## 1. Introduction



*Tense* roughly means reference to the time at which events
take place, or at which processes or states hold. English, for
example, clearly distinguishes between past and non-past tense as in
(1a) and (1b) and (1c). 


* 1.

(a)

John promised to pay ten pounds.
* (b)

I promise to pay you ten pounds.
* (c)

The student will work on his thesis.



In English, *verb* forms are typically used to signal the time
when an action or event occurs or a state holds. Thus in (1a), John's
promise was given before the present time, in (1b) the promising is
simultaneous with the present time and (1c) says that the student's
work on his thesis will occur at some time after the present. It
should be noted that the verb forms used to express temporal
information may also be used to signal information that is not purely
temporal. For instance, the present tense form in *John walks*


characterizes a certain habit of John. And in the statement *The
train departs at five o'clock tomorrow*, the present verb form
clearly has a futurate meaning. So English distinguishes between past
and non-past but not between future and non-future. Moreover, many
linguists exclude future as a pure tense, because the auxiliary
*will* may be used to express volition as in *He will go
swimming in dangerous
 water*.[1]



Apart from *absolute* tense--exemplified by
(1a)-(1c)--where the reference point from which the
location in time is evaluated is the present, there is also

*relative* tense where the reference point is not necessarily
the present but may be given by context. Thus in a sentence like
*The student had worked on his thesis* the student's work took
place at a time before a reference point in the past, in contrast to
(1c) where the reference point is the present. Other instances of
relative tense in English are the present perfect and the future
perfect in (2a) respectively (2b). 


* 2.

(a)



I have lost my watch.
* (b)

The student will have finished his thesis next month.



The notion of *aspect* according to Comrie refers to
"[the] different ways of viewing the internal temporal
constituency of a situation" (1976: 3). It is customary to
distinguish between *lexical* and *grammatical* aspect.
The following are examples concerning lexical aspect. Since Vendler
(1967) linguists distinguish at least four aspectual classes. These
are states like *know*, activities like *run*,
accomplishments like *cross the street* and achievements
exemplified by a verb like *recognize*. Accomplishments
describe the internal temporal constituency of a situation in a more
detailed way than (say) activities. *Crossing the street*, for
example, includes the starting of a crossing activity which goes on
for some time and involves a result state that is characterized by
being on the other side of the street. No such elaborate descriptions
are necessary for activities or achievements. There are grammatical
means of distinguishing between these aspectual classes. For example,
states and achievements cannot occur with the progressive while
activities and accomplishments can; more about this below in
 Section 5.


 These differences are customarily considered to be lexical
differences and this area is therefore dubbed *lexical* aspect.




A further crucial aspectual difference is that between perfective and
imperfective aspect. Comrie characterizes these notions as follows.




> 
> 
> ...perfectivity indicates the view of the situation as a single
> whole, without distinction of the various separate phases that make up
> that situation, while the imperfective pays essential attention to the
> internal structure of the situation. (Comrie 1976: 16) 
> 
> 
> 



In English this difference is often expressed by *grammatical*
means, for instance by past tense versus past progressive. This is an
instance of grammatical aspect. Slavic language often have an
elaborate grammatical system to signal the difference between
imperfectivity (ipv) and perfectivity (pv). Here is an example from Russian. 


* 3

(a)

Petr
  

Peter  pisal\(\_{ipv}\)/\*napisal\(\_{pv}\)
  

was-writing/\*wrote  pis'mo,
  

letter,  kogda

  


when  on
  

he  byl
  

was  prervan
  

interrupted  telefonnym

  

by-telephone  zvonkom.

  

call.
* (b)

Petr
  

Peter  \*pisal\(\_{ipv}\)/napisal\(\_{pv}\)
  

\*was-writing/wrote   pis'mo
  

letter  i

  

and   otnes

  

brought   ego
  

it   na
  

to   poctu.

  

post-office.



Sentence (3b) presents the writing of a letter as a single whole and
it is clear that this activity was finished when the letter was brought to the
post office. By contrast, (3a) focuses on an ongoing activity of
letter writing in the past which is not completed because of the
interruption. In fact it is consistent with the truth of (3a) that the
letter was never completely written. 



The natural language categories *tense* and *aspect*


embody the linguistic encoding of time. From a typological point of
view these categories are typical verbal
 categories.[2]
 This means that if these categories are morphologically realized in a
language then these morphemes attach to the verb. Verbs usually
express events, processes, actions or
 states[3]
 and the temporal morphemes locate these eventualities in time. Of
course this does not mean that languages that lack such morphemes are
not able to express temporal relations. They just have to choose other
means for this purpose. One famous example of a language not
containing temporal morphology is Mandarin Chinese. Often tense and
aspect cannot be clearly separated. For example the Quiche
prefix \(x-\) denotes a completed (aspect) action in the past
 (tense).[4]




Typologists also observed an asymmetry between past and future. Most
languages that possess temporal morphology use these means to
differentiate between past and non-past. There are, however, a few
languages which use temporal morphology to distinguish between future
and non-future. 



We will not go into further details concerning the linguistic
realization of tense and aspect, but present a brief and necessarily
incomplete review of major philosophical and semantic theories of
these notions. 


## 2. Temporal Logic



Temporal logic was introduced by Arthur Prior (see for instance Prior
1967). Here we will sketch only propositional temporal logic (for a
more thorough introduction the reader is advised to consult Gamut
1991: chapters 2 and 3) and the entry on
 temporal logic.


 Temporal logic introduces operators \(\bG\), \(\bH\), \(\bF\) and
\(\bP\), which are similar to the modal operators \(\Box\) and \(\Diamond\).
We summarize the intuitive meaning of these operators in Table 1. 




Table 1.| \(\bG\phi\) | it is always going to be the case that \(\phi\)  |
| \(\bH\phi\)  | it always has been the case that \(\phi\)  |
| \(\bF\phi\)  | it will at some stage in the future be the case that
\(\phi\) |
| \(\bP\phi\)  | it was at some stage in the past that \(\phi\)  |



By adding the above operators to propositional logic we extend
propositional logic to propositional temporal logic. Let \(q\)
abbreviate the sentence *Sam is working*, then we are able to
express the following verb tenses in temporal logic: 


* 4.

(a)



\(q\)   Sam is working.
* (b)

\(\bF q\)   Sam will work.
* (c)

\(\bP q\)   Sam worked.
* (d)

\(\bP\bP q\)   Sam had worked.
* (e)

\(\bF\bP q\)   Sam will have worked.
* (f)

\(\bP\bF q\)   Sam would work.



A model \(\sM\) for propositional temporal logic consists of a
nonempty set \(T\) of moments of time, an earlier than relation
\(\prec\) and a valuation function \(I\) which for each moment of
time and each proposition letter \(q\) assigns a truth value
\(I\_{\sM ,t}(q)\). The following
definition characterizes \(I\_{\sM ,t}(\phi)\)
for the temporal operators. 



**Definition 1.**
*Let \(\sM\) be a model, then \(I\_{\sM ,t}(\phi)\) is defined by
the following clauses:* 
1. \(I\_{\sM ,t}(\bG\phi) = 1\) *iff for all* \(t' \in T\)
*such that* \(t \prec t' : I\_{\sM ,t'}(\phi) =
1\).
2. \(I\_{\sM ,t}(\bH\phi) = 1\) *iff for all* \(t' \in T\)
*such that* \(t' \prec t : I\_{\sM ,t'}(\phi) =
1\).
3. \(I\_{\sM ,t}(\bF\phi) = 1\) *iff there exists* \(t' \in T\)

*such that* \(t \prec t'
 : I\_{\sM ,t'}(\phi) = 1\).
4. \(I\_{\sM ,t}(\bP\phi) = 1\)
 *iff there exists* \(t' \in T\)
 *such that* \(t' \prec t
: I\_{\sM ,t'}(\phi) = 1\).




These clauses are analogous to the clauses which define the modal
operators \(\Box\) and \(\Diamond\)--with \(\bG\) and \(\bH\)
corresponding to \(\Box\), and \(\bF\) and \(\bP\) to \(\Diamond\)--but
instead of possible worlds the valuation function takes moments of
time as arguments. Therefore principle (5) is valid in any model,
since the modal analogue is valid for \(\Box\) independently of the
accessibility relation. 


* 5.

\(\bG(\phi \rightarrow \psi) \rightarrow(\bG\phi \rightarrow \bG\psi)\) (and analogously for \(\bH)\).



However, principle (6), which is often considered as valid for
\(\Box\), is presumably not valid for \(\bG\). 


* 6.


\(\bG\phi \rightarrow \phi\)



Formula (6) is equivalent to \(\phi \rightarrow \bF\phi\). The latter
formula says that if \(\phi\) is the case then \(\phi\) will be the case,
which is intuitively incorrect. Suppose that \(\prec\) is irreflexive;
this assumption makes sense since it means that no time point is
earlier than itself. But under this assumption, Principle (6) is
obviously wrong. For extensions of temporal logic and more examples
concerning the correspondence between properties of models and the
validity or invalidity of formulas see Gamut 1991: chapter 2.



This concludes our brief remarks about temporal logic. We now turn to
an alternative approach due to the philosopher Hans Reichenbach. 


## 3. Speech Time, Event Time and Reference Time



The notions *speech time*, *event time*, and

*reference time* were introduced by Reichenbach (1947) in order
to distinguish simple past and present perfect or, more generally,
absolute and relative tense. According to Comrie (1985), who refined
Reichenbach's system (in Chapter 6 of his book on tense) speech time
and event time are sufficient for the analysis of *absolute
time*; i.e., present, (simple) past and future. But for
*relative tense*--of which the present perfect is but one
example--reference time is required. Let us explain these notions
by applying them to distinguish between past and present perfect. 



Consider the following examples from Steedman (1997). 


* 7.

(a)

I have lost my watch.
* (b)



Yesterday, I lost my watch.



It has often been observed that the present perfect in English has
present relevance. For instance the continuation of (7a) with *but
I have found it again* is infelicitous in English; the German
translation of this sequence is acceptable, showing that the German
perfect is more like a past tense (but see Kamp et al. 2015 for an
extensive discussion, in Other Internet Resources). The same
continuation is fine for sentence (7b). In this sense the perfect is a
relative tense and the past an absolute tense in English. Let \(E\)
be short for event time and let \(S, R\) stand for speech
time and reference time respectively. 




|  |  |
| --- | --- |
| Simple Past  | Present Perfect  |
| *I saw John*  | *I have seen John* |
| a long right arrow over E,R space S | a long right arrow over E space R,S |



For the simple past, both event time and reference time are situated
on the time line before speech time. In case of the perfect, R and S
are simultaneous and E is earlier than both R and S. Intuitively,
reference time represents the perspective from which an eventuality is
perceived on the time line. This is not a purely semantic theory of
tense like temporal logic, rather it is a pragmatic theory of tense.
In particular, reference time (in contrast to event time) has to be
mutually known by communication partners. When using the present
perfect, reference time is known since it coincides with speech time.
However, reference time is not necessarily known when the past tense
is used, since it is just required to be earlier than speech time and
to coincide with event time. This may explain why sentences in the
past sound strange when uttered out of the blue. Steedman (1997)
presents the following examples: 


* 8.

(a)

\*Chapman breathed a sign of relief.
* (b)



When Nixon was elected, Chapman breathed a sigh of relief.



The past-tensed sentence (8a) uttered out of the blue is infelicitous,
while sentence (8b) is fine, since in this case the
*when*-clause introduces the reference time.



This is by no means the complete story of the perfect. For example,
Comrie (1976) distinguishes four typical uses of the perfect: the
perfect of result, the experiential perfect, the perfect of persistent
situation and the perfect of recent past. For an extensive recent
discussion the reader is advised to consult Kamp et al. 2015 (in Other
Internet Resources). 





The following table summarizes the positions of event time, reference
time and speech time for other tenses: 




Table 2.| E, R, S  | Sam is working.  |
| S -- E, R  | Sam will work.  |
| E, R -- S  | Sam worked.  |
| E -- R -- S | Sam had worked.  |
| S -- E -- R | Sam will have worked. |
| R -- E -- S | Sam would work.  |



Comrie (1985) and Gamut (1991) point out that this cannot be a
complete account for temporal constructions that occur in natural
languages. For instance the temporal profile of the sentence *Sam
would have worked* cannot be analyzed by using a single reference
time. This is one reason for the extended requirements for a formal
theory of tense proposed by Comrie (1985: chapter 6). Furthermore, 
Blackburn and Jorgensen (2016) combine the key insights of Prior 
and Reichenbach in the framework of hybrid tense logic suggesting that 
the two approaches are fully compatible with each other.


## 4. From Events to Time


In event semantics following Davidson (1967) events are commonly
taken as ontological primitives. This raises the philosophical
question how events and times relate to each other. A construction,
due to Russell, Wiener and Kamp, shows that the time line can be
constructed from events. We will briefly sketch this construction
before we move on to lexical and grammatical aspect.


Russell remarked:



> 
> 
> 
> Even if there be a physical world such as the mathematical theory of
> motion presupposes, impressions on our sense-organs produce sensations
> which are not merely and strictly instantaneous, and therefore the
> objects of sense of which we are immediately conscious are not
> strictly instantaneous. Instants, therefore, are not among the data of
> experience and must be either inferred or constructed. It is difficult
> to see how they can be validly inferred; thus we are left with the
> alternative that they must be constructed. (Russell 1914: lecture
> IV)
> 
> 
> 


Kamp (1979)
took up Russell's ideas and slightly modified them. The construction
takes event structures as primitives from which the structure of time,
i.e., a total order, can be derived. 



An event structure \(\langle E, P, O\rangle\) simply
consists of a set of events \(E\) and the relations \(P\)
(*precedes*) and \(O\) (*overlaps*) and is
characterized by axioms A1-7. 



\[\begin{align}
\tag\*{A1.}
 &\forall x\forall y(P(x, y) \rightarrow \neg P(y, x)) \\
\tag\*{A2.}
 &\forall x\forall y\forall z(P(x, y) \wedge P(y, z) \rightarrow P(x,z)) \\
\tag\*{A3.}
 &\forall x(O(x, x)) \\
\tag\*{A4.}
 &\forall x\forall y(O(x, y) \rightarrow O(y, x)) \\
\tag\*{A5.}
 &\forall x\forall y(P(x, y) \rightarrow \neg O(x, y)) \\
\tag\*{A6.}
 &\forall x\forall y\forall z\forall t(P(x, y) \wedge O(y, z) \wedge P(z,
t) \rightarrow P(x, t)) \\
\tag\*{A7.}
 &\forall x\forall y(P(x, y) \vee O(x, y) \vee P(y, x))
\end{align}\]



Given an event structure satisfying these axioms, the set of instants
and their strict linear ordering can be constructed according to
definition 2. The idea is to identify an instant with the maximal set
of pairwise overlapping events. Instants are thus conceived of as
abstractions over events that happen (or go on) simultaneously. Events
linked by the precedence relation will give rise to distinct instants.




**Definition 2.** *Let \(\langle E, P, O\rangle\) be a
structure which satisfies axioms A1-7.*
1. An instant of \(\langle E, P, O\rangle\) is a maximal subset of
pairwise overlapping events, i.e., \(i\) is an instant of \(\langle E,
P, O\rangle\)iff
	1. \(i \subseteq E\)
	2. For any \(a, b \in i : O(a, b)\)
	3. if \(e \in E\) but \(e \not\in i\) then there is
	\(d \in i\) such that \(\neg O(d, e)\)
2. Let \(I\) be the set of instants of \(\langle E,
P, O\rangle\)
3. For \(i\_1, i\_2 \in I\) put \(i\_1 \lt i\_2\) iff there
are \(e\_1 \in i\_1\) and \(e\_2 \in i\_2\) such
that \(P(e\_1,
e\_2)\)





We will first illustrate the Russell-Kamp construction by way of an
example and then show that \(\langle I,\lt \rangle\) has the
properties specified in Theorem 1: it is a strict linear ordering
which gives rise to an interval structure. 



**Example 1.** *Let \(E = \{a, b, c, d, e\}\) and \(P\) be the
set of ordered pairs \(\{(a, c), (a, d), (a, e), (b, e), (c,
e)\}\). Given A7, the relation \(O\) consists of the set of pairs
\(\{(a, b), (b, c), (b, d), (c, d), (d, e)\}\). \(E\) contains three
maximal subsets of pairwise overlapping events, that is, instants
\(i\_1 = \{a, b\}\), \(i\_2 = \{b, c, d\}\) and \(i\_3 = \{d,
e\}\). Furthermore, \(i\_1\), \(i\_2\) and \(i\_3\) are linearly ordered
as \(i\_1 \lt i\_2 \lt i\_3\) because \(a \in i\_1\), \(c \in i\_2\) and
\(P(a, c)\) hence \(i\_1 \lt i\_2\), and \(c \in i\_2\), \(e \in i\_3\)
and \(P(c, e)\) hence \(i\_2 \lt i\_3\), and similarly for \(i\_1 \lt
i\_3.\)* 




A proof of the following theorem can be found in (Kamp 1979: 379).




**Theorem 1.** *If an event structure \(\langle E, P,
O\rangle)\) satisfies A1-7, then the structure \(\langle I,\lt
\rangle\) constructed according to definition 3 is a strict linear
ordering, i.e., the following three conditions hold:* 


\[\begin{align}
&1.\ \forall i\_1, i\_2 (i\_1 \lt i\_2 \rightarrow \neg i\_2 \lt i\_1)
 &\text{(asymmetry)} \\
&2.\ \forall i\_1, i\_2, i\_3 (i\_1 \lt i\_2 \wedge i\_2 \lt i\_3 \rightarrow i\_1 \lt i\_3)
 &\text{(transitivity)} \\
&3.\ \forall i\_1, i\_2 (i\_1 \ne i\_2 \rightarrow i\_1 \lt i\_2 \vee i\_2 \lt i\_1)
 &\text{(totality)}
\end{align}\]


Once having constructed the set \(I\) it is possible to define time
intervals and these, in turn, can be used to measure the temporal
extent of an event. In Example 1, for example, \(a\) *goes
on* in the interval [\(i\_1], b\) in
[\(i\_1, i\_2], c\) in
[\(i\_2], d\) in [\(i\_2,
i\_3\)] and \(e\) in [\(i\_3\)].
Corollary 1 guarantees that these intervals can be constructed (Kamp
1979: 379). 



**Corollary 1.** *For each \(e \in E\), the set \(e' =\{i \in I
\mid e \in i\}\) is a non-empty interval of \(\langle I,\lt
\rangle.\)*




Proof: If \(i\_1, i\_2 \in e'\) and \(i\_1 \lt i \lt i\_2\), it has to be
shown that \(e \in i\), too. Suppose it is not. Then there is \(d \in
i\) such that \(\neg O(d, e)\), hence \(P(d, e)\) or \(P(e, d)\). In
the first case \(i \lt i\_1\) and in the second case \(i\_2 \lt
i\). Both yield a contradiction.\(\Box\) 



We cannot give further details here, but refer the interested reader
to Thomason (1986) who discusses how to construct time as a
continuum--isomorphic to the real numbers--and uses a
different construction originally proposed by Walker (1947). Empirical
evidence for events being ontological primitives will be presented in
 section 8
where we review psychological studies on event perception. We will now
move to lexical and grammatical aspect, that is the expression of event
types and how a given event is conceptualized, which form basic
concepts in linguistic theory.


## 5. Lexical and Grammatical Aspect



Linguists distinguish four or five lexical aspectual classes or
*Aktionsarten.* The following four were introduced by the
philosopher Zeno Vendler (1967: chapter four) on the basis of earlier
work by philosophers such as Kenny, Ryle and Aristotle (see also the
 entry on events). For a formal
definition of *Aktionsart* the reader is referred to van
Lambalgen and Hamm (2005: 85 ff). 




Table 3.| *States* | *Activities* | *Accomplishments* | *Achievements* |
| --- | --- | --- | --- |
| know  | run  | cross the street  | recognize  |
| be beautiful | swim  | draw a circle  | reach  |
| believe  | walk  | paint a picture  | find |



In addition, linguists often assume that verbs like *flash*,
*spot* and *blink* form an extra class--the class
of semelfactives or points (see Smith 1991). 



A useful notion for distinguishing *Aktionarten* is the
*event nucleus* introduced by Moens and Steedman (1988). The
event nucleus is constructed from a preparatory phase, a culminating
event and a consequent phase. Activities only refer to the preparatory
phase, states only to the consequent phase, achievements to the
culminating event and to the consequent phase and finally
accomplishments to all three parts of the event nucleus. 



![preparation event consequent](nucleus.png)



Vendler proposed several linguistic tests for distinguishing these
verb classes. We will present only the most important ones here; for a
much more comprehensive list the reader is referred to Dowty (1979: 60).
The first test separates non-statives from statives. Only non-statives
occur in the progressive. 


* 9.

(a)

\*Sam is knowing the answer.
* (b)

Sam is running.
* (c)


Sam is writing a letter.



Accomplishment verbs prefer \(in\)-adverbials as temporal
modifiers, whereas activity verbs allow only
*for*-adverbials.


* 10.

(a)

 ?Sam wrote a letter for an hour.
* (b)



 Sam wrote a letter in an hour.
* (c)

 Sam swam for an hour.
* (d)

 \*Sam swam in an hour.



Achievement verbs are usually infelicitous with

*for*-adverbials but allow the combination with
\(in\)-adverbials. 


* 11.

(a)

Sam reached the summit in five hours.
* (b)

\*Sam reached the summit for five hours.



The last test we mention here concerns the different entailment
patterns of activities and accomplishments in the progressive.
Activities in the past progressive entail their past reading but
accomplishments in the past progressive don't. 


* 12.

(a)

Sam was pushing a cart. \(\Rightarrow\)
* (b)

Sam pushed a cart.


* 13.

(a)

Sam was building a house. \(\not\Rightarrow\)
* (b)


Sam built a house.



This is an instance of the famous imperfective paradox. Of course,
taken literally this is not a paradox. Nevertheless the entailment
patterns (12) and (13) pose a significant problem for any formal
theory of tense and aspect. For instance, the first formal proposal
for truth conditions of sentences in the past progressive stated that
a sentence S in the past progressive is true if and only if there
exists an open interval before the speech time at which the sentence
without the progressive is true. According to definition 1.4 this
immediately validates pattern (12) but without qualifications this
truth definition does not account for pattern (13). Sections
 6
 and
 7
contain extensive discussion of the imperfective paradox. 



Vendler thought of the *Aktionarten* as lexical properties of
verbs. That this position is dubious was pointed out by many
linguists, in particular by Dowty (1979) and Verkuyl (1993). For a
simple case, consider the verb *drink*. If we assume that being
an activity is a lexical property of this verb, then surely by
combining it with the noun *wine* this property is preserved
for the complex phrase *drink wine*. However, if we combine the
verb *drink* with the noun phrase *a bottle of wine* we
turn an activity into an accomplishment. The converse problem arises
if we assume that being an accomplishment is a lexical property of

*drink*. This phenomenon of aspectual reinterpretation was
dubbed *coercion* in Moens and Steedman (1988). In
 Section 6.1 
we will sketch a systematic solution for these types of coercion
 in mereological semantics (Krifka 1989, 1992). 



*Aktionsart* is not even definitely fixed at the VP-level. The
verb *arrive* in 


* 14.

(a)
Chapman arrived.
* (b)
\*Chapman arrived all night.



is an achievement, as demonstrated by the ungrammaticality of (14-b).
But if we choose a bare plural as subject it is turned into an
activity and sentence (15) is grammatical. 


* 15.

Visitors arrived all night.



Therefore a final decision about aspectual class cannot be reached
below the sentence level. Steedman's famous sentence (16) 


* 16.


It took me two years to play the "Minute Waltz" in
less than sixty seconds for one hour without stopping.



shows that coercion can be iterated and that aspectual class can
switch back and forth in this iteration process. We close this section
with two further examples of aspectual coercion. In the first one a
stative verb is transformed into an activity.



The verb *resemble* is a stative verb and therefore (17a) is
unacceptable but (17b) is fine. 


* 17.

(a)

\*She is resembling her mother.
* (b)

She resembles her mother.



However, if we add *more and more every day* to (17a) thus
forming 


* 18.

She is resembling her mother more and more every day.




we get an acceptable result. This is due to the phrase *more and
more every day* which coerces a state into an activity. Our last
example is about temporal modification with *for*-adverbials.
As pointed out above accomplishments are usually rather bad with
*for*-adverbials. They require \(in\)-adverbials. But
consider sentence (19) from van Lambalgen and Hamm (2005). 


* 19.

Pollini played Opus 111 for two weeks.



This sentence seems to be fine. However it is not interpreted as an
accomplishment any more but is reinterpreted as an iterated activity.
To get this reading, a lot of non-linguistic knowledge is required.
First, one has to guess that Opus 111 refers to Beethoven's last piano
sonata and moreover one has to be aware that this piece lasts about 25
minutes. Under these assumptions, sentence (19) says that Pollini
played Opus 111 repeatedly within a time span of two weeks. Formal
accounts of coercion phenomena are contained in Egg (2005), van
Lambalgen and Hamm (2005) and Steedman (1997) among others. The reader
is also advised to consult Steedman's updated manuscript, *The
Productions of Time* (see the Other Internet Resources).
Processing studies of coercion are reviewed in Bott (2010); see also
 Section 8.



## 6. Intensionality versus Events




The most influential approach to solving the imperfective paradox is
due to Dowty (1979). The basic idea is to treat the progressive as a
modal operator \((\Box)\) but restrict its domain to inertia worlds.
The notion of an *inertia world* is characterized informally.
An inertia world is exactly like the real world up to the time of
evaluation; after this it may differ from the real world but is
assumed to be as similar to the real world as possible. Given this
notion, the following definition introduces truth conditions for
sentences in the progressive. 



**Definition 3.** PROG\((\phi)\) *is true with
respect to interval \(I\) and world \(w\) if and only if \(\forall w'
\in IE(w)\) there exists an interval \(I' \supseteq I\) such that
\(\phi\) is true with respect to \(I'\) and \(w'\). In addition \(I\)
should not be the final subinterval of \(I'\). \(IE(w)\) is the set of
inertia worlds for \(w\).*




Definition 3 does not yet account for the different entailment
patterns of activities and accomplishments. Two additional assumptions
are required. The first concerns activities; the second,
accomplishments. 


* 20.

If \(\phi\) is an activity that is true with respect to interval
\(I\), then \(\phi\) is true with respect to any sufficiently large
subinterval of \(I\).



Accomplishments like *Mary draw a circle* are split up into two
parts, an activity part *Mary draw* that satisfies principle
(20) and a result part which is characterized by the existence of a
circle. Both parts are connected by a causality relation

*CAUSE*. 


* 21.

*CAUSE*[Mary draw, A circle comes into existence]



Thus an accomplishment has a richer internal structure than an
activity. 



With these additional requirements, the inference patterns for
activities and accomplishments follow. Let us first show that *Mary
was pushing a cart* implies *Mary pushed a cart*. Assume
that the first sentence is true in \(w\) with respect to speech time
\(S\). Then *Mary be pushing a cart* is true in \(w\) with
respect to an interval \(I\) before \(S\). According to Definition 3,
*Mary push a cart* is then true in every inertia world \(w' \in
IE(w)\) with respect to an interval \(I' \supseteq I\). Because of
postulate (20), *Mary push a cart* is true in every \(w' \in
IE(w)\) with respect to interval \(I \ (I\) is a subinterval of
\(I')\). Now, the definition of inertia worlds implies that

*Mary push a cart* is true in \(w\) with respect to interval
\(I\). This means that *Mary pushed a cart* is true in
\(w\) with respect to speech time \(S\). 



A completely analogous argument shows that *Mary was drawing a
circle* implies *Mary drew*. But postulate (20) only holds
for the first part of accomplishments; the result part is only
required to hold in every inertia world \(w'\) with respect
to a larger interval \(I'\), normally one after speech time.
Since the inertia worlds after \(S\) may differ from \(w\) and
since the subinterval property (20) is not required to hold for the
result parts of accomplishments, the stronger conclusion that *Mary
drew a circle* does not follow. 



Many researchers assume that inertia worlds introduce a notion of
*normality* for the semantic analysis of the progressive. The
thunderbolt in example (22) from Landman (1992) seems to break the
normal development of the real world. Dowty's analysis seems to be
tailored for cases like this one. 


* 22.

Mary was crossing the street, when a thunderbolt from heaven
struck her down.




However, objections have been raised to the normality interpretation
of inertia. The first one is attributed to Frank Vlach in Ogihara
(1990). Consider sentence (23) 


* 23.

John was crossing the street, when he was hit by a truck.



and a situation in which the truck is only a few centimeters away from
John. Moreover the truck's speed is such that it is impossible for it
to stop before hitting John. In this situation, the normal course of
events is such that John will never reach the other side of the
street. Therefore, given definition 3, sentence (23) should be false
under these circumstances. This is intuitively incorrect. A similar
objection was raised by Bonomi (1997).



On the other hand, the normality interpretation seems to be at least
questionable. Consider the following example from Naumann and
Pinon (1997). 


* 24.

Rebecca was running across the minefield.



This does presumably not mean that when you are running across a
minefield, you normally will eventually get to the other side.



Note that these objections all make use of a particular informal
interpretation of the notion *inertia*. Judging the validity of
these objections to Dowty's theory in a precise way therefore requires
an explicit theory of this notion with exact empirical predictions.




Dowty analyzes the progressive as an intensional construction. Does
the progressive satisfy the standard philosophical tests for
intensional constructions, that is, the invalidity of substitution of
co-designative proper names and the impossibility of pulling the
existential quantifier out of the context created by the intensional
construction (see Bealer and Monnich (1989) for a more detailed
discussion)? Let us consider accomplishments, for example sentence
(25). 


* 25.

 Jackson Pollock was painting Ruth Rendel.



Given that Ruth Rendel and Barbara Vine are co-designative proper
names, sentence (25) implies that *Jackson Pollock was painting
Barbara Vine*. Therefore the first test fails. Co-designative
proper names are substitutable *salva veritate* in progressive
constructions. The second one, however, applies, since 


* 26.

 Carlos was building a house.



does not imply that there is a house that Carlos was building. These
tests therefore don't provide a clear answer to the question whether
the progressive is an intensional construction. Even if the answer is
yes, the progressive is certainly a different type of intensional
construction than, for example, propositional attitudes like
*believe* or *doubt*. 



An extensional alternative was developed by Parsons (1989, 1990).
Parsons, like Davidson (1967), assumes that first-order quantification
over events is possible. But unlike Davidson, Parsons assumes
quantification over eventualities in the sense of Bach (1986), which
includes events proper but also states, processes and so on. Like
Dowty, Parsons supposes that eventualities are split up into a
development phase and a culmination phase. In order to represent this
difference in first-order logic two new predicates are introduced,
*Cul*\((e, t)\) and *Hold*\((e,
t)\). The intuitive meaning of the first is that eventuality
\(e\) culminates at time \(t\). The second one says that
\(e\) is either an eventuality in its development phase or that
\(e\) is a state. Given these assumptions a sentence like *Mary
drew a circle* is translated into first-order formula (27),
\(S\) is again short for speech time. 


* 27.


\(\exists x(Circle(x)\,\wedge\) \(\exists e\exists I[I \lt S \wedge
Drawing(e)\ \wedge\) \(Subject(e, Mary)\ \wedge\) \(Object(e, x)\ \wedge\) 
\(Cul(e, I)])\)



The semantic effect of the progressive consists in transforming events
into states. 



If '\(A\)' is an event verb, then 'be
\(A\)-ing' is to be treated semantically as a state verb;
otherwise, 'be \(A\)-ing' is to be treated the same as
'\(A\)'. (Parsons 1989: 222) 



With this assumption the logical representation of *Mary was
drawing a circle* is (28). 


* 28.

 \(\exists x(Circle(x)\,\wedge\) \(\exists e\exists I[I \lt S \wedge
Drawing(e)\ \wedge\) \(Subject(e, Mary)\ \wedge\) \(Object(e, x)\ \wedge\) 
\(Hold(e, I)])\)



Obviously (27) does not follow from (28). But (28) does imply that
there is a circle Mary was drawing. Of course this circle may not be
complete. This means that Parson's ontology is bound to include
incomplete objects. For a discussion of incomplete objects see for
instance Baggio and van Lambalgen (2007).



According to the quotation above, the progressive doesn't change
anything in the case of activities. Therefore *Mary pushed a
cart* and *Mary was pushing a cart* are predicted to be
equivalent. 



Parsons' theory rests on the difference between the predicates

*Cul* and *Hold*. This difference is left to intuition.
No axiomatisation of these predicates is given. This is one of the
objections made in Zucchi (1999). Zucchi also notes that under certain
circumstances it is possible to derive a specific version of the
imperfective paradox in Parsons' theory. A careful discussion of
Dowty' and Parsons' approaches and a combination of the two systems is
contained in Landman (1992).


Another approach to the imperfective paradox and - more generally -
non-culminating accomplishments was proposed by Copley and Harley
(2015). They abandon event semantics and propose instead a framework
using situations and forces as primitives. Forces lead to transitions
between situations. Their theory is motivated by the observation that
across languages, non-culminating accomplishments are
crosslinguistically in fact a very common phenomenon. Interestingly,
these non-culminating readings are per default conveyed by less complex
expressions than complete events. Often, it is the perfective and not
the imperfective that is marked linguistically. Prima facie, this is a
problem for analyses such as Dowty's which assume that the imperfective
paradox in accomplishments hinges on an additional operator such as
PROG. This is where Copley and Harley's causal analysis comes into
play. Their force-theoretic account does not run into this problem
because causally efficacious forces always take effects ceteris
paribus, for instance, an effect may not occur do to the occurrence of a
counter force preventing the effect. They develop a compositional
semantic framework linking forces to the argument structure of the
sentence. At the present stage of the theory, however, some of the
basic concepts also call for axiomatisation. For instance, it is
crucial to gain a proper understanding of what it means for a situation
to be a ceteribus paribus successor situation for some initial
situation. For this purpose, their semantic framework must be connected
to a theory of causality connecting linguistic meaning to cognition and
causal reasoning via a conceptual level integrating the two.


We will come back to the imperfective
paradox in
 Section 7
 where we combine linguistic theory with event theories from artificial
 intelligence. This will give us a formalization of
 *ceteris paribus* from which a
 new solution to the imperfective paradox will follow.


### 6.1 Nominal Reference and Temporal Constitution


Let us now show that events are also useful for a systematic
account of certain types of coercion that were mentioned
above in Section 5.


Intuitively
there is a close link between the nominal predicates *wine* and *a bottle
of wine* and their verbal counterparts (29-a), (29-b).


* 29.

(a)
drink wine
* (b)
drink a bottle of wine


 
Whenever there are two entities to which *wine* applies, this
predicate applies to their collection as well. The predicate *a
bottle of wine* does not have this property. Whenever there are
two different entities to which *a bottle of wine* applies,
this predicate does not apply to their collection. The first predicate
is *cumulative* (see Definition 4) the second is
*quantized* (see Definition 5). A similar argumentation applies
to the difference between the activity *drink wine* and the
accomplishment *drink a bottle of wine*. It was observed by
many linguists that the combination of a verb like *drink* with
a cumulative predicate like *wine* yields an atelic predicate
like (29-a), whereas its combination with a quantized predicate yields
a telic predicate like in (29-b).



In order to account for these intuitions in a precise way Krifka assumes
that the parts of the structure \(\mathcal{M} = (\mathfrak{O},
\mathfrak{E}, \mathfrak{T})\) consisting of objects \(\mathfrak{O}\),
events \(\mathfrak{E}\) and times \(\mathfrak{T}\) are each structured
as complete join semi-lattices without a bottom element (see Gratzer,
1978). This means that we have a two place relation \(\sqcup\) (join)
and relations \(\sqsubseteq\), \(\sqsubset\), and \(\bigcirc\) (part,
proper part and overlap) that impose part-whole relations (mereologies,
see the entry mereology)
on the sets of objects, events and times. We cannot list all the
properties that must hold for these structures to serve as admissible
interpretations of natural language but we illustrate these properties
by giving two examples.


* \(\forall x, y \exists z(x \sqcup y = z)\) (completeness)
* \(\forall x, y (x \bigcirc y \leftrightarrow \exists z (z \sqsubseteq x \wedge z \sqsubseteq y))\) (overlap)

Given this algebraic background it is possible to state precise definitions of cumulative and quantized predicates \(P\).


**Definition 4.** *Cumulative (CUM) reference:*   
 
\(\forall P (CUM(P) \leftrightarrow \forall x, y(P(x) \wedge P(y) \rightarrow P(x \sqcup y)))\).




**Definition 5.** *Quantized (QUA) reference:*   
 
\(\forall P(QUA(P) \leftrightarrow \forall x, y (P(x) \wedge P(y) \rightarrow \neg y \sqsubset x))\).




Definitions 4 and 5 now allow for a formal descripion of the
referential difference between *wine* and *a bottle of
wine*.


* 30.

(a)
*wine* \(\subseteq \mathfrak{O} \wedge
CUM\)(*wine*)
* (b)
*a bottle of wine* \(\subseteq
\mathfrak{O} \wedge QUA\)(*a bottle of wine*)


For verbal predicates Krifka assumes that they are event denoting
and moreover that thematic relations
like *agent*, *theme*, etc are represented as in
(31-b). Thematic roles of verbs are crucial since they determine which
verbs give raise to the *drink wine* versus *drink a bottle
of wine*
 distinction.[5]


* 31.

(a)
Mary runs.
* (b)
\(run(e)\wedge Agent(e, Mary)\)


It
is now straightforward to fix the reference of the verbal predicates in
a way that exhibits their simlarity to the respective nominal
predicates.


* 32.

(a)
*drink wine* \(\subseteq \mathfrak{E} 
 \wedge CUM\)(*drink wine*)
* (b)
*drink a bottle of wine* \(\subseteq
\mathfrak{E} \wedge QUA\)(*drink a bottle of
wine*)



Cumulativity of nominal predicates corresponds to atelicity in the
verbal domain and a quantized nominal predicate corresponds to a telic
verbal predicate.



There is still something missing. How can the impact of the reference
of the nominal predicates on the aspectual class be derived formally?
We will present only the basic idea of Krifka's construction here. Let
\(w\) be the predicate corresponding to *wine* and let \(e\) be the event predicate representing *drink wine*. The basic idea is that an object is changed by the influence of such an event in a *gradual*
manner. Since we know that \(w\) is cumulative this predicate also
applies to proper parts of \(w\), say \(w'\). Then given that *drink  wine*
can be applied to \(e\) it should also be possible to apply it to a
proper part of \(e\), say \(e'\). Quantized predicates show a different
behaviour. Since *a bottle of wine* is quantized no proper part of it is *a bottle of wine* and therefore no proper part of \(e\) can be described as *drink a bottle of wine*.
To account for this intuitive explanation in formal terms it is
necessary to specify certain preservation properties for thematic roles
since these roles relate the algebraic structure of events and the
algebraic structure of objects. This means that notions like *summativity*, *uniqueness of objects* etc. have to be defined for relations between events and objects; i.e. thematic roles.



Here we will illustrate these concepts with only one example, *summativity (SUM)*.



**Definition 6.**   

\(\forall R [SUM(R) \leftrightarrow \forall e, e', x, x'[R(e, x)
\wedge R(e', x') \rightarrow R(e\sqcup e', x \sqcup
x')]]\).




Summativity says that the sum-operation \(\sqcup\) on
events and objects is preserved under summative relations on events and
objects. For instance if we have two distinct events of *drinking a bottle of wine* we get an event of *drinking two bottles of wine*.



With this formal background it is now possible to state exactly when a
predicate of events corresponding to a verb-phrase is cumulative, telic
etc.. Again we will just state one result. Let \(\phi = \{ e|\exists
x(\alpha(e) \wedge \delta(x) \wedge \theta(e, x))\}\) be such a
predicate where \(\alpha\) represents a verb - say *draw* - \(\delta\) the nominal predicate - *pictures* - and \(\theta\) a thematic relation - say *theme*.
It then follows that one set of conditions for \(\phi\) being
cumulative is that \(\delta\) is cumulative and \(\theta\) is
summative. An example is the verb phrase *draw pictures*. Similar results characterize, *iterative*, *telic*, *atelic* etc. readings.



This concludes our glimpse of events in linguistic semantics; Readers interested in a more comprehensive treatment of this issue may find Champollion (2014, see *Other Internet Resources*) a useful contemporary source. We now come back to the imperfective paradox and turn to an
event calculus that was developed in artificial intelligence.


## 7. Event Calculus



In the following section we will show that the imperfective paradox is
an instance of the frame problem prominent in Artificial Intelligence
(AI). To this end we will explain the event calculus, which is an
extension of McCarthy's situation calculus (McCarthy 1977) developed
by Kowalski and Sergot (1986). Then we will indicate how the event
calculus leads to a formalization of natural language tense and aspect
by analysing the imperfective paradox. 



Consider narrative (33). 


* 33.

Yesterday morning Vincent loaded the gun. Then he smoked a
cigarette. After smoking he pointed the gun at Fred and pulled the
trigger.



Assuming that shooting at somebody with a loaded gun will lead to the
death of that person, we will interpret the discourse in a way that
Fred eventually gets killed. Arriving at this inference is, however,
not a trivial task. In (33) the frame problem is exemplified in the
form of the Yale Shooting Scenario (Hanks and McDermott 1986). To
demonstrate this, we will start with a very simple calculus--a
variant of the *Simple Event Calculus* by Shanahan (1997) which
only incorporates a very basic notion of instantaneous change. Later
we will add axioms in order to deal with continuous change and then
return to the imperfective paradox. In short, we will argue that the
paradox can be viewed as an instance of the frame problem. 


### 7.1 The Yale Shooting Scenario



Suppose we have three kinds of actions/events--*Load*,
*Smoke* and *Shoot*--and three time dependent
properties, the fluents *Alive*, *Loaded* and
 *Dead*.[6]
 Besides actions and fluents we will also need instants of time in our
basic ontology. The predicates listed in Table 4 are used to express
actions and their effects and to locate them in time. 





| *Formula* | *Meaning* |
| --- | --- |
| *Initiates*\((\alpha , \beta , \tau)\) | Fluent \(\beta\) starts to hold after action \(\alpha\) at time
\(\tau\) |
| *Terminates*\((\alpha , \beta , \tau)\) | Fluent \(\beta\) ceases to hold after action \(\alpha\) at time
\(\tau\) |
| *Initially*\((\beta)\) | Fluent \(\beta\) holds from time 0  |
| \(\tau\_1 \lt \tau\_2\) | Time instant \(\tau\_1\) is before time instant
\(\tau\_2\)  |
| *Happens*\((\alpha , \tau)\) | Action \(\alpha\) happens at time \(\tau\)  |
| *HoldsAt*\((\beta , \tau)\) | Fluent \(\beta\) holds at time \(\tau\)  |
| *Clipped*\((\tau\_1,\beta ,\tau\_2)\) | Fluent \(\beta\) is terminated between times \(\tau\_1\) and
\(\tau\_2\)  |


Table 4.
Predicates of the simple event calculus (from Shanahan 1999)




The predicates are related to each other by a set of four simple
axioms (34)-(37). In these axioms, all variables are assumed to be universally
quantified with maximal scope. We will come to the semantics of the
implication when we have introduced the complete scenario. 


* 34.

 \(Initially(f) \rightarrow HoldsAt(f, 0)\)
* 35.

 \(Initially(f) \wedge \neg Clipped(0, f, t) \rightarrow HoldsAt(f, t)\)
* 36.

 \(Happens(a, t\_1) \wedge Initiates(a, f, t\_1) \wedge t\_1 \lt t\_2\ \wedge\)
 \(\neg Clipped(t\_1, f, t\_2) \rightarrow HoldsAt(f, t\_2)\)
* 37.

 \(Happens(a, t) \wedge t\_1 \lt t \lt t\_2 \wedge Terminates(a, f, t) \rightarrow\)
 \(Clipped(t\_1, f, t\_2)\)



 The first three axioms state the conditions
under which a fluent can hold at a time \(t\): either it holds
right from the start or it is initiated at a time \(t\_1\)
before \(t\) without a terminating action occurring in between the
two. The fourth axiom defines the predicate *Clipped*\((t\_1,
f,
t\_2)\) saying that a fluent \(f\) is clipped
between times \(t\_1\) and \(t\_2\) if it is
terminated by some action \(a\) happening between
\(t\_1\) and \(t\_2\). Simplifying a whole
lot, the axioms (30)-(33) formalize the notion of instantaneous
change, such as two balls colliding or somebody dying. Moreover, they
embody a notion of inertia: fluents continue to hold unless
terminated.






Next, we have to translate the discourse (33) into the Simple Event
Calculus. The first set of formulas (38)-(40) states what the
mentioned actions do. These formulas provide a crude sketch of the
lexical meaning of *load*, *smoke* and *shoot*.
The effect of a *Load* action is to make the fluent
*Loaded* hold, a *Shoot* action makes *Dead* hold
(and *Alive* not hold) with *Loaded* being a
precondition. Finally, *Smoke* is assumed to have no effects
(thus, there are no rules involving *Smoke*). 


* 38.

\(Initiates(Load, Loaded, t)\)
* 39.

\(HoldsAt(Loaded, t) \rightarrow Initiates(Shoot, Dead, t)\)
* 40.


\(HoldsAt(Loaded, t) \rightarrow Terminates(Shoot, Alive, t)\)



(33) comprises a *Load* action followed by a *Smoke*
action followed by a *Shoot* action. Using four arbitrarily
chosen time points \(T\_1\)-\(T\_4\) the
discourse can be represented as follows: 


* 41.

\(Initially(Alive)\)
* 42.

\(Happens(Load, T\_1)\)
* 43.


\(Happens(Smoke, T\_2)\)
* 44.

\(Happens(Shoot, T\_3)\)
* 45.

\(T\_1 \lt T\_2 \lt T\_3 \lt T\_4\)




From this little theory we want to derive
*HoldsAt*(*Dead*, \(T\_4)\). Unfortunately,
this sequent is not valid in the classical sense of being true in all
models which satisfy the theory. Think for example of a situation in
which the gun obscurely becomes unloaded while Vincent is smoking. In
such a model, the precondition of the shooting action is clearly not
met and thus the action will lack any effects. Note that nothing rules
out such a model since it is entirely consistent with our theory. We
thus have to further constrain the models under consideration.
Crucially, we have to find a way to deal with the non-effects of
actions; this is the famous *frame problem*. 



How to proceed? From the fact that no terminating action was
mentioned, we want to conclude that no terminating event occurred. To
do so, we have to strengthen the assumptions of the theory in a way
that only those events are assumed to occur which have been explicitly
stated in the discourse. The discourse model should be minimal,
linking discourse understanding intimately to *closed world
reasoning*. It is important to note that this strategy forces
reasoning to be non-monotonic: adding further premises to a theory can
make inferences invalid that were valid before (see the entry on 
non-monotonic logic). 


### 7.2 The completion of a program



There are different techniques for formalizing this line of reasoning;
one is circumscription (e.g., Shanahan 1997). Here, we will use the
*completion of a logic program* technique, adopting the

*Event Calculus* of van Lambalgen and Hamm (2005). Taking a
closer look at the formulas of the theory
\(((34) \wedge \ldots \wedge (45))\), an important feature to notice is that
formulas come in two variants: they are either *facts*
consisting of a simple predicate expression or they express
*rules* with a conjunction of potentially negated formulas to
the left of the implication sign and a single positive atomic
expression to its right (in logic programming the former is called the

*body* and the latter is called the *head* of a clause).
We will now illustrate how the completion of a simple logic program is
computed. Consider the description of a situation where the gun gets
loaded at time 1 and a shooting event happens at time 10. This
situation is stated in the following program: 


* 46.

\(Happens(Load, 1)\)
* 47.

\(Happens(Shoot, 10)\)



The uncompleted program does not yet rule out intervening events. It
is, for example, consistent with an *Unload* event occurring at
time instant 9. The completion of the program should tell us that (46)
and (47) were the only events. The completion is computed according to
the following
 procedure.[7]
 We start with the facts in (46) and (47). Both of these are
*Happens* formulae. The completion of the program intuitively
corresponds to an assertion that, given this program,
*Happens*\((e, t)\) can only mean *Load*
occurring at 1 or *Shoot* occurring at 10. We therefore first
substitute variables for the constants and write the following
disjunction: 


* 48.


 \((e = Load \wedge t = 1) \vee(e = Shoot \wedge t = 10)\)



Then, we universally quantify over the variables \(e\) and \(t\)
and strengthen the implication to a bi-implication: 


* 49.

 \(\forall e\forall t(Happens(e, t)\ \leftrightarrow\) \((e = Load
\wedge t = 1)\ \vee\) \((e = Shoot \wedge t = 10))\)



From (49) it follows that there were no intervening events.
Strengthening the implications to bi-implications makes it impossible
that a head can be true without the enabling conditions being
 met.[8]
 The uniquely determined model of (49) is the minimal model of the
logic program consisting of (46) and (47). 



We can now come back to the Yale Shooting Problem in discourse (33).
From the completion of the logic program
((34))\(\wedge \ldots \wedge\)((45)) it follows (non-monotonically) that
*HoldsAt*(*Dead*, \(T\_4)\). Enriching the
program with further information and computing the new completion,
however, could clearly lead to cancellation of this inference. 


### 7.3 Introducing tense information



If we want to use these ideas to model natural language discourse, we
need to incorporate tense. In van Lambalgen and Hamm (2005) this is
done by adopting the classic tripartition into *reference
time*, *event time* and *speech time* due to
Reichenbach (see
 section 3).
 



Recall that above, when we translated discourse (33) into Shanahan's
Event Calculus, we arbitrarily chose instants of time
\(T\_1 ,\ldots ,T\_4\) to locate actions
in time. This is not how it is done in natural language discourse. The
first sentence (*Yesterday morning Vincent loaded the gun*)
states that within some time interval \((=\) reference time) before now \((=\)
speech time) there was a time \(t'\) at which a loading event
happened \((=\) event time). The second sentence (*Then he smoked a
cigarette*) is linked to this event by the connective
*then*, stating that at some later time \(t''\) a
smoking event happened with \(t''\) also being temporally
located before *now*. Analogously for the third sentence: there
is an event time \(t'''\) at which *Shoot* happens and
\(t'' \lt t''' \lt\)
*now*. Note that when locating events in time it is
always done by existential statements of the form *there is a
time*. So far, however, we have only been dealing with universally
quantified variables. Clearly, a formula like \(\forall t\).*Happens*(*Load*,
\(t)
\wedge t \lt\) *now* cannot be used to represent
tense since it would state that *Load* happens at all times in
the past. We have to find a way to introduce existentially quantified
information. 



Intuitively, it suffices to introduce new entities into the domain of
discourse. This approach lies at the heart of dynamic semantic
theories such as Discourse Representation Theory (Kamp and Reyle 1993,
and see also the entry on
 Discourse Representation Theory)
 or File Change Semantics (Heim 1982). We will need an update
procedure that introduces new actions/events into the discourse model
but is compatible with the non-monotonic semantics introduced so far.
Following van Lambalgen and Hamm (2005) we will use *integrity
constraints*--a device from database theory (for a more
complete and formal treatment the reader is referred to Kowalski 1995
and van Lambalgen and Hamm 2005: ch. 8)--to implement the notion
of a minimal update of a discourse model. Consider discourse (50).



* 50.

 The gun is loaded. Vincent is ready to shoot.



In interpreting the first sentence, we want to update the situation
model in a way that makes the sentence true. The tense information
will be interpreted as a goal to minimally update the situation model
with a fluent *Loaded* that holds now. In our framework
semantic interpretation is thus closely linked to planning, in that
finding an interpretation for a sentences requires coming up with a
sequence of actions that makes the goal succeed. 


* 51.

 Make the query \(?HoldsAt(Loaded, now)\) succeed.




To resolve this integrity constraint we will use the axioms of the
Simple Event Calculus plus the world knowledge about the consequences
of a load action stated in (52). 


* 52.

 \(Initiates(Load, Loaded, t)\)




In logic programming, the sort of reasoning required to resolve (51)
is carried out by a derivation procedure called *resolution*.
We will use resolution here in a non-standard way, i.e. logic
programming combined with integrity constraints (for details see van
Lambalgen and Hamm 2005). It starts with the formula that has to be
made true in the discourse model. Resolution proceeds by identifying
rules which have the query as their consequent and substituting the
consequent with the antecedent conditions of the rule, making the
antecedent formulas new queries themselves. The resolution stops when
the query cannot be further resolved, that is, when a plan has been
computed whose preconditions are all fulfilled given an appropriate
update of the discourse model. To get an impression of how this works
we will illustrate the resolution of (51). First, we have to compute
the completion of the program and whenever possible substitute the
variables with constants, in our case *Loaded* and

*now*. In logic programming this substitution is done
automatically via unification. Here is the completion of the
*HoldsAt* predicate which forms the head of two
axioms--(35) and (36). 


* 53.


 \(HoldsAt(Loaded, now)\,\leftrightarrow\) 
 \((Initially(Loaded) \wedge \neg Clipped(0, Loaded, now))\,\vee\)
 \((Happens(a, s) \wedge Initiates(a, Loaded, s) \wedge s \lt now \wedge\)
 \(\neg Clipped(s, Loaded, now))\)




Since the theory doesn't contain the statement
*Initially*(*Loaded*), closed world reasoning yields
that at time 0 the gun is not loaded
\((\neg\)*Initially*(*Loaded*)). No matter how we update
the discourse representation the first disjunct can never succeed. We
thus have to move on to the second disjunct. In this case, the
database is searched for an action \(a\) and a time \(s\) such
that *Initiates*\((a, f, s)\),

*Happens*\((a, s)\) and
\(\neg\)*Clipped*\((s\), *Loaded*, \(t)\). On the
sole basis of the information provided in the discourse the subquery
?*Happens*\((a, s)\) will fail, because in discourse
(50) there is no information about a load action. However, since
integrity constraints are intended to be made true, the database will
be updated with a clause *Happens*(*Load*,
\(s) \wedge s \lt\)

*now* which makes the second disjunct true and the query
succeed. 


### 7.4 The imperfective paradox reconsidered



In the following, we will analyze the imperfective paradox as an
instance of the frame problem (cf. Stenning and van Lambalgen 2005;
van Lambalgen and Hamm 2005; Stenning and van Lambalgen 2008; Baggio
and van Lambalgen 2007; Baggio et al. 2008). Both activities and
accomplishments involve continuous change and we therefore have to add
additional predicates and axioms to properly deal with gradually
changing objects which are under the influence of an external force.
Table 5 introduces two new predicates; axioms A1-A5 provide a
general theory of instantaneous and continuous change (from van
Lambalgen and Hamm 2005: 40). As can be easily seen, the axiom system
extends that of Shanahan (1997). 





| *Formula* | *Meaning* |
| --- | --- |
| *Releases*\((e, f, t)\) | Fluent \(f\) starts to change due to event \(e\) at time
\(t\) |
| *Trajectory*\((f\_1, t,
f\_2, d)\) | If fluent \(f\_1\) holds from t until \(t + d\), then \(f\_2\) holds at \(t + d\)  |


Table 5.

Predicates for continuous change




\[\begin{align}
\tag\*{A1.}
 & Initially(f) \rightarrow HoldsAt(f, 0) \\
\tag\*{A2.}
 & HoldsAt(f, r) \wedge r \lt t\ \wedge \\
 &\quad \neg \exists s \lt r HoldsAt(f,s) \wedge \neg Clipped(r, f, t) \rightarrow HoldsAt(f, t) \\
\tag\*{A3.}
 & Happens(e, t) \wedge Initiates(e, f, t) \wedge t \lt t'\ \wedge \\
 &\quad \neg Clipped(t, f, t') \rightarrow HoldsAt(f, t') \\
\tag\*{A4.}
 & Happens(e, t) \wedge Initiates(e, f\_1, t) \wedge t \lt t'\ \wedge \\
 &\quad t' = t + d \wedge Trajectory(f\_1, t, f\_2, d) \wedge \neg Clipped(t,
f\_1, t') \rightarrow HoldsAt(f\_2, t') \\
\tag\*{A5.}
 & Happens(e, s) \wedge t \lt s \lt t'\ \wedge \\
 &\quad (Terminates(e, f, s) \vee Releases(e, f, s)) \rightarrow Clipped(t, f, t') 
\end{align}\]



Axioms 4 and 5 define continuous change. Axiom 4 defines the
*Trajectory* predicate. To see what it says, let's consider a
situation of *running a mile* where \(f\_1\) is
instantiated by *running* and \(f\_2\) by
*distance*\((x)\). Should *running* be true during
the whole interval from \(t\) until \(t'\) then *distance*\((a + 1)\) will be true
at \(t'\). The value of \(x\), that is, the actual
distance actually run at each time instant will be determined by the
law of the process under consideration (here, running speed). Axiom 5
defines the predicate *Clipped*\((t,
f, t')\)
which is true if \(f\) is terminated or released by an event
\(e\) which happens in the time interval between \(t\) and
\(t'\). Thus *Clipped* covers both instantaneous and
continuous change. The axioms provide a general theory of what can
change and what stays constant.



For concrete situations, we also need the specific temporal and causal
relationships. This kind of information is specified in so called
*scenarios* representing lexical meaning. The lexical meaning
of the accomplishment *run a mile* corresponds to scenario
(54). Like the axioms, the scenario takes the form of a logic program.
All variables are universally quantified. 


* 54.

(a)

\(Initially(distance(a))\)
* (b)

\(Initiates(start, run, t)\)
* (c)

\(Releases(start, distance(x))\)
* (d)

\(Initiates(finish, distance(a + 1), t)\)
* (e)

\(Terminates(finish, run, t)\)
* (f)

\(HoldsAt(run, t) \wedge HoldsAt(distance(a + 1), t) \rightarrow\)
 \(Happens(finish, t)\)
* (g)

\(HoldsAt(distance(x), t) \rightarrow \)
 \(Trajectory(run, t, distance(x + g(d)), d)\)



Scenarios formalize the event nucleus of Moens and Steedman (1988).
Every accomplishment takes the same form of scenario, the only
differences being that the individual scenarios involve different
preparatory processes (*run*, *build*, etc.), incremental
themes (*distance*\((x)\), *house*\((x)\), etc.)
and resultant states. Furthermore, it is obvious that the preparatory
process *run*, an activity, is a proper part of scenario (54).
The simpler activity scenario can be easily arrived at if we remove
clauses (d)-(f) from (54). We are now in the position to solve
the imperfective paradox avoiding the problems discussed in
 section 6.
 Both a past progressive activity sentence as in (55a) and a past
progressive accomplishment sentence (55b) will trigger a discourse
update as stated in integrity constraint (55c). 


* 55.

(a)

John was running.
* (b)

John was running a mile.
* (c)

\(?HoldsAt(run, t) \wedge t \lt now\) succeeds.



The discourse update of the completed program (54) yields that at some
point after \(t\) John will achieve the culmination and will have
run one mile (see van Lambalgen and Hamm 2005: 61ff. for a proof).
Obviously, the same integrity constraint in combination with the
activity in (55a) only licenses the inferences that there was some
past running activity. In the case of an accomplishment the course of
events dramatically changes if we add information to the
representation as in (56b). Now, both (56a) and (56b) only allow the
inference that there was a running event but, arguably, John didn't
reach his goal in (56b). Without going into further details it should
be clear how this non-monotonic inference works. The derivation works
analogously to the simpler instances of the frame problem discussed
above. 


* 56.

(a)

John was running, when he was hit by a truck.
* (b)


John was running a mile, when he was hit by a truck.



The solution to the imperfective paradox just outlined makes reference
to the intentions of the agent involved in the event; therefore the
notion of a goal or intention is built into the system right from the
start. Note, however, it does not involve possible worlds, or
primitive notions like inertia worlds; instead it is based upon
minimal models and non-monotonic reasoning. This yields additional
benefits. Firstly, the Event Calculus will compute a minimal model in
cases where an integrity constraint can be satisfied. Moreover, the
number of construction steps required for this yields a precise
complexity measure which (for example) allows us to derive predictions
for cognitive processing. Secondly, the non-monotonic nature of the
Event Calculus can be used to model the incremental construction of a
temporal model with inferences which hold locally at a discourse
segment \(s\_n\) but which can be undone at some
later discourse unit \(s\_{n+1}\). This is a
necessary prerequisite if we think of interpreting sentences like
(56b) from left to right assuming incremental interpretation (see
e.g., Baggio and van Lambalgen 2007 and Bott 2010 for implementations
of the Event Calculus as a processing model). In the next section we
will review psychological and psycholinguistic work on the
interpretation of aspect which will provide empirical motivation for
the outlined analysis. 


## 8. Cognitive aspects



Due to limitations of space, we will not discuss psychological and
psycholinguistic studies of tense but will focus solely on processing
studies of aspect. Moreover, within this topic we limit ourselves to
studies of the adult system. The interested reader is referred to
Dickey (2001) and the references therein for psycholinguistic work on
tense as well as to the entry on
 the experience and perception of time

 and the contributions in Grondin (2008) for psychology of time in
general. 



Psychologists have recently begun studying event perception and its
neural basis (e.g., Zacks and Tversky 2001; Zacks, Tversky, and Iyer
2001; Zacks et al. 2006). The underlying question behind this line of
research is whether events play a role in how we cognitively structure
and remember changing states of affairs and how they are represented.
Zacks, Tversky, and Iyer (2001) use an analogy from the domain of
objects to the domain of events to define an event as a "segment
in time at a given location that is conceived by an observer to have a
beginning and an end" (p.30). They claim that "the ability
to identify the parts of events and their relationships constitutes a
distinct perceptual process" (p.30) which they call *event
structure perception*. Just as an object is an ontological
primitive in the spatial domain, so an event is an ontological
primitive in the spatio-temporal
 domain.[9]
 Objects are recognized by shape, color etc. and have boundaries in
space. Analogously, events have boundaries in time, but are also
bounded in space. An event of buttering toast, for instance, happens
at a particular time, but also in a particular location in space and
is therefore spatially bounded. Continuing the analogy, both objects
and events can be identified and categorized using hierarchical
relations. 




In Zacks, Tversky, and Iyer's experiments, participants segmented an
ongoing activity (like *washing dishes*) while watching it on
film by pressing a key to mark "natural and meaningful"
unit boundaries. The grain at which participants segmented the
activity was manipulated between subjects: one group was asked to mark
the largest meaningful units (coarse grain size), the other group the
smallest units (fine grain size) which still could be considered to be
complete events. The placement of perceived event boundaries provides
information about the psychological status of events and their
partonomic hierarchy. If the stream of action is perceived as
consisting of discrete events, participants should place event
boundaries consistently and this is what Zacks, Tversky, and Iyer
(2001) found. Further, if participants make use of partonomic
hierarchies, coarse event boundaries should be aligned with fine
boundaries, since the end of each superordinate event is also the end
of its last subevent. Again, this hypothesis was corroborated by a
strong hierarchical bias effect. Events thus seem to be
psychologically real and to be hierarchically structured. 


### 8.1 Lexical Aspect



During the last decade psycholinguists have started to investigate the
cognitive reality of lexical aspect and how it is processed. The
existing studies can be sub-classified into studies concerning
*Aktionsart simpliciter* and those concerned with shifts from
one aspectual class to an other, i.e., *aspectual coercion*. In
the following, we will provide a brief overview of both areas of
research. 



McKoon and Macfarland (2002) were among the first to study processing
consequences of decompositional analyses along the lines of Dowty
(1979). They provided evidence from reading times and lexical decision
times that accomplishments \((\alpha\) CAUSE \(x\) BECOME IN-STATE)
are inherently more complex than achievements \((x\) BECOME
IN-STATE). Similarly, Gennari and Poeppel (2003) compared eventive
predicates (which included accomplishments, achievements and
activities) to statives and found that the former were more complex to
process than the latter. Brennan and Pylkkanen (2010) extended
this line of research to psychological verbs and compared
accomplishments (e.g., *scare*) with statives (e.g.,

*cherish*) using reading time methods and
magnetoencephalography (MEG). They also included a comparison of the
simple psychological statives with modified sentences that required
coercion (e.g., *within half an hour, the child cherished the
precious kitten*). Reading times and MEG data indicated that
accomplishments were more complex to comprehend than statives and,
moreover, that the enhanced semantic complexity led to different MEG
components than aspectual coercion. Finally, Coll-Florit and Gennari
(2011) compared durative states and punctual event predicates (mostly
achievements) and observed longer reading times of statives than of
event predicates. They attributed this effect--which runs counter
to what would be expected under a decomposition analysis--to the
fact that durative situations occur in semantically more diverse
contexts and elicit more diverse associations than event predicates
which may modulate the required processing effort. Taken together, the
existing studies provide evidence for complexity differences between
the aspectual classes, lending support to decompositional analyses.
However, not all issues are entirely solved yet and further research
is required. 



The second line of research concerns aspectual coercion. Here, the
main research question has been whether aspectual coercion is a costly
operation. Existing research has almost exclusively focused on one
coercion type, i.e., coercion of point action verbs into an iterative
interpretation (but see Brennan and Pylkkanen 2010 and Bott 2010
for exceptions). Again, the findings are mixed. Early studies have
provided evidence for coercion costs employing secondary tasks such as
*cross-modal lexical decision* or *stop making sense
judgments* (Pinango, Zurif, and Jackendoff 1999; Todorova
et al. 2000). Pickering et al. (2006) used the same materials as in
the experiments mentioned above, but tested a coerced meaning during
ordinary reading without an additional task. In two self-paced reading
and two eyetracking experiments, they found aspectual coercion to be
no more difficult than their aspectual control conditions. This lack
of effect let them propose the aspectual underspecification
hypothesis, stating that the aspectual representation stays
underspecified during normal reading. Brennan and Pylkkanen
(2008) challenged this view and reported a coercion effect of coercion
sentences like (57a) as compared to aspectual controls (57b) both in
self-paced reading and in MEG; on the basis of a rating study they had
carefully selected clear instances of point action verbs. Their MEG
study revealed activation in the anterior midline field, a MEG
component that has been observed for other non-aspectual cases of
coercion, too. 


* 57.

(a)

Throughout the day, the student *sneezed* in the back of the classroom.
* (b)

After twenty minutes, the student *sneezed* in the back of the classroom.



Complicating matters, Bott (2008, 2010) applied the same norming
procedures, but did not find evidence of coercion cost in iterative
point action verbs in German. With other types of aspectual coercion,
however, there were clear indications of processing difficulty
indicating that the underlying processes differ between coercion
types. Furthermore, the study provided evidence that at least in some
cases (achievement ~ accomplishment) coercion operations are not
triggered by an aspectual mismatch, but can proceed smoothly by
enriching the aspectual representation with additional eventualities.
This was shown in an event-related potentials (ERP) study in which a
double dissociation of ERP components was elicited by sentences with
aspectual coercion as compared to sentences that contained an
aspectual mismatch.


### 8.2 Grammatical Aspect



Grammatical aspect has recently received increased interest in
psycholinguistics (for a comprehensive review see Madden and Ferretti
2009). One line of studies has investigated (e.g., Ferretti
et al. 2007) the accessibility of event participants in English
progressive and simple sentences. They provide evidence that, in line
with the linguistic description provided above, the progressive
presents an event from the inside, making participants, instruments
and places fully accessible, whereas the simple forms present events
as complete units and the event participants are less accessible.
Using picture selection/verification, Madden and Zwaan (2003) found
that participants were faster and more likely to choose a picture
showing a complete event, rather than a picture depicting an ongoing
event, after they had read a sentence with an accomplishment verb in
the simple past. This indicates that English speakers encode
accomplishments in the simple past as complete events. This finding
was supported by a study by Anderson et al. (2008). They compared
accomplishments describing a path such as *Tom jogged/was jogging
to the woods and then stretched when he got there*. Participants
listened to these sentences while using a computer mouse to drag and
drop a human character in a visual scene. When hearing a sentence in
the past progressive, many drops took place at the beginning and the
center of the path, whereas in the simple past most drops were at the
end of the path. Interestingly, the differences were by no means
categorical. Even in the simple past condition, the character was in
some trials positioned well before the end of the path. This indicates
that accomplishments in the simple past are consistent with incomplete
events. 



Another study relevant for the discussion in the previous sections is
an ERP study on the imperfective paradox by Baggio et al. (2008). They
investigated the processing of Dutch imperfective sentences such as
*het meisje was een brief aan het schrijven* (*the girl was
writing a letter*) which were either followed by a

*when*-clause that made the culmination unlikely (when her
friend spilled coffee on the paper) or by a sentence which was
compatible with the attainment of the goal (when her friend spilled
coffee on the tablecloth). The first kind of sentences elicited larger
sustained anterior negativities compared to sentences that were
compatible with the culmination. Moreover, the amplitude of the
negativity was correlated with the frequency with which participants
responded that the culmination was not reached. These findings
indicate that a progressive sentence triggers a default inference to a
complete event which can be canceled again, if the context so
requires. 



Grammatical aspect is subject to clear crosslinguistic differences.
This raises the question of whether the grammatical system has an
influence on how we process language. In von Stutterheim et al. (2009)
the authors reported findings of a production study comparing event
descriptions of English, German and Dutch speakers that were elicited
while they were watching and describing a silent video clip. Their
study revealed clear differences between productions from the three
languages. English speakers used the progressive to start event
descriptions well before the endpoint was visible (e.g., *A car is
going down a lane ... to a farmhouse*). Germans showed a
different behavior, relating their descriptions to the endpoint of a
motion event. It thus seems as if production is constrained by the
language in which it is realized. Recently, von Stutterheim et al.
(2012) extended this line of research to a sample of seven languages
and showed that the aspectual properties of a language influence how
speakers conceptualize events. It's not only that speakers of
different languages talked differently about motion events (e.g.,
mention of endpoints), their language also had a clear influence on
their viewing behavior while watching the movies and influenced their
memory capacity for those parts of the scene corresponding to
endpoints. 


As for comprehension, Bott and Hamm (2014)
investigated how the aspectual system of a language influences
processing difficulty. They compared coercion of German and English
accomplishments which were modified by *for \(x\) time*
adverbials. (58) is a sample item. 


* 58.

(a)

*Der Architekt errichtete das Monument zwei Jahre lang, aber dann wurde ihm der Kredit gekundigt.*
* (b)


The architect built the monument for two years, but then the bank canceled his credit.



In the German experiments, condition (58a) was read as fast as a
non-coercing control condition with an *in x time* adverbial.
This was different in English where the authors found coercion to
cause a substantial slowdown of reading speed. They interpret this
crosslinguistic difference as follows: languages which have the
grammatical means to express an aspectual difference via alternative
forms--the progressive vs. simple forms in English--enforce
immediate aspectual commitment, whereas languages that lack this
grammatical feature have to leave it underspecified. In other words,
German readers will leave it to the context whether an accomplishment
expresses a complete or incomplete event, whereas English readers
immediately strengthen an accomplishment in the simple form into a
complete event interpretation. They take this strengthening to be a
pragmatic process that is due to competition between alternative
grammatical forms.


In an eyetracking during reading study, Bott and Gattnar (2015)
followed up on the observed cross-linguistic variation in the
processing of grammatical aspect comparing an aspect language, Russian,
with the non-aspect language German. They investigated whether the
differences in the aspectual systems of the two languages affected the
time course of mismatch detection in the case of lexical aspectual
mismatch.All test sentences contained clear instances of (in Russian always perfective) transitive achievement verbs such as *win*, *find*, *reach*, *notice*, etc. which were modified by a semantically mismatching *for*-adverbial (e.g., *for three hours*) or a semantically fitting *ago*-adverbial (e.g., *three hours ago*)
in aspectual control conditions. Crucially, both Russian and German
have relatively free word order making it possible to test for effects
of mismatching vs. matchingadverbials in the word order *subject verb object adverbial* but also in the order *adverbial verb subject object*. The results show that Russian readers immediately detect the aspectual mismatch
irrespective of whether they have seen the subject or the object
argument yet. This is different in German where aspectual mismatch
detection only started after the verb plus all its obligary arguments
had been encountered. This result was modeled by Bott and Sternefeld
(2017) in an incremental event semantics drawing on insights from
mereological semantics as proposed by Krifka (1989, 1992); see
 also
 section 6.1 (and the entry
 mereology).
The incremental semantic derivations show that in a non-aspect language
like German, lexical aspect really is a property of whole sentences and
strongly depends on the compositional interpretation of the verb, its
arguments, and their thematic roles. In an aspect language like
Russian, on the other hand, the grammatical aspect imposes itself
strong constraints on lexical aspectual interpretation. These
psycholinguistic results thus nicely fit semantic proposals for
grammatical aspect that have been proposed in the framework of
mereological semantics (see, e.g., Filip 2008, a.o.).



To summarize, the existing studies lend empirical support to the
aspectual distinctions made in the semantic and philosophic
literature. Moreover, the grammatical system of a language influences
the way language is processed and may even influence the way we
experience the world. 


## 9. Conclusion



We introduced and discussed several important semantic, philosophical
and technical concepts and theories of temporality and at least
indicated how these concepts and theories are related to cognition.
However, we did this exclusively from a sentence internal perspective.
Tense and aspect are important means for organizing discourse as
 well.[10]
 To conclude our overview, we present French examples due to Kamp and
Rohrer (see van Lambalgen and Hamm 2005, ch. 9) and indicate the role
tense and aspect play in discourse organization. Consider sentence
(59) in which the *passe simple* (PS) occurs four times.




* 59.

*Pierre se leva, monta dans sa chambre, ferma la porte et alluma la
radio.*
  

Pierre got up, went up to his room, closed the door, and switched
on the radio.



First the PS in (59) provides temporal information; all events are
located in the past. But the PS conveys aspectual information too. It
says that the internal constitution of the events is not important and
this means that PS expresses perfectivity. PS imposes a view of the
events "from the outside". This is then taken to explain
why multiple uses of the PS imply a succession of the events described
as witnessed in example (59). This means that Pierre getting up
precedes his going up to his room and this event precedes closing the
door which finally precedes turning the radio on. Thus, the PS
structures the above little discourse in a particular way. Early
discourse representation theory explains this effect by assuming that
the PS introduces a new reference point after a previously introduced
one.



The discourse function of the French imparfait is different. Example
(60a) shows that, like the past in English, the
 imparfait[11]
 is not felicitous when uttered out of the blue. 


* 60.

(a)

\**Il faisait chaud.*
  

It was hot.
* (b)

*Il faisait chaud. Jean ota sa veste.*
  

 It was hot. Jean took off his sweater.



Kamp explains these facts in the following way. The imparfait does not
introduce its own reference time and in (60a) there is no previous
reference point given. Therefore (60a) is out. In (60b) the sentence
in the passe simple *Jean ota sa veste* introduces
a reference time that can be used to anchor *If faisait chaud*.
The imperfective aspect of the imparfait is then explained by the fact
that the PS event happens while the sentence in the imparfait holds.
Thus passe simple and imparfait not only provide temporal
information but they also serve as means to structure discourse in
different ways.



Of course these brief remarks about the discourse function of tense
and aspect barely touch the tip of the iceberg. For more information
on the discourse semantics of French tense and discourse organization
in general, the reader is advised to consult de Swart and Corblin
(2002), Asher and Lascarides (2003) and van Lambalgen and Hamm (2005,
especially chapter 9). This short note on discourse structure
completes our article on tense and aspect.