tichý - do we need interval semantics?

PAVEL TICH~(

D O W E N E E D I N T E R V A L S E M A N T I C S ?

Many sentences - as, for example, ' John played the piano for an hour yesterday ' - involve explicit reference to intervals of time. Surely every-

body would agree that they cannot be properly analyzed without invoking intervals; everybddy, therefore, is an interval semanticist in this sense.

However , the term ' interval semantics ' has recently been used in a

narrower sense, to denote a certain modification of what is known as

possible-world semantics. It is this modification that this article is concer- ned with.

1. INTERVAL SEMANTICS

As well known, the basic idea of possible-world semantics is that the actual world is one of many possible worlds, and that propositions, properties,

relations, individual concepts, and the like can be explicated as set- theoretical entities over a logical space consisting of couples of the form

(W, T}, where W is a possible world and T a time. A proposit ion is

explicated as the class of those world/t ime couples at which the prop-

osition is true, a proper ty as a function associating each world/t ime couple

with the class of individuals instantiating the proper ty at that couple, and

an individual concept as a function associating with each world/t ime couple the individual which embodies, or is picked out by, the concept at

that couple. The concept of Miss America , to use an illustration given in

Dowty, Wall, and Peters,

is that function which for each index (i.e. each world/time couple) gives the person picked out by the name ('Miss America') at that world and time. (1981, p. 176)

The proponents of interval semantics argue that this approach is in need of

reform. On their view, the logical space of world/instant couples is too narrow and must be supplanted by a logical space of couples of the form

(W, I) , where W is a world and I is an (uninterrupted) interval of time.

The idea is usually credited to Bennet t and Partee, who, according to Dowty

made the fundamental revision of taking truth of an atomic sentence at an interval as basic. That is. . . an index is taken to be an ordered pair consisting of a possible world and an interval, and an interpretation function [to] assign to each constant a function from the set of all indices to an appropriate extension. (1979, p. 138)

Dowty accepts the proposal in his book. K a m p reports that

Linguistics and Philosophy 8 (1985) 263-282. 0165-0157/85.10 O 1985 by D. Reidel Publishing Company

264 PAVEL TICH~(

[a semantic description] which has received much attention in recent work on tense and aspect, relates to the temporal elements with respect to which such semantic notions as truth and satisfaction are (recursively) defined. For Cocchiarella these elements are simply instants; but in subsequent work it has been claimed that such semantic concepts must be analysed as relations between expressions and intervals rather than between expressions and moments of time. (1979, p. 392)

Cresswell states that

until very recently a temporal index [at which a sentence is assessed for truth and falsity] would have been thought of as a single moment or instant of time, but certain authors have considered that the appropriate index is rather a time interval... This paper accepts the idea. (1977, p. 7)

Barry Richards makes the same assumption:

a proposition.., is a function from worlds and intervals to truth-values. (1982, p. 68)

This switch from instants to intervals may seem a minor alteration at first

sight; but a little reflection reveals that it necessitates a radical revision of

some of the most basic semantic intuitions.

To begin with, the intervalist proposal requires a radical revision of our

intuitive notion of proposition. As ordinarily understood, a proposition is

the sort of thing that one can assert. Moreover, once a proposition is

asserted, it is a matter of objective fact whether the assertion is correct.

For example, when someone asserts that

(1) It is 30 °C

it is in principle possible to objectively determine whether he is right,

because at the moment of assertion, it either is 30 °C or it is not. This

simple account of asserting is neatly explicated in ordinary possible-world

semantics, where a proposition is thought of as a class of world/instant

pairs. Every act of assertion determines a unique world/instant pair,

namely the one which consists of the world ~which happens to be actualized

and of the instant which happens to be current when the assertion is made.

The assertion is then correct just in case this distinguished pair is a

member of the proposition.

Nothing so simple can be said on the intervalist approach, which takes a proposition to be a class of world/interval pairs. One cannot say that an

assertion of such a proposition is correct just in case the couple consisting

of the world which is actualized and of the interval which is current when

the assertion is made is a member of the proposition. For there is no such

thing as the interval current when the assertion is made. Any assertion is

made in the middle of no end of current intervals. It is unclear which of them is relevant in evaluating the assertion.

To illustrate, consider the class of couples of the form (W, I), where W

D O W E N E E D I N T E R V A L S E M A N T I C S ? 265

is any world and I any interval of exactly 30 minutes' duration. This, according to the intervalist, is a perfectly good proposition, one which is true in any world 'at' any half-hour interval. Now suppose that someone tries to assert that proposition, perhaps by uttering

(2) It is half-an-hour,

or in any other way. 1 How do we decide whether he is right? What objective condition must be satisfied in order for the assertion to be correct? There is obviously no such condition, hence the speaker did not state anything. The class of world/interval pairs in question thus does not represent a proposition in the ordinary sense of the term. And this is why (2), as distinct from (1), is a piece of unintelligible nonsense.

Interval semanticists themselves never discuss expressions like (2), despite the fact that 'sentences' of that sort illustrate the essence of their proposal in a pure form. But the comments just made about (2) apply with equal force to 'sentences' which they do discuss; for example

(3) John is in Boston twice

(Richards, 1982, p. 98) and

(4) John sleeps for an hour

(Dowty, 1979, p. 333). (3) is supposed to express the class of world/interval couples ( W, I) such that John is in Boston twice in W 'at' I. But when (3) is asserted, which world/interval couple must be in that class in order for the assertion to be acceptable? (3) is clearly no more intelligible than (2): when someone tells us that John is in Boston, it is absurd to ask 'How many times?'; and an answer to an absurd question is itself absurd. Similarly, (4) is supposed to express the class of world/interval couples (W, I) such that I is a one-hour interval 'at' which John sleeps in W. But which interval must John sleep 'at' in order for a given assertion of (4) to be true?

Some intervalists (for example, Richards) try to resolve the difficulty by imagining that the maker of an assertion somehow deictically selects a definite current interval of time with respect to which he intends the assertion to be evaluated. This, however, is implausible on at least two counts. For one thing, it is difficult to see how this interval selection can possibly be communicated. (Note that the sentence 'John is in Boston twice today' cannot express the same intervalist proposition as (3) does, for it is presumably true or false, as the case may be, at any of today's one-instant intervals.) Besides, the interval would often have to be selected with an unrealistically high degree of accuracy. Suppose, for

266 PAVEL TICH'Y

example, that John's nap is exactly one hour long and someone asserts (4) in the middle of it. Then on Richards's theory, the statement will be false not only if the selected interval is ever so slightly shorter or longer than an hour, but also if it is of the right length but ever so slightly shifted backwards or forwards vis-a-vis the actual duration of the nap.

Other authors (e.g., Kamp, 1979, p. 392) seem to assume that the proper temporal index of evaluation is always the interval whose only element is the instant of assertion. On that assumption, any assertion of (3) or (4) would be invariably incorrect. But then it would seem to follow that any denial of (3) or (4) is invariably correct. This, however, hardly agrees with linguistic practice: it is hardly correct to say 'John does not sleep for an hour' in the middle of John's one-hour n a p )

A natural solution - one which is in full agreement with linguistic practice - would be to declare that neither (3) nor (4) is a well-formed sentence. But interval semanticists are understandably disinclined to take this way out, for it is a hallmark of their approach to write such sentences without attaching the asterisk of impropriety.

But even assuming that the question of the proper evaluation interval of an assertion has been somehow settled, there still remains a problem as to what particular proposition (i.e., which particular set of world-interval couples) an asserted sentence expresses. This is because it is unclear what the extensions of various quite simple lexical items at a world-interval pair are supposed to be.

Consider 'yesterday'. On the world/instant approach there is no problem. The extension of 'yesterday' in world W at moment T is simply the latest interval which is a day and precedes T. But once instants have been traded for intervals, no such simple answer is forthcoming. What is the extension of 'yesterday' 'at' an interval which straddles two days? What is its extension 'at' a year-long interval, say 'at' 1983? Dowty and Richards try to solve the problem by relegating 'yesterday' to a class of expressions whose extension at a world/interval index is not a function of that index at all and is determined in an index-independent way. (It is therefore somewhat confusing that Dowty calls yesterday an indexical constant. This terminological confusion is hardly alleviated by Dowty's treatment of 'now', a constant which one would expect to come from the same box as 'yesterday'. For the extension of 'now' at a world-interval index is, according to Dowty (1977, p. 333), uniquely determined by the index: it is identical with the temporal component of the index. And this, he says, is what makes that constant fully indexical.) It is of the essence of possible- world semantics to represent the meaning of a term as a function from world/time indices to appropriate extensions. When the temporal corn-


ponent of the index is construed as an interval, 'yesterday' can no longer be treated that way.

But let us set 'yesterday' aside and consider 'Miss America'. As mentioned above, on the world/instant approach, the meaning of the term is represented by a function which takes a world and a moment to whoever holds the title in that world at that moment. It is far from clear, on the other hand, which function from world/interval pairs to individuals is supposed to represent the meaning of the term on the intervalist approach. Given that a fresh Miss America is selected every year, what is to be the value of the function at a pair consisting of the actual world and an interval in the middle of which an annual selection is made? Should perhaps the term 'Miss America' be treated the way Richards and Dowty treat 'yesterday', namely as taking semantic value independently of the world/interval index?

Similar comments apply to property words. It is obvious, for example, which function from world/instants to classes of individuals represents the meaning of 'is in Boston' on the punctualist approach. But if the representing function is to be defined at world/intervals, we have a problem whenever the interval in question is of positive duration. Since people constantly cross Boston's city boundary in both directions, it is unclear which individuals should belong to the value of that function 'at' such an interval: those who spend all of it within the boundary, or s o m e of it, or perhaps m o s t of it?

Pending a resolution of all these problems, it remains unclear what is the meaning of a sentence like 'Miss America is in Boston'. The interval semanticist wants to represent its meaning as the class of world/interval pairs at which the sentence is true. But consider any interval which includes either a crowning of Miss America or a crossing by Miss America of Boston's city limits. Is the sentence true or false at that interval? Either option seems at odds with our intuitions. Should we perhaps posit further truth values? Some intervalists (e.g., Kamp, 1979, p. 397) indeed suggest that perhaps we should.

We thus see that the innocent looking step of replacing instants by intervals amounts in fact to a major revision of some basic principles underlying possible-world semantics.Thus the reasons which motivate the step should be very compelling indeed. What are they?

2. MOTIVATION

The following are three representative arguments offered by leading

interval semanticists by way of motivating the new approach.

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Argument 1 (Dowty, 1979, p. 138). Consider

(5) John took an hour to draw a circle.

The truth of this sentence depends on the length and temporal location of intervals which John spends drawing a (full) circle. But such an interval cannot be defined as one consisting of moments at which John draws a (full) circle. Since drawing a circle takes time, there are no such moments, hence no intervals at all would fall under such a definition. Therefore, the sentence 'John draws a circle', embedded in (5), must take truth-values at intervals, not moments.

Argument 2 (Dowty, 1979, p. 139). Consider

(6) John played the piano for an hour.

The truth of this sentence depends on the length and temporal location of intervals which John spends playing the piano. But an interval of this sort cannot be defined as a maximal interval consisting of moments at which it is true to say that John plays the piano. An hour of John's piano-playing can legitimately contain short brow-wiping or nose-blowing breaks, during which John does not play the piano. Such an hour would not fall under that definition. Therefore, the sentence 'John plays the piano', embedded in (6), must take truth-values at intervals, not just moments.

Argument 3 (Kamp, 1979, p. 392-3). Consider

(7) Fritz always writes an article in less than a month.

The truth of this sentence depends on the length of the intervals which Fritz spends writing an article. But an interval of this sort cannot be defined as a maximal interval consisting of moments at which it is true to say that Fritz writes an article. Since Fritz may start on an article before he has finished another, some intervals falling under this definition would be too long. Therefore, the sentence 'Fritz writes an article', embedded in (7), takes truth-values at intervals, not just moments.

Do the premises of Dowty's and Kamp's arguments establish the conclusions?

Each of the examples illustrates the fact that for an interval to have a property is not the same thing as for each instant in the interval to have that property. The fact is obvious, the properties of being one hour long, and having passed, being trivial cases in point: a one-hour interval does not consist of one-hour instants, and an interval which has not passed yet maY nevertheless contain many instants which have already passed. Dowty's and Kamp's arguments simply provide further cases: the property of being taken by John to draw a circle, the property of being devoted by


John to playing the piano, and the property of being spent by Fritz writing a (complete) article. But these observations nowise necessitate the authors' conclusion that the atomic sentences embedded in (5), (6), and (7) have to be evaluated at intervals of time rather than moments of time. All the invoked data can be equally well accounted for by assuming that (5), (6), and (7) involve quantification over time intervals. Instead of constru- ing the predicates 'draw a circle', 'play the piano', and 'write an article' as signifying properties which individuals instantiate at intervals, they can be construed as relations between individuals and intervals. Drawing a circle, for instance, can be construed as a relation R which obtains between an agent and an interval just in case the agent spends no more and no less than that interval drawing a full circle. (5) can then be analyzed as saying that there exists an interval I such that I is past, I is one hour long, and John bears R to I. On this construal, (5) as a whole is true or false, and the embedded (open) subclause 'John bears R to 1' satisfied or counter- satisfied, at moments of time, not intervals.

This, in fact, is the style of analysis proposed by Barry Taylor in Taylor (1977). On his theory, the word 'stab' signifies a relation, Stab, between the stabber, the stabbed and the interval taken up by the stabbing. His paraphrase of 'Brutus is stabbing Caesar', for example, is then

Stab(b, c, now)&(3 t)(nowE3 t&Stab(b, c, t)),

where ' t ' ranges over intervals of time and r- is the subinterval relation. Just like the formula as a whole, its subclause 'Stab(b, c, t)' evaluates at instants, not intervals. (In fact, for any particular value of t, 'Stab(b, c, t)' is an 'eternal' sentence true either at all instants or at none.) Taylor's approach is thus an antithesis of what goes under the name 'interval semantics.' The fac t that some interval semanticists (e.g., Dowty, 1979, p. 166) treat Taylor as an honorary member of the intervalist movement shows that they are not fully aware of the gravity of their own proposal. Taylor is an interval semanticist only in the trivial sense of entertaining intervals at all: his logical space is the conventional one.

However, interval semantics and Taylor's theory do have a common source: the realization, on the part of the proponents, that the extension of a non-stative verb in a world cannot be exhaustively described in the form of a function which associates individuals with classes of instants. Circle- drawing, for example, is not fully described by such a function, for an individual who draws circles the way a chain-smoker Consumes cigarettes would have to be associated with a class of instants forming a long, uninterrupted interval, from which the break-down into one-circle intervals could not be recovered.

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3. I N T E R V A L S A R E N O T E N O U G H

But interval semanticists are mistaken if they think, as they seem to do, that the extension of a non-stative verb can be adequately described by a function which associates individuals with classes of continuous intervals. Cons ider 'draw a circle' again. Is the circle-drawing history of the world captured by associating each individual with the class of minimal uninterrupted intervals during which the individual succeeds in drawing a full circle in that world?

Imagine that John, who has taken up circle-drawing as a hobby, started, at the beginning of 1983, to draw an unusually large circle which he did not complete until the end of the year. The circle-drawing function will record this fact by associating John with a class of intervals one of which is

the whole of 1983. But this uninterrupted year-long interval yields no information concerning the distribution of times which John devoted to the project. Thus the function does not enable us to decide the truth-value of, say,

(8) In 1983 John drew a circle in his spare time

even if we know how John divided his year between work and leisure. Since a circle can be drawn in fits and starts, a function describing the circle-drawing activities in a world should associate individuals with classes of broken as well as unbroken intervals.

But a little reflection reveals that the admission of broken intervals would still not be enough. An individual can display two distinct circle-

drawing behaviours over the very same interval of time, solid or broken. He can draw one circle with his right hand and another circle with his left foot, taking exactly the same time over both. A function which associates each individual with the class of intervals during which the individual draws a circle does not tell us, as regards a given interval o~ this sort, whether it is one during which the individual accomplished one, two, or even more circle-drawing feats. One may know the function and yet be at a loss to determine the truth-value of the sentence

(9) John drew a circle no more than once yesterday.

Examples of this sort suggest that an adequate description of the circle- drawing activities taking place in a world must take the form of a function associating each individual not with the (continuous or broken) time- stretches he spends drawing a circle but with those of his behaviours which amount, in that world, to drawing a circle.

A behaviour is a kind of episode. What is an episode? An episode is best conceived as a series of momentary basic events happening over some


stretch of time. It can be identified with the class of instantaneous stages which make it up, and the stages can in turn be identified with propositions stating their occurrence. Thus, a particular fall of rock X can be identified with the class of true propositions of the form

X is in place P at time T,

for every instant T within the duration of the fall. The individual propositions in the class are rather like individual frames in a film clip, except that film frames are temporally discrete and contain information relating not only to the episode depicted but also to the background against which it unfolds. 3

A circle-drawing behaviour by John is an episode of this sort. It may consist of a series of positions of John's right hand (i.e., of propositions of the form: John's right hand is in such-and-such position at such-and-such time). Another circle-drawing behaviour may consist of a series of positions of John's left foot. These will be two numerically distinct episodes even if they take up exactly the same interval of time.

The inadequacy of the intervalist approach is most strikingly manifested in its inability to deal with truncated behaviours. Suppose that John was struck by lightning in the middle of drawing a circle. Although the circle was never finished, it is nevertheless true to say that

(10) When the lightning struck, John was drawing a circle.

How can one account for this in a theory which represents the extension of 'draw a circle' by associating individuals with time intervals? Since the class of intervals associated with John relative to the actual world will contain only intervals during which John in fact produces a complete circle, no member of that class will correspond to the actual circle-drawing effort thwarted by the lightning. The duration of that effort will, of course, be an initial segment of an interval during which John draws a full circle in some other possible world. But the mere circumstance that John spends an interval drawing a circle in some unactualized possible world has by itself no bearing on what John does during an initial segment of that interval in the actual world. It may have a bearing, however, if the alternative (unactualized) world satisfies some further condition. What condition?

According to Dowty (1977, p. 57) and Dowty (1979, p. 146) it is the condition of being exactly like the actual world up to the time of the interruption. Dowty suggests, in other words, that on the assumptions of the example, (10) is true because there is an unactualized possible world which is exactly like the actual world up to the time of the lightning strike and in which John succeeds in completing the circle. But is there such a

272 P A V E L T I C H Y

world? It seems obvious that in a world which is exactly like the actual world up to the time of the lightning strike, the strike also takes place and aborts John's drawing effort. (There may be possible worlds, of course, in which a lightning strike is powerless to deflect people from finishing their drawings, but such worlds are not governed by causal principles wh ich govern the actual world, and are not, therefore, exactly like the actual world at any time.)

Dowty's condition is thus clearly too strong and needs weakening. The alternative world in which John finishes the actually unfinished circle need not coincide with the actual world, up to the interruption, in all respects. All that need be required is that it contain the episode which constitutes John's actual unfinished effort, say a certain series of positions of John's right hand. Since the irreversible atmospheric process which actually eventuated in the lightning strike is no part of that episode, it need not occur in the alternative world. Now if in the alternative world the episode in question extends into a full-blown circle-drawing behaviour by John, John is correctly said to have been actually drawing a circle when the lightning struck. In other words, (10) is actually true because a piece of behaviour which he actually displayed just prior to the lightning strike, is completable into an (unactualized) circle-drawing episode.

This rather obvious idea cannot, of course, be formally implemented if the extension of 'draw a circle' is represented by associating individuals with classes of intervals rather than classes of episodes. On the interval approach, Dowty's unsatisfactory solution seems the only available one.

But granted that, as the foregoing considerations seem to show, the formal explication of verbs must proceed in terms of episodes rather than intervals, the question still remains exactly what role should be assigned to episodes in semantic theory. Should interval semantics be perhaps super- seded by episode semantics, in which the truth-value of a statement would be relative to world and a particular episode? The absurdity of such a proposal is manifest. The only tenable conclusion is that statements containing verbs like 'draw a circle' must be analyzed as involving reference to relations between individuals and episodes and quantification over episodes.

Each episode uniquely determines the time stretch that it takes up; call it the running time of the episode. Now suppose O relates each individual to his circle-drawing behaviours. The truth-condition of

(11) John draws a circle

can be given in terms of Q within the conventional world/instant space: (11) is true (now, at the very present moment) iff John bears Q to at least


one episode whose running time is current. Thus the truth of (11) will often depend on John's conduct over an interval a part of which is already in the recent past and another part still in the immediate future. But this does not alter the fact that it is the present, momentary truth-value of (11) that is at issue: just like the circumstance that the velocity of a moving body at time T depends on the positions of the body shortly before and shortly after T, does not alter the fact that the velocity is the momentary velocity of the body at the instant T.

The truth-condition of (5) can be explained in terms of Q very much the way it was explained above in terms of the individual/interval relation R: (5) is true iff there is at least one episode E such that the running time of E is a past one-hour interval and John bears O to E. Since the running time of an episode may be scattered, it is easily seen how the truth-condition of (8) can be stated in terms of O. But more importantly, (9) presents no problem either: (9) is true iff John bears O to no more than one episode whose running time is included in yesterday. This truth-condition is not satisfied if yesterday John drew two circles simultaneously. Finally, (10) is true iff the lightning strike occurred during the running time of an (actual) episode forming an initial segment of an episode to which John bears O in some world or other. 4

Things, incidentally, are even more complicated. John can draw two circles at the same time not only by drawing one with his hand and the other with his foot. If he has two pencils suitably attached to the opposite ends of a stick, he can draw two circles using just one hand. In this case only one bodily movement is involved, yet two drawings of a circle take place. Or, to take a less contrived example, if John throws a stone and kills two birds, only one behaviour on John's part is involved, yet two bird-killings take place. This shows that a particular killing of a bird is characterized not by a single episode, but by an ordered couple of episodes: a particular avicidal behaviour on the part of the killer, which may be called the labour episode of the killing, and the death of a particular bird, which may be called the upshot episode. Two such couples may share their labour episode and yet be distinct. The extension of a verb like kill a bird or draw a circle in a world thus cannot be described by associating each individual with the class of bird-killing behaviours exhibited by the individual in the world, but with the appropriate class of labour-episode/ upshot-episode couples.

Several independent linguistic phenomena point towards such explication of verbs. Consider the progressive again. Suppose that John succeeded in pleasing Henrietta by surface mail: he wrote a nice letter which she received two days later. There may nevertheless be no time at

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which it was appropriate to say 'John is pleasing Henrietta'. For when he was writing the letter, he was not pleasing Henrietta yet (she may have been utterly miserable at that time); but two days later, when Henrietta was duly pleased, John may have been no longer engaged in pleasing her: he may have been busy writing to Eunice, or he may have been dead; in neither of the two cases would he have been correctly reported to be pleasing Henrietta. On the other hand, when John pleases Henrietta by giving her a piggy-back ride, then at any time during the ride when Henrietta is already pleased, John is correctly said to be pleasing her. The explanation which suggest itself is that the progressive is correctly used only during the temporal overlap, if any, of the labour episode (i.e., the writing or the burdened running) with the upshot episode (Henrietta's pleasure). When there is no overlap, as in the pleasure-by-correspondence case, the progressive is appropriate at no time.

The present construal of verbs also makes it possible to give a natural explication of the intuitive distinction between achievement verbs and performance verbs. Draw a circle, kill a bird, and please Henrietta are achievement verbs because with them the upshot episode (i.e., the achievement) is always materially disjoint from the labour episode (i.e., the effort). Other verbs allow for, or even require, material overlap. Quarrel with Henrietta is a case in point. The quarrelsome behaviour that John exhibits when quarrelling with Henrietta is an inalienable part of the result, the quarrel as a whole.

With some verbs, like stick out one's tongue, the labour coincides entirely with the upshot. When John is reported to have stuck his tongue out, there is no implication that he achieved anything over and above the tongue-sticking behaviour itself. (One may, of course, achieve various things by sticking out his tongue: upset Henrietta, extinguish a candle, and the like. But unlike upset Henrietta or extinguish a candle, the verb stick

out one's tongue is incapable of reporting any such behaviour-transcend- ing achievements.) Verbs of this sort are known as performance verbs. A performance verb is thus one which associates individuals with couples of episodes whose components coincide.

It should be obvious that no such distinction can be drawn if the extension of a verb is represented by associating individuals with classes of intervals. But note that it would be no help to replace intervals with couples of intervals. Push a cart is clearly an achievement verb, the pushing effort being materially disjoint from the achievement, i.e., the motion of the cart. Yet temporally the two episodes coincide. Thus replacing episodes with their running times would clearly obliterate the distinction in kind between push a cart (an achievement verb) and stick out

one's tongue (a performance).


But what all the above arguments show is merely that ordered couples of episodes have to be invoked in any semantically adequate account of common sentences. They do not begin to show that sentences take truth-values with respect to such couples, that the couples must be promoted to the status of indices of evaluation. When John sticks out his tongue while drawing a circle, two labour/upshot couples take place. But surely we do not want to say that John draws a circle is true relative to (or 'at') one of the two couples and false relative to the other. The sentence is true relative to the instant at which it is affirmed because there exists an appropriate circle-drawing episode couple both components of which are in progress at that instant.

The same goes for the arguments presented by Dowty, Kamp, and others in favour of intervals. All they establish is the undeniable fact that intervals play an important role in analyzing many ubiquitous sentences.

They do not begin to show that sentences take truth-values relative to intervals. And we have seen why it is not a good idea to assume that they do. For one thing, a switch from instants to intervals as indices of evaluation opens a Pandora 's box of artificial problems lacking any contact with linguistic intuition. Besides, it breeds unrealistic expectations concerning the amount of Semantic burden that intervals can bear.

4 . T H E S I M P L E P A S T

Interval semanticists, we have seen, are strongly impressed by the fact that some properties of intervals cannot be reduced to the properties of instants belonging to those intervals. There is, for example, no property such that for any interval I to be one in which John draws a circle, it is necessary and sufficient that each instant of I have that property. It is this very observation which has prompted the intervalist revolution.

It is odd, therefore, that the same theorists fail to appreciate a fact which, although distinct from the one just mentioned, bears a strong family resemblance to it. What I have in mind is the equally undeniable fact that some properties of intervals cannot be defined by specifying what sort of subintervals they contain. Some interval properties can, of course, be so defined. For example, for any interval I to be one during which John draws a circle at least once, it is necessary and sufficient that there exist a (proper or improper) subinterval of J during which John draws a circle at least once. But there is no property such that for any interval I to be one during which John draws a circle exactly twice (or every 5 minutes, or hal]: the time) it is necessary and sufficient that there exist a subinterval of I which has that property. In particular, the property of being an interval in which John draws a circle exactly twice (or every 5 minutes, or half the

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time) does not fill the bill. Yesterday, for example, may fail to be an interval during which John draws a circle exactly twice (or every 5 minutes, or half the t ime) despite containing subintervals during which John does draw a circle exactly twice (or every 5 minutes, or half the time).

Interval semanticists overlook this fact in defining the past and future tenses. For definiteness, let us consider the simple past. (The comments which follow carry easily over to the future tense.) A simple-past state-

ment tells us something about a past interval of time, traditionally known as the reference time: yesterday, last year, the reign of Queen Victoria, before World War II, and the like. It tells us that the reference time satisfies some condition or other. Thus the sentence

(12) John drew a circle (at least once) yesterday

tells us that yesterday satisfies the condition of containing a subinterval in which John drew a circle. (Those who take the view that a past-tense statement is perfectly intelligible without an explicitly specified or tacitly understood reference time (see, e.g. Hintikka (1982), p. 9), should test it by phoning the Weather Bureau and asking 'Did it rain?', to see whether

the answer will be an unqualified 'Yes'.) In many past-tense statements the relevant condition is of the same form

as in (12): an interval satisfies the condition iff it contains a subinterval of a specified sort. But in other past-tense statements the condition is not of that form. As we have seen, the condition that the sentence

(13) John drew a circle exactly twice yesterday

places on the reference time is not one of containing a subinterval of some particular sort.

Yet the way the intervalists explicate the past tense is based on the assumption that all reference-t ime conditions are of the former kind. On Richards's theory (1982, pp. 99-100) the past tense of a base proposition A with reference time J has the logical form

Past(J(A)),

where for any proposition A,

J(A) is true in W at (interval) I iff I = J and there is some subinterval I ' of I such that A is true in W at I ' ;

and

Past(A) is true in W at I iff there exists an interval J < I such

that A is true in W at J.


On Dowty's theory (1979, pp. 325, 327) the past-tense proposition is of

the form

Past(J, A ),

where for any proposition A and interval J,

Past(J,A) is true in W at I iff J < I and there exists a subinterval I ' of J such that A is true in W at I'.

It is easy to check that these two analyses ascribe to past-tense statements the same truth-condition. Both Richards's Past(J(A)) and Dowty's Past (J, A) are true in a world 'at' an interval iff J is before that interval and A is true 'at' at least one of J 's subintervals. But we know already that this truth-condition cannot be correct. For suppose that yesterday John drew a circle first thing in the morning, then shortly before lunch, and again after dinner. Then on the Dowty-Richards truth-condition not only (12) but (13) too is true, for yesterday does contain a subinterval (namely, the a.m. part of that day) in which John drew a circle exactly twice.

Thus in general, a past tense statement does not say that the reference time contains a subinterval of a certain sort. Rather, it says that the reference time itself is of a certain sort. More particularly, it says that the reference time is an interval in which the underlying proposition (say, that John draws a circle) is true with a certain frequency. It is the function of phrases like 'exactly twice', 'every five minutes', and 'half the time' to indicate the frequency in question; these phrases are thus fittingly called frequency adverbs.

In order for the underlying proposition to be true, during the reference time, with a certain frequency, it is not in general enough that it be true with that frequency in a proper subinterval of the reference time. In some special cases it is, as with the frequency adverb 'at least once'. But 'at least once' is only one of a wide range of frequency adverbs and it is a mistake to build its meaning into the past-tense operator (Dowry) or into reference-time operators (Richards). One has to assume, rather, that a definite frequency adverb, F, occurs in the logical structure of every past-tense statement, and that the structure of such a statement is

Past(F(A), J).

Dowty and Richards were misled by the syntactic fact that the frequency adverb 'at least once' is often suppressed in the surface structure.

It might be argued that if this analysis is correct then interval semantics is inevitable. For F(A) must clearly be true or false of intervals of time, not


just instants. Such an argument, however, depends on the tacit premise that the result of applying a frequency adverb to a proposition is another proposition. The premise is hardly uncontroversial. It is hardly obvious that

(14) John draws a circle exactly twice (every five minutes, half the time)

represents a self-contained proposition. One can draw a circle quickly, in someone's company, or reluctantly; each of these is a manner of drawing a circle. But is exactly twice, every five minutes, or half the time a manner of drawing a circle? Common sense would have it that it is not John that may or may not draw a circle exactly twice, every five minutes etc., but rather a time interval that may or may not be one in which John draws a circle exactly twice, every five minutes, etc. In other words, (14) does not express a self-contained proposition but rather something which yields a proposition only when applied to a time interval, as in 'Yesterday is a day in which John draws a circle exactly twice' (which is j u s t an unidiomatic paraphrase of (13)). It is thus natural to say that (in each world) a f requency adverb transforms a proposition into a class of intervals, not into a proposition. The adverb 'at least once' , for example, takes the proposition John draws a circle to the class, say Ca, of intervals in which John draws a circle at least once, while 'exactly twice' takes the same proposition to the class, say C2, of intervals in which John draws a circle exactly twice. (Clearly C2 c Ca, but not vice versa.)

The past-tense relation Past itself works in the following way: Let C be a class of intervals and J a particular interval. If the whole of J is in future, then Past(C, J) is undefined (i.e., truth-valueless); otherwise Past(C, J) is true or false according as the non-future part of J is or is not a member of C. Assuming, as above, that yesterday was a day on which John drew a circle thrice, yesterday is a member of C1, but not of (72; thus it is that (12) is true and (13) is false. The sentences 'John drew a circle (at least once) tomorrow' and 'John drew a circle exactly twice tomorrow' are both truth-valueless because the whole of tomorrow is in the future. 5

5 . T H E P R E S E N T P E R F E C T

The way Dowty and Richards treat the present perfect is flawed by an error which is closely analogous to the error we detected in their treatment of the simple past.

As we have seen, the fact that in past-tense sentences the frequency adverb 'at least once ' is often syntactically null misled Dowty and Richards


into assuming that its import is present in past-tense statements in general. Now in present-perfect sentences it is the frequency adverb 'throughout' or 'continuously' that is often syntactically null. Dowty is misled by this fact into the parallel assumption that the import of this adverb is invariably present, or at any rate whenever the present perfect tense occurs in combination with 'since' or 'for'.

The theory proposed in Dowry (1979, Sec. 7.5) yields the following truth-condition for a statement of the form

(*) X has q~ed since (time) K:

(*) is true in W at I iff for every interval J, if J is later than K and I is a final subinterval of J, then X q~s in W at J.

(A subinterval I of J is J 's final subinterval iff for any instant T in I, I also contains all instants of J which are later than T.) Thus, according to Dowty, the sentence

(15) John has slept (continuously) since midnight

is true at 1 a.m. just in case John sleeps 'at' every (uninterrupted) interval which starts some time between midnight and 1 a.m. and ends at the latter time.

This seems intuitively acceptable, but the acceptability is due to the fact that in (15) the operative frequency adverb (whether syntactically represented or just tacitly understood) is 'continuously'. The inadequacy of Dowty's truth-condition comes to light as soon as 'continuously' is replaced by some other frequency adverb, as in

(16) John has slept intermittently since midnight.

For it may well be true at 1 a.m. that John has slept intermittently since midnight, even if he has slept solidly for the last 5 minutes. In fact, if Dowty's truth-condition were correct, (16) would be false at 1 a.m. come what may, for many an interval which starts after midnight and ends at 1 a.m. is obviously too short for John to sleep, wake up, and fall asleep again, all within the space of it.

Dowty's truth-condition can, incidentally, be shown inadequate even without resort to frequency adverbs other than 'continuously'. Suppose that I say, at 1 a.m., that

(17) John has waltzed (continuously) since midnight.

On Dowty's theory, I cannot possibly be right. For while on the subject of waltzing, Dowty tells us that 'any interval at which x takes less than three steps is not an interval at which x waltzes is true' (Dowty, 1979, p. 171).

280 PAVEL TICH~"

But if so, then even if John actually gave the hour between midnight and 1 a.m. over to relentless waltzing, the short interval in which he took the last two steps just before 1 a.m. is not one 'at' which he waltzes. Thus on Dowty's definition, (17) is false at 1 a.m.

Richards's truth-condition for statements of the form (*) combines the mistake that he and Dowty make in connection with the simple past with the mistake we have just detected in Dowty's definition of the present perfect. On Richards's theory,

(*) is true in W at I iff I contains a subinterval J such that K is the initial bound of J and X q~s in W at every subinterval of J.

As a result, Richards gets the worst of both worlds. For imagine that as a matter of fact, John spent the time between 1 t p.m and 1 a.m. dropping off and waking up again, say once every five minutes, and that midnight is one of the times at which he happened to be asleep. Then on Richards's theory the interval I = [11 p.m., 1 a.m.] is one at which (15) is true (since John was asleep throughout a short subinterval of I starting at midnight) and at which (16) is false (since any subinterval of I contains intervals, for example, one-instant intervals, in which John is not intermittently asleep). This, of course, is the exact reverse of what one would expect. (It is easy to check, incidentally, that Richards's definition makes (16) false come what may.)

The inadequacies of both definitions come from a common source. Instead of granting frequency adverbs their rightful place in present- perfect statements, Dowty and Richards try to incorporate the meaning of particular frequency adverbs either in the definition of the tense itself or in the definitions of reference-time operators like since or [or. It is thus easy to manufacture counter-examples by picking adverbs indicative of some other frequencies.

Just like the simple past or future, the present-perfect tense enables us to say something about a reference-time interval. It enables us to say that the base proposition comes true in the interval with a certain frequency. But since there is a wide range of frequencies, many of them incompatible with others, the tense itself, as well as reference-time operators like since, for etc., must be frequency-neutral. A particular frequency adverb must be assumed to be present in the logical structure of each present-perfect sentence. The structure of such a sentence is thus isomorphic with that of a simple-past sentence:

Perf(F(A), J),

where A is the base proposition, F a frequency adverb and J the reference


time. The present-perfect relation Peff itself works in the following way. Let C be a class of intervals and J any particular interval. If J does not include the present moment, then Peff(C, J) is undefined; otherwise Perf (C, J) is true or false according as the non-future part of J does or does not belong to C.

Since is like after: it transforms an interval I into the interval which consists of I and all moments later than I. Thus in both (15) and (16), the reference time is the interval which starts at midnight and never ends. The frequency adverb 'continuously' transforms the proposition John is asleep into the class, say D1, of intervals throughout which that proposition is true. The adverb 'intermittently' transforms the same proposition into the class, say/)2, of intervals where instants at which the proposition is true alternate with instants at which it is false. 6

If John slept solidly between midnight and 1 a.m., (15) was true and (16) false at 1 a.m., because the interval [midnight, 1 a.m.] (i.e., the non-future part of [midnight, eternity]) is a member of D1, but not a member of D2. If, on the other hand, John spent between midnight and 1 a.m. falling asleep and waking up again then (15) was false and (16) true at 1 a.m. because the interval [midnight, 1 a.m] is not a member of D 1 and is a member of D2. The sentence 'John has slept (continuously) yesterday' and 'John has slept intermittenly tomorrow' are both truth-valueless because neither yesterday nor tomorrow contain the present moment.

Although intervals play a crucial role in this explanation, neither the present-perfect statements themselves, nor the base propositions embedded in them, need be assumed to evaluate at intervals rather than moments of time.

N O T E S

i I do not wish to imply that every proposition mus t be expressible by a sentence. This is not the case on the punctual is t approach either. But if the class of world/intervals couples whose second components are ha l f -an-hour long should count as a proposition, (2) would sugges t itself as a natural way to express it. Syntactically, 'It is ha l f -an-hour ' comes from, say, 'Fred sleeps for hal f -an-hour ' the way 'It rains ' comes from 'Fred sleeps while it rains'. On the intervalist account , the analogy is not only syntactic, but semantic as well. In order for 'Fred sleeps while it rains' to be true at an interval, the interval mus t satisfy two conditions: Fred mus t sleep at it, and it mus t rain at it. While in order for 'It rains ' to be true at an interval, only the second condition mus t be satisfied. Quite analogously, in order for 'Fred sleeps for hal f -an-hour ' to be true at an interval, the interval mus t satisfy two conditions: Fred must sleep at it and it mus t be hal f -an-hour long. While in order for 'It is ha l f -an-hour ' to be true at an interval, only the second condit ion mus t be satisfied. 2 K a m p suggests that

in order to arrive at a correct recursive characterizat ion of the condit ions under which assertoric u t terances of certain complex sentences are true at the t imes at which they are


made we must allow the recursion to pass through intermediate stages at Which subexpressions of those sentences are evaluated with respect to intervals rather than to instants. (1979, p. 392) Eventually however there is a return to notions involving instants, such as in particular that of the truth of a complete sentence at the instant it is used. (p. 384)

Thus according to Kamp, whenever a sentence is used it is to be evaluated with respect to the instant of use.

Now it is not clear whether some of the subexpressions which according to Kamp occasion the resort to intervals qualify as sentences in their own right. If not, then what Kamp says amounts to a fiat repudiation of the basic idea of interval semantics, namely that sentences take truth-values relative to intervals rather than instants. If, on the other hand, some of those subexpressions do qualify as sentences, they will be false whenever used in their own right (rather than as subexpressions of larger sentences). So their negations will be true. 3 For a formal treatment of the notion of episode see Tich~ (1980b), Section 3. 4 For a formal definition of the progressive tense, see Tich~ (1980b), Section 11. 5 The past-tense relation is defined in a more formal manner in Tich~ (1980a), Section 5. 6 The present-perfect relation is defined in a more formal manner in Tich~) (1980a), Section 6.

R E F E R E N C E S

Cresswell, M. J.: 1977, 'Interval Semantics and Logical Words', in Christian Rohrer (ed.), On the Logical Analysis of Tense and Aspect (Tuebingen).

Dowty, D. R.: 1979, Word Meaning and Montague Grammar (Reidel, Dordrecht). Dowty, D. R., Wall, R. E., and Peters, S.: 1981, Introduction to Montague Semantics

(Reidel, Dordrecht). Hintikka, Jaakko: 1982, 'Temporal Discourse and Semantical Games', Linguistics and

Philosophy 5, 1-22. Kamp, Hans: 1979, 'Events, Instants and Temporal Reference', in R. B~iuerle et al. (eds.),

Semantics From Different Points of View (Berlin). Richards, Barry, 1982, 'Tense, Aspect, and Time Adverbials', Linguistics and Philosophy 5,

59-107. Taylor, Barry: 1977, 'Tense and Continuity', Linguistics and Philosophy 1, 199-220. Tich~, Pavel: 1980a, 'The Logic of Temporal Discourse', Linguistics and Philosophy 3,

343-369. Tich~,, Pavel: 1980b, 'The Semantics of Episodic Verbs', Theoretical Linguistics 7, 263-296

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tichý - do we need interval semantics?

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