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<Paper uid="E87-1043">
  <Title>THE EXTENDED FORMALISM Cyclic Iteration</Title>
  <Section position="3" start_page="0" end_page="270" type="metho">
    <SectionTitle>
THE CORE FORMALISM
A Temporal Model
</SectionTitle>
    <Paragraph position="0"> T~e formalist tha: ~ill oe use: here has oee~ de~ineo e~pilcitiy i~, van Eynce, aes TomDe Q Maes ~5). irk th!s p~per i will on!y give a s~or~ In~ormai present~zion of tDe formailS~ CO%Cemtratlrlg On th~se partS ~ICh will De neeOe~ In the se~onO pert,  The model COnSIStS of a set of linearly oroerep irtsrvals.</Paragraph>
    <Paragraph position="1"> An interval ~s a continuous set of time points on the time axis :</Paragraph>
    <Paragraph position="3"> A la limite it might consist of one moment o6</Paragraph>
    <Paragraph position="5"> For an~ pair of ~ntervaie one can Oeflne tnelr interlection as the set of tlme points which the'/ share:</Paragraph>
    <Paragraph position="7"> Tn~s set m~g~t also be empty, as in I J it Is, furthermore, possible to define some b~narv relations between intervais, such ae</Paragraph>
    <Paragraph position="9"> T~ese relations are also useO in Bruce (1972).</Paragraph>
    <Paragraph position="10"> A Format for Representation For the semantic analysis of the temporal expressions i wili start from tne assumption t~at every sentence can be analvseo in two parts : the temporal informal:on expressa~ by the tenses.</Paragraph>
    <Paragraph position="11"> auxiliaries anO ao~erbials on t~e one hano. anp ~as~o atempora! proposition on the ot~er hand.</Paragraph>
    <Paragraph position="12"> (b; the cat sat on t~e mat w~.i. for instance, be analyseo in a basic proposition &amp;quot;the cat sit on the met&amp;quot; and the ~n~ormation conveyed Dv the past tense.</Paragraph>
    <Paragraph position="13"> The relation between both is established in two steps : the basic proposition is first relateO to the interval ~or whicn it is said to be true, the socalled time of event (E), and then this interval is related to the time Of speech ~S) :</Paragraph>
  </Section>
  <Section position="4" start_page="270" end_page="274" type="metho">
    <SectionTitle>
3 E :,E,S) ~ AT(E,the cat sit on the mat)3
</SectionTitle>
    <Paragraph position="0"> This formula states that &amp;quot;the cat sit on the mat&amp;quot; i~ true at an interval E which precedes the tlme of speech S.</Paragraph>
    <Paragraph position="1"> Following Reichenbach (1947) I will furthermore assume that the relation between the time of event and the time of speech is mediated by a thiro kind o~ interval, namely the time of reference (R), So, instead of the simple ReI(E,S) we w111 have a composite ReI~E,R) &amp; RefeR,S).</Paragraph>
    <Paragraph position="2"> Ne.t to thls relational information tn~ tempore: expreeslons can also give specific informatlon about the iocatlon or the length of the reie~ant intervals. This is typically Oone by means o~ t:me aOverbiais, such as &amp;quot;next year&amp;quot;, &amp;quot;in the spring':, &amp;quot;for t~o years&amp;quot;, &amp;quot;till Christmas&amp;quot;, etc. T~is in~ormatlon will be represented bY means o~ one-place preOicates over intervsls : Freo(E) and Pred~), ~n exception ~s tc be ~ade here +or the time o, speec~. ~nose precise location or length is never spec~fleo b, iinQo~stic means, bu\[ rather bv pragmatic factors. A possible way to reelect tn~s-In the &amp;oc~,allsm is to trest it as an unbouno variable.</Paragraph>
    <Paragraph position="3"> In sum, the general format for the representatlon of temporal information looks as  As it stanos this format is not adequate yet fo ~ the ~epresentatlon of sentences like (8~ last year they played chess every week (e~ he was always late The basic propoe~tions &amp;quot;they play chess&amp;quot; and &amp;quot;he oe late&amp;quot; do not hold for one particular time of event E, but rather for a set of intervals wnicn are spread in time in some way specifieo by &amp;quot;every week&amp;quot; in (8) and &amp;quot;always&amp;quot; in (9). In the following part I will introduce an exter.oeO formalism which can OeaI with these typos  &amp;quot;caiiv&amp;quot;, &amp;quot;every Monday&amp;quot;, &amp;quot;each year&amp;quot;, etc. In ~virk e.a. (1972) they are callao periodic frequency adverbials.</Paragraph>
    <Paragraph position="4"> For the analysis of these adverbials I first IntroOuce the notion Crams time. The frame time ie the interval which contains all the instances of the event describeo in the basic proposition. In (8~ last year they played chess every week t~e ~rame time is last year. In the general forma~ t.e frame time occupies the same place as the time c~ event in non-iterative interpretations (= the E-i~tervai~, ~ext, I de~ine a set of distinct, non-overlapping subintervals ~I~ which are all part o+ the frame time. In (8~, these intervals have a length of one week each. This gives the following</Paragraph>
    <Paragraph position="6"> similar analysis can be found in Stump (198i where t~e aoverbial frequency ad~ectlvee (P) ere given the following truth condition : F~' is true in a world w at an interval I i~4 ,~m is true in w at non-overlapping subintervals o$ i distriOuteO throughout i ~t perioOs of a speci$ieo length I. &amp;quot; \[Stump 1981, 226\] 5t~mp s i-interval corresponds to my frame time. and his non-overlapplng subintervals correspond to my I-intervals.</Paragraph>
    <Paragraph position="7"> As a representation of (B) this formula is not sufficient, though, since the instances of chess pla~ing do not have to take a whole week for (B~ to. be true. A more adequate paraphrase is to say that every week contained at least one subinterval (e~ during which they played chess :</Paragraph>
    <Paragraph position="9"> An argument in favor of this refinement is that languages have special means for specifying the etimes. In ~I(' last year she arrived at ~ c clock every da~ the aoverbia2 &amp;quot;at eight o C/ioc~&amp;quot; denotes the locatlOn 04 t~e e-intervai ; B Notice tha~ the pro~artlee of e are constant within 'the 4tame time : the aoverDial &amp;quot;st eight o clot!&amp;quot; specities t~e time of each oC/ her arrlvals cf last year.</Paragraph>
    <Paragraph position="10"> The general format for the representation of</Paragraph>
    <Paragraph position="12"> where P is replacec ov the head o4 a periooic ~requencv aoverbial, specifying the location or the iengtn o~ I  R is oef~neo with respect to S : ReI~R,S~ E as defined with respect to R : ReI(E,R~ I is defineo w~th respect to E : ~(I,E) and e is oefineo with respect to I : c(e.I~ As it stands, the format does not provioe any means for stating a direct relationship between the intervals inside the frame time ~I and e~ ano the intervals outside the frame time (S anO R~. As consequence, the formal~sm predicts that temporal adverbials w~ich are in the scope o~ a frequency adverbial (: the e-specifiers~ cannot refer ba~K to the speech t~me or the eeference time: * Rei(e,S) and * Rel(e,R~, gooo p;ece of evidence for this hypothesis ~s pr~ioed by the WHEN-aoveroiais. In general one can distinguish two kinde of those adverbials : t~e relational ones, which express a relation Oetween the reference time and the speech time, such as &amp;quot;~esterday&amp;quot; a'nd &amp;quot;tomorrow&amp;quot;, and the non-relational ones, which identify the location o~ an :nterval without any reference to the speech t~me, suc~ as &amp;quot;between 8 and 9&amp;quot; and &amp;quot;at two o clock&amp;quot;. The interesting thing now is t~at only the latter adverbials can occur in the scope of a frequency adverbial. Compare :iI~ she arrived every day between 8 anq 9 e *(12~ she arrived every day yesterday e The fact that the relational WHEN-adverbials cannot occur in the scope of a frequency aoverb~al prcviOes some positive evioence ~or not inciuolnSS direct relations between e ano S in the formal~em. The chaln-like structure of the representation format Is, hence, i~nguistically motivated.</Paragraph>
    <Section position="1" start_page="272" end_page="274" type="sub_section">
      <SectionTitle>
Temporal Quantifiers
</SectionTitle>
      <Paragraph position="0"> The format Oeveloped for the analysis of cvclic iteratlon can also be useo for the analysis o~ the temporal ~uantifier$, such as &amp;quot;miway~&amp;quot;, &amp;quot;scmetlmes&amp;quot;, &amp;quot;never&amp;quot;, &amp;quot;seldom&amp;quot; ano &amp;quot;often&amp;quot;. The ~rmetion they proviOe is less specific than the ona p~ovioed by the period frequency aOverb~ais, ar, d t~s should be refiecteO in their representation.</Paragraph>
      <Paragraph position="1"> As a starting point I take the general ~ormat ~or the representation o~ sentences w~th a  .=,is sixfold dzvis:on is taken beer from Lewis ~1975).</Paragraph>
      <Paragraph position="2"> This analysis account~ for the anomaly of sentences like o ,13} we sometimes played chess every wee~  These sentences are eemantlceiiy anomalous oecauee t~e sa~e ~ino o* In*ormation. namely the v~iue o~ ~. is epec~lec twice. This leaps to :~cons~etenc~ ~ ~13) and (14} where the Qve~ia~ie IB s~l~ to be both universal anO non~r;vers~i at tme same time, and it leaos to pleonasm in (15~ where the Q-variable is twice sago to Oe u~,iversal.</Paragraph>
      <Paragraph position="3"> The ne, t question is whethe,&amp;quot; thP temporal quant!~iers introduce any extra-conqitions on those Intervals, ouch ms c~l,E), ~I=~ and P~i~. The f~rst of t~ese conditions appears to Pe relevant : the temporal quantifiers are ~ndeeo interpreteO wi~ respect to some given frame time. In ~x he was al~ays late &amp;quot;always&amp;quot; ooesnot oenote AL~ possible intervals. but onl~ all possibie intervals ~n the past. The conoit~on that the subintervals may no~ overlap does not seem to be relevant, though, in (16, quaOratlc equations are aIweye s~mple  the Instances for whlon &amp;quot;quadratic equations Pe ~imple&amp;quot; are true are no~ temporally ordereo at all. it, is m~gnt indicate, Ov the way, that the iobjects ~re not necessarily intervals, but rather cases or occasions wnlcn can but need no: be given m temporal interpretation (of. Lewis 1975i.</Paragraph>
      <Paragraph position="4"> The third conOition concerns the properties of t~e I-objects. In the case of the periodic * ,equency aOverblals the relevant properties concern the location or the length of the interval. In the case of the temporal guantiflers one could think of specifying a relevance conoiticn~ for a sentence like ~ he was always late ones not mea= that he was late at any possible occasion in the past, Put rather that he was late on all occasions on which his being late or timel~ could nave mattered.</Paragraph>
      <Paragraph position="5"> in Aqv~st, Hoepelman &amp; Rohrer (1980) one can ~ind a proposal to incorporate this information in the semantic representation, but I will not adopt t~is proposal here, since the conditions o~ the ,non)relevance of the occasions are typicaliv determined O~ pragmatic factors, in ~:&amp;quot; he always leaves o~-~ twelve the relevant occasions (1) could just as well oe all occasions on which he leaves as all occasions on Wnlch ne leaves for work as a!i occasions on ~hish he leaves for watching the home game of nls ~avourlte footOaii team.</Paragraph>
      <Paragraph position="6"> As a result of the foregoing reductions ar~o changes the general format for analysing tempo, al cuantifiers looks as follows :</Paragraph>
      <Paragraph position="8"> ,here O is replaced by any of {V, 3, &amp;quot;3, Most, Few, Many} M is replaced by some time adverbial which specifies the location or the length of e (if there is anv~ Habituality The sentences oiscusse~ so far all contain an explicit indication of iteration. !he presence of SL~Ch an IndlCatlOn Is, however, not necessary for deriving an iterative interpretation. Take, /or instance, (in~ he leaves at twel~e This sentence cannot only mean tnat he will leave at twelve, but also that he has the habit of leaving ~-* twelve.</Paragraph>
      <Paragraph position="9"> in the representation of interpretation the time adverbial specifiss the t~me of reference :  in the representation of tne habitual i~terpretation~ on tne other hand, tne time adverolal shouls be tal~en to specify the multiple e-tlme, for the sentence Ooes no~ report on one o~ his ieavzngs at twelve, out rather on several of socn :ea,es. As a representation of this interpretation I propose :</Paragraph>
      <Paragraph position="11"> (19~ he leaves at twelve is t-eaten as synonymous with (20, he usually leaves at twelve If this is felt to be undesirable, one cam introGuce a special quantifier for marking habituaiitv, but at this moment ~ do not see an~ reason for SUCh a move.</Paragraph>
      <Paragraph position="12">  The general format for the representation of  The assl~nment of these meanings to particuiar sentences is fairly straightforward when the sentence contains a frequency adverOial or a temporal quantifier, but if there is none o~ those~ then the sentence is amOiguous Oetween a non-lterative and an habitual interpretation ~cf. the two interpretations of &amp;quot;he leaves at tweive&amp;quot;~. It, practice there are some oisambiguatlng ~. I* the basic proposition (p) denotes a state, ~r. er, the sentence can not have an habitual ir~erpreta~ior~ Compare :i;~ ne leaves at twelve ,21 ne is in jei! ~1~ can be interpreted as meaning that he has the naPlt of leaving at twelve, bu~ (21i cannot Oe interpreted ms meaning that he has the habit of bel=g in jail.</Paragraph>
      <Paragraph position="13"> ~, Certain verb forms can biock the Oerivation o~ one of t~s two possiole interpretations. Compare ~2~ he is drinking coffee 12\]) he drinks coffee (22, can Oenote a single instance of drinking as wei&amp;quot; as a recent habit of him to drink:: coffee ~cf. in the sense of &amp;quot;he is. drinklng coffee nowadays&amp;quot;). (2;,, on the other hand, can only denote a habit; it cmnnot be used to report on a single instance o~ drinking.</Paragraph>
      <Paragraph position="14"> This demonstrates the need to distingulsn oi4ferent types of verb forms : the ones that will aiways elicit an habitual interpretation, the ones that block the derivation o~ an habitual interpretation, and the ones that admit both kinds of interpretmtions. The firs~ are unequivocall~ \[+habitual\], the second C-habitual\[ and the last will be given the feature \[+/-habitual\].</Paragraph>
    </Section>
  </Section>
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