they bothcontribute existential quantifiers. However, it is instructive to showhow contextualization and existential closure apply to the verb,because this will be needed in Section 2.7.2.

The contextualization operation (12a) is a function of type (it)it,and is thus of thewrong type to combine with the verb via functionalapplication; it can however combine via functional composition.

(48) Functional composition: fa o gP' =df LrP.f(g(r))[p, a, z are variables over types]



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(49) cry - AIA .ic A I(i) o AxAi.cry(x)(i)= Ax.[IU i.ic 1A I(i)]([,Axi.cry(x)(i)](x))= AxAi.ic IA cry(x)(i)The existentially closed meaning of the verb cry is likewise arrived atthrough functional composition of the existential closure operation(13a) with the contextualized meaning of the verb.

(50) cry -+ ;I.3i[I(i)] o Ax2i.i iA cry(x)(i)= Ax.[AI3i[I(i)]]([AxiU.i c i A cry(x)(i)](x))= Ax.3i[i c tA cry(x)(i)]The subject some executive is a familiar (nominal) generalizedquantifier; it applies to the VP meaning to give themeaning of thesentence some executive cried, with a free temporal context variable.

(51) some executive - AP.3x[exec(x) A P(x)](52) some executive cried -PP.3x[exec(x) P(x)](Ax.3i[i iA cry(x)(i)])

=3x[exec(x) A 3i[i c i A cry(x)(i)]]Finally, the temporal PP applies. Recall the discussion in Section2.6.3, which established that themeaning of during each meeting isidentical to that of each meeting (20). This is a temporal generalizedquantifier, and it applies to the Ai-abstract over themeaning of someexecutive cried.

(53) some executive cried during each meetingLJ.Vi[(iC IAmeeting(i)) - J(i)](Ai.3x[exec(x) A 3i'[i' c A cry(x)(i')]])= Vi[(i c iAmeeting (i)) -

3x[exec(x) A 3i'[i' c iA cry(x)(i')]]]The resulting meaning is that for each meeting there is an executivewho cried during that meeting. Executives may vary with meetings,which is precisely the reading we want.2.7.2. Narrow scope temporal generalized quantifiersIn order to get a narrow scope reading for the temporal generalizedquantifier in the sentence some executive cried during each meeting,



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 563the temporal PP has to apply to the verb before the subject does. Asin the previous cases of temporal modification, the temporalgeneralized quantifier meaning of themodifier has to combine withan abstract over the temporal context variable of the existentiallyclosed meaning of the verb (50). Since the latter is of a type higherthan t, simple prefixation of )i to the verb meaning will not work.Rather, the lambda abstract over the temporal context variable isadded to the verb meaning as the last lambda operator.

(54) A temporalmodifier denoting a temporal generalizedquantifier T of type (it)t combines with a constituent witha denotation sli ... Asn. of type a1... ant for some n > 0via functional application or composition (whichever isappropriate) of T to themeaning isl ... isni.4 (the stringof lambdas may be empty).

When the modified constituent is of type t, the rule results inprefixation of Ai as we have seen so far.When the rule is applied to apredicate of type et, the abstractor 2i skips over the initial Ax; this canbe thought of as functional composition of the abstraction operationitself (thanks to Fred Landman for this observation).The meaning of the VP cried during each meeting is derivedthrough functional composition of the temporal generalized quantifierwith the lambda abstract over the temporal context variable ofthe verb meaning (55); the VP then combines with the subject

meaning (51) to give themeaning of the complete sentence (56).(55) cried during each meeting

AJ.tV4(i Ameeting(i)) - J(i)] o xiA.3i[i'c A cry(x)(i)]= Ax.[AJ.Vi[(i c i A meeting(i)) -- J(i)]]([x^ 1.3i'[i'c _ A cry(x)(i')]](x))= Ax.Vi[(i c IAmeeting(i)) -> 3i'[i' c iA cry(x)(i')]]

(56) some executive cried during each meetingAP.3x[exec(x) A P(x)](Ax.Vtl(i C Ameeting(i)) -> 3i'[i' c i A cry(x)(')]])

= 3x[exec(x) AV4(i C IAmeeting(i)) -> 3i[i' c iA cry(x)(i)]]]This derivation entails the existence of one particular executive whocried at each meeting, which is the desired narrow-scope reading ofthe temporal generalized quantifier.



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2.9. Temporal Modification of NounsThe sentence Bill cried after eachmeeting on a Tuesday has a readingwhich is true if there were several Tuesday meetings and Bill criedafter each of them; the crying intervals themselves do not have to beon Tuesdays. This reading arises when the PP on a Tuesday modifiesthe noun meeting rather than the verb cried (Pratt and Francez 2001;Francez and Steedman to appear).

(61) Bill cried after [each [meeting on a Tuesday]]The discussion of the semantics of this structure will lead to a generalconstraint on semantic derivations, which will be important in rulingout certain undesired readings in Section 3.We startwith themodified common noun in (61). A Aioperator isadded to the contextualized meaning of the noun meeting (18)according to the rule (54), and this composes with the temporal generalized quantifier on a Tuesday, yielding the representation in (62).

(62) meeting on a TuesdayLJ.31ii c A tues(i) A J(i)] o Ai'L.i' c i Ameeting(i')- 2i'.3i[i c i A tues(i) A i' c iAmeeting(i')]The temporal property (62) is true of an interval if it isboth ameetingand on a Tuesday. The temporal determiner each applies to the aboverepresentation creating a temporal generalized quantifier (63); this is

modified by the preposition after, and the resulting temporalgeneralized quantifier (64) applies to the sentence Bill cried (65).

(63) each meeting on a Tuesday)J.Vi'[3i[i c i A tues(i) A i'c iAmeeting(i')] - J(i')]

(64) after each meeting on a Tuesday,J.Vpi'[3i c A tues(i) A i'c iAmeeting(i')] -> J(aftert(i'))]

(65) Bill cried after each meeting on a Tuesday --Vi'[3Ii[iC A tues(i) A i'c i Ameeting(i')] -3i" [i"' after (i') A cry(bill)(i")]]

The final translation is indeed the desired reading: each intervalwhichis a meeting on a Tuesday is followed by an interval of Bill crying.The contextual parameter of after isthe temporal context variable i



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which stands for the overall context, so the crying intervals are notnecessarily on any Tuesday.

Adjunction of other temporal modifiers to common nouns isproblematic. The following examples show what happens when thecommon noun meeting ismodified by the PPs on each Tuesday and onno Tuesday.(66) meeting on each Tuesday

,J.Vi[(i c tA tues(i)) - J(i)] o i'Al.i'. iAmeeting(i')- i'.Vti(ic i A tues(i)) -- i'c i Ameeting(i')](67) meeting on no TuesdayUJ.-31ii c i A tues(i) A J(i)] o ri'A.i' c i Ameeting(i')= Ai'.-3i[i c tA tues(i) A i'c i Ameeting(i')]

The temporal property (66) is true of an interval if it is ameeting andis included in each Tuesday in the context; the temporal property (67)is even more bizarre - it is true of any interval which isnot ameetingon a Tuesday Both properties lead to incorrect truth conditionswhen they combine with a determiner and a temporal preposition tomodify a sentence.

(68) Bill cried after ameeting on each Tuesday3i'[Vtf(i c i A tues(i)) - i'c iAmeeting(i')] A3i"[i"c after#(i') cry(bill)(i")]]

(69) Bill cried after ameeting on no Tuesday3-i'[-3i[i c A tues() A i'c iAmeeting(i')] A3i"[i"c afteri(i')A cry(bill)(i")]]

The formula in (68) cannot be true if the context i includes two ormore non-overlapping Tuesdays, which may ormay not be desirable.If the context contains just one Tuesday the formula can be true, and ifthere are no Tuesdays in the context then the condition (66) is triviallysatisfied and the sentence is true as long as Bill cried, even if there areno meetings at all. This is obviously incorrect. As for (69), it is true ifBill cried after an interval which is either not ameeting, or not on a

Tuesday, or not in the context of evaluation. This is sheer nonsense.It is debatable whether the constituents (66) and (67), and consequently the sentences (68) and (69), have any meaning at at all. Iwilltherefore not try to find another way of assigning meanings to these



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structures, but rather be content with blocking the meanings wepresently get. The fault with the above representations is that thetranslation of meeting together with its context variable are separatedfrom their determiner by the determiner of the noun Tuesday; anintervening determiner also exists in (62)-(65), but since thisdeterminer is existential it causes no ill effects. Iwill therefore blockthe unwanted representations by stipulating the following constrainton semantic derivations.

(70) A universal or negative generalized quantifier may notapply to a contextualized temporal property before adeterminer does.

This stipulation is not pretty, and begs for further investigation. Itwould be nicer ifwe could say that the restriction (70) applies to allmodifiers and rule out such structures in the syntax, but this wouldrequire an alternative means to get themeaning (65). I therefore leavethe stipulation as it is.We should also note that temporal modifiers adjoin not only tocommon nouns but also to fullNPs.

(71) Bill cried after [[ameeting] on each Tuesday]In this structure the PP on each Tuesday takes scope above thedeterminer of a meeting, in a similar fashion to what we see in asentence like Two volunteers from each class decorated the room. Theresulting reading is that on each Tuesday there is a (separate) meeting, after which Bill cried, with the crying not necessarily on Tuesday.This comes through straightforward application of our semantics; Iwill not give the full derivation here, since structures like thiswill notplay a role in our subsequent discussion.

2.10. SummaryThe basic theory of temporal modification involves the followingoperations.Contextualization (12a) applies to all temporal nouns, and all verbs.Existential closure (13a) applies at the top of the clause to all temporally modified clauses (as well as other clauses that are nottemporal modifiers). It always applies to contextualized properties.



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An implicit temporaldeterminer (27) applies at the top of the clause to alltemporalmodifier clauses, and to temporal common nouns that lack anexplicitdeterminer. talwaysappliestocontextualized roperties.Temporal modification (54) involves abstraction over the temporalcontext variable of the temporally modified constituent.Quantificational arguments and temporal modifiers (nominal andtemporal generalized quantifiers) may take scope over the existentialclosure operation (13a); this has the effect of applying existentialclosure at a lower level, close to the verb. However, a universal ornegative generalized quantifier may not intervene between a contextualized temporal property and its determiner (70).

3. TEMPORAL CLAUSES AND QUANTIFIER SCOPE

A temporal clause may have a quantificational subject, as in sentence(72), repeated from Section 1.

(72) A secretary cried after each executive resigned.Under one reading of the sentence (possibly themost salient one), thesentence is true if each resignation is followed by the crying of apossibly different secretary. The subject of the temporal clause takesscope above the subject of thematrix clause, inwhat appears to be aviolation of a locality constraint: quantifiers do not normally takescope outside of adjunct clauses (cf. Huang 1982). This is one of afamily of readings which Iwill call dependent-time readings, where thetimes at which the matrix clause is evaluated depend on quantifierswithin the temporal clause. Iwill show that dependent-time readingsdo not violate the adjunct island constraint: they arise from localscope relations within the adjunct clause, where the quantificationalargument takes scope above the implicit temporal determiner. (Issuespertaining to the relative scope of temporal determiners and quantificational arguments were noted in Pratt and Francez (1997, pp.9-11) but not applied to temporal clauses.)

Dependent-time readings will be discussed in Section 3.2 below,following the discussion of the simpler single-time readings inSection3.1. Section 3.3 looks at restrictions on the readings produced in thepreceding sections, and Section 3.4 explores the complicated cases oftemporal clauses that have internal temporal modifiers.



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 5693.1. Single-Time ReadingsThe single-time reading of (72) entails that all the executives resignedat the same time. This reading obtains when contextualization andthe implicit temporal determiner apply to the entire temporal clause,above its subject. Before contextualization, the quantificational subject each executive has to apply to the verb resigned. This results in atypemismatch, because the verb (73) is of type eit while the subject(74) expects an argument of type et.

(73) resigned AxAii.resign(x)(i)(74) each executive - AP.Vx[exec(x) -> P(x)]

Pratt and Francez (2001) solve themismatch through an operation of"pseudo-application", which makes the temporal lambda term of theverb invisible to the subject. Iwill use amore traditional approach oftype shifting.We can either shift the type of the verb according to thescheme in (75) or shift the type of the subject according to the schemein (76).

(75) Shifting the type of the verb: A relation R of type aztapplies to a generalized quantifier of type (at)t by raising totype((at)t)zt:

R6' | fT((t)'Ut.T(Asa. R(s)(u))(76) Shifting the type of the subject: A generalized quantifier Tof type (at)t applies to a relation of type azt by raising totype (azt)zt:T(t')'t ; Ra`tAtu`.Tf(AS.R(s)(u))

These type-shifting schemes are nothing more than temporal versionsof the ones needed in extensional, non-temporal semantics for combining a transitive verb of type eetwith a quantificational direct objectof type (et)t, allowing the object generalized quantifier to skip over thesubject slot of the verb. Both schemes are used in the literature forverb-object constructions: the standard approach to transitive verbsinMontague Grammar is that they accommodate generalized quantifier objects (Montague 1973), and suchmeanings are considered byPartee and Rooth (1983) to be the result of a type-shifting scheme like



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(75); a raised version of the generalized quantifier, corresponding tothescheme(76), isproposed inSteedman(2004).Whichever type-shiftingchemewe choose, thecombination f thesubject and the verb results in the following temporal property as theraw sentence meaning.(77) each executive resigned A.i.Vx[exec(x) -> resign(x)(i)]

Contextualization (12a) and a determiner meaning (27) turn theabove meaning into a temporal generalized quantifier (78), and arefollowed by the application of the preposition after.

(78) each executiveresigned;J.3i[i c i A Vx[exec(x) -> resign(x)(i)] A J(i)]

(79) after each executive resigned;J.3t1ic iAVx[exec(x)-> resign(x)(i)]A J(afteri(i))]Finally, the temporal clause applies to the Li-abstract over thematrixclause, whose meaning is derived by applying the generalized quantifier a secretary to the contextualized and existentially closedmeaning of cry (50).

(80) a secretary cried ]3x[secretary(x) A 3i[i c i A cry(x)(i)]](81) a secretary cried after each executive resigned -

;J.3i[i c i A Vx[exec(x) -> resign(x)(i)] A J(after(i))](Ai.3y[secretary(y)3[i'[i i A cry(y)(i')]])= 3i[i c iA Vx[exec(x) -> resign(x)(i)] A3y[secretary(y) A 3i'[i c afteri(i) A cry(y)(i')]]]

The result is indeed a single-time reading - there is a time interval atwhich all the executives resigned, and a secretary cried after that.

3.2. Dependent-Time ReadingsIf the implicit existential determiner applies to the verb of (72) beforethe subject does, we get a reading where resignation times vary withthe resigning executives. Contextualization (12a) and a determiner

meaning (27) can apply to the the raw verb by using a scopemechanism on the subject, as discussed in Section 2.7.1.



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QUANTIFICATIONAL ARGUMENTS INTEMPORAL ADJUNCT CLAUSES 571

(82) resigned -* lxLJ.31i c A resign(x)(i) A J(i)]The quantificational subject applies using one of the type-shiftingschemes (75) or (76); the resulting temporal generalized quantifier ismodified by the preposition after.

(83) each executiveresignediJ.Vx[exec(x) ->31li c i A resign(x)(i) A J(i)]]

(84) after each executive resignedA/.Vx[exec(x) -->3i[i c IA resign(x)(i) A J(afteri(i))]]

At this point we can already see how the quantificational relations willend up. The universal quantifier contributed by the subject NP outscopes the temporal existential determiner; since the temporal propertyvariable J lieswithin the scope of the temporal existential determiner, italso lieswithin the scope of the subject. The variable J will get replacedwith themeaning of thematrix clause, so thematrix clause itselfwill fallunder the scope of the subject of the temporal clause.The temporal clause can take scope over the entire main clause(Section 2.7.1). In this case the temporal generalized quantifier (84)applies to the Al-abstract over the matrix clause meaning (80).

(85) a secretary cried after each executive resignedAJ.Vx[exec(x) -> 3ili c i A resign(x)(i) A J(afteri(i))]]

(;L.3y[secretary(y) A 3i'[i' c A cry(y)(i')]])= Vx[exec(x) -> 3i[i c i A resign(x)(i) A3y[secretary(y) A 3i'[i' afterr(i) A cry(y)(i')]]]]

We get a representation of the desired meaning - each executiveresigned, and following that resignation a secretary cried; the secretariesmay vary with the executives. The end result is that the subjectof the temporal adjunct clause has scope over the subject of themainclause, but this is achieved through local operations: within thetemporal clause, the subject takes scope over the temporal determiner; additionally, the entire temporal clause takes scope over thematrix subject. Thus, the representation (85) is not in violation oflocality constraints.The temporal clause may also modify thematrix predicate ratherthan the whole matrix clause. In this case the matrix subject takesscope above the temporal clause (Section 2.7.2).



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(86) a secretary cried after each executive resigned= 3y[secretary(y) A Vx[exec(x) ->3t1ic tA resign(x)(i) A 3i'[i' c afters(i) A cry(y)(')]]]]This reading entails that therewas one secretary who cried, but foreach executive this secretary cried at a different time, namely after theexecutive'sesignation. eadings(85)and (86)arebothdependent-timereadings, because the times at which thematrix predicate is evaluateddepend on the subject of the temporal clause.Arnim von Stechow has told me that dependent-time readings arenot possible inGerman, and this judgment has been confirmed byother speakers. In terms of our theory, thismeans that German doesnot allow a quantificational argument to take scope over the temporaldeterminer. Martina Wiltschko has noted an interesting property inthis regard: the German complementizer nachdem 'after' is composedof themorphemes nach 'after' (the preposition) and dem 'the'; itmaybe, then, that the temporal determiner is not implicit inGerman, butmorphologically realized on the complementizer. Morphology cannotbe thewhole story, because theGerman complementizer bevor 'before'does not have amorphologically identifiable determiner; but it too isdistinct from the preposition vor 'before'.Recall from Section 2.4 thatan important motivation in keeping the temporal determiner meaningseparate from the preposition was the desire to keep the semantics ofbefore and after the same whether they take NPs or sentences ascomplements. Since German complementizers are morphologicallydistinct from the corresponding prepositions, itmay be that theyembody the determiner meaning as well; this will explain why quantificational arguments in German do not take scope over thedeterminer of a temporal clause, and hence why German lacksdependent-time readings.

3.3. Restrictions on Scope Taking3.3.1. Downward-monotone quantifiersThe difference between single-time and dependent-time readings liesin the relative scope of the implicit temporal determiner and thegeneralized quantifier within the temporal clause. Thus, both singletime and dependent-time readings are predicted when the subject ofthe temporal clause is a negative generalized quantifier like noexecutive.



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(87) Bill cried after no executive resigned.a. Single time: 31ii c i A --3x[exec(x) A resign(x)(i)] A3i'[ c after(i) A cry(bill)(i')]]

b. Dependent time: --3x[exec(x) A 314ic i A resign(x)(i) A3i[i' c afteri(i) A cry(bill)(')]]]

The single-time reading (87a) states that Bill cried after some point intime during which no executive resigned, while the dependent-timereading (87b) states that there is no executive whose resignation wasfollowed by Bill's crying. These readings are logically independent: ifBill never cried, the former is false and the latter is true; if Bill criedtwice, once after an executive resigned and again after a time inwhichno executive resigned, then the former is true and the latter is false.

Dependent-time readings that depend on the quantifier no are verydifficult: many speakers refuse to accept (87b) as a reading of (87),and those who do accept it are only inclined to do so when there is astrong supporting context, for instance when no contrasts withanother quantifier.

(88) As opposed to Mary, who cried after every executiveresigned, Bill cried after no executive resigned.

The difficulty with (87b) isnot with themeaning itself,which isperfectlycoherent; nor is there a general problem with temporal dependency onnegative quantifiers - sentence (89) has the expected meaning.

(89) Bill cried after no resignation.-3i[i c i A resignation(i) A 3i[i c after((i) A cry(bill)(i)]]

The difficulty with the dependent-time reading in (87)must thereforeoriginate in the temporal clause itself: it is difficult for the negativegeneralized quantifier to take scope above the temporal determiner.

Negative inversion can serve as a test for whether thematrix clauseisunder the scope of a negative quantifier. When a negative adverbialis preposed, an auxiliary is added to themain clause.(90) After no resignation did Bill cry. (meaning: (89))

A preposed temporal clause with a negative quantifier but withoutnegative inversion only has a single-time reading (91). This iswhat weexpect, because single-time readings ensue when the quantificational



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argument of the temporal clause does not take scope over the mainclause.(91) After no executiveresigned,Bill cried. (meaning: 87a))

With negative inversion we would expect a dependent-time reading,because these are the readings in which the negative generalizedquantifier takes scope over the main clause. But (92) has the samestatus as (87b) - unacceptable tomany, difficult for some.

(92) ?After no executive resigned did Bill cry.(meaning 87b)difficult)Other downward monotone generalized quantifier arguments alsoresult inmarginally acceptable dependent-time readings.

(93) Bill cried after few executives resigned.?"there are few executives such that Bill cried after theyresigned"; cf. Bill cried after few resignations.(94) Bill cried after at most three executives resigned.?"there are at most three executives such that Bill cried after

they resigned"; cf. Bill cried after at most three resignations.Since dependent-time readings come from a wide-scope interpretation of a quantificational argument relative to the implicit temporaldeterminer, we can state this observation as a constraint onsemantic derivations: downward monotone quantifiers generally donot take scope over quantifiers that originate in higher syntacticpositions. This fits with an observation by Liu (1990), reported bySzabolcsi and Zwarts (1993, p. 237), that downward monotoneobject generalized quantifiers generally do not take scope over thesubject.

We should also note that the restriction (70) from Section 2.9prohibits negative generalized quantifiers from taking scope betweenthe contextualization operation and the implicit temporal determiner.The restriction is needed for the same reasons that led to its originalformulation: such a quantifier would lock the free temporal contextvariable under a negation operator, resulting in a nonsensical reading.



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(95) Bill cried after no executive resigned*31--3x[exec(x) A iC i A resign(x)(i)] A3i'[i'c after'(i)A cry(bill)(i')]]

The representation in (95) is true if Bill cried after any interval whichis outside the context of evaluation; our restriction (70) successfullyrules out this derivation.3.3.2. Surface order and scopeThere are also restrictions on the relative scope of the temporal clauseand quantifiers in thematrix clause. The semantics allows the variousscope-taking elements to combine in any order, but not all orders areequally acceptable. As with other constructions that involve scopetaking elements, surface order is often preferred (for recent experimental evidence see Anderson 2004). Thus, some of the speakers Ihave consulted dislike the wide-scope reading of each executive in(72). For these speakers it has about the same status as the widescope reading of every teacher in the sentence a student likes everyteacher: dispreferred, but possible. A dependent-time reading ismuchmore natural when the subject a secretary is replaced with a differentsecretary or the same secretary, which forces the matrix subject totake low scope in order to allow a sentence-internal interpretation ofdifferent and same (Carlson 1987; Barker 2004).

(96) A different secretary cried after each executive resigned.(97) The same secretary cried after each executive resigned.

Also, when context or world knowledge make the single-time readingunlikely, a dependent-time reading with wide scope for the embeddedquantifier is very easily accessible.

(98) A dove is released after each dignitary concludes her speech.The reading of (98)which ismost compatible with our knowledge ofthe real world is that the dignitaries conclude their speeches at different times, and after each speech a different dove is released.When the temporal adjunct ispreposed it tends to takewide scope,and so do quantifiers within the temporal adjunct. Thus, on the dependent-time reading of sentence (99), it isnaturally understood thatthe secretaries may vary with the executives.



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578 RON ARTSTEIN

existentially closed sentence. I call this a temporally dependentreading, and these are discussed Section 3.4.2.3.4.1. Aggregate readingsSentence (104), repeated below, is true if Bill resigned after a certainperiod, inwhich John disappeared every Friday.

(104) Bill resigned after John disappeared every Friday.We want to subordinate the temporal context of themain clause to thecontext of the temporal clause. We start with the familiar clause-typemeaning for the temporal clause, derived by applying the temporalgeneralized quantifier every Friday (106) to the contextualized andexistentially closed meaning of the sentence John disappeared (107).

(106) every Friday --* {JVi[(i iA friday(i)) - J(i)](107) Johndisappeared 3i'[i'c iA disappear(john)(i')](108) John disappeared every Friday

VII(i 5 IA friday(i)) -- 3i'[i' c iA disappear(john)(0')]]The next step is an "aggregating" operation of abstraction over thetemporal context variable of the temporal clause. This gives a temporal property - the set of intervals which can serve as a context forthe temporal clause.

(109) John disappeared every Friday;i.V4t(i c i A friday(i)) -> 3i'[' C iA disappear(john)(i')]]

This property is contextualized and an existential determiner is applied (110), followed by the preposition after (111). The resultingtemporal generalized quantifier modifies the main clause (112).

(110) John disappeared every Friday -;J.31li c_ A Vi'[(i' c iA friday(')) ->3i"[i" i'A disappear(john)(i")]]A J (i)]

(111) after John disappeared every FridayAJ.3]ii c iA Vi'[(i' c iA friday(i')) ->3i"[i" i'A disappear(john)(i")]]AJ (afteri(i))]



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(112) Bill resigned after John disappeared every Friday3i4i c A Vi'[(i' c iA friday(i')) -3i"[i" i'A disappear(john)(i")]]A 3i"[i"' C after(i) A resign(bill)(i"')]]The representation of the sentence has the desired truth conditions:there exists an interval which serves as a context in which Johndisappeared every Friday, and after this interval, Bill resigned.3.4.2. Temporally ependent eadingsSentence (105), repeated below, is true if Bill's dad came home everyevening and Bill did his homework every day before his dad's arrival,though the homework does not necessarily have to be done in theevening.

(105) Bill did his homework before daddy came home everyevening.

Neither the aggregate reading nor the cascaded reading have thedesired interpretation: the aggregate reading (113) says that Bill didhis homework once, before a period inwhich daddy came home everyevening; the cascaded reading (114) entails that Bill did his homeworkevery evening (notice the parameter i on the temporal functionbefore). For simplicity, I assume theVPs did his homework and camehome translate as predicate constants (type eit), and that the worddaddy is a proper name (type e).

(113) Aggregate: 31iic iA Vi'[(i' c iA evening(i')) -3i"i" c i'A come-home(dad)(')]]A 3i"'[i"' c befores(i) A do-hw(bill)(i"')]](114) Cascaded: Vi[(i c IA evening(i)) -

3i'[i' c iA come-home(dad)(i') A3i"[i" before{i')A do-hw(bill)(i")]]]While (113) and (114) are both valid readings of the sentence, themost salient reading is the one described at the beginning of thispassage, where the temporal generalized quantifier every evening isinside the temporal clause and the evaluation times of the matrixpredicate depend on it. The obvious way to derive this temporallydependent reading is to allow temporal generalized quantifiers to



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QUANTIFICATIONAL ARGUMENTS IN TEMPORAL ADJUNCT CLAUSES 581characterization of the temporally modified clauses which allowdependent readings will have to wait for a further study. In theremainder of the section Iwill survey the data in order to reach somepreliminary onclusions.We start by noting that there is a clear contrast between sentence(120), with a universally quantified argument, and (121), with auniversally quantified temporal modifier.

(120) John got angry after each executive cried.(121) John got angry after Bill cried during each meeting.

The salient reading of (120) is a dependent reading, where John getsangry after each cry

artstein, ron - 2005 - quantificational arguments in temporal adjunct clauses

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