11

Tense
Atle Grønn and Arnim von Stechow

11.1 Introduction 313

11.2 Desiderata 314
11.3 Tense in compositional semantics 316
11.4 Referential tense 322
11.5 Quantification and adverbials 324
11.6 The perfect 326
11.7 (In)definite tense 328
11.8 (In)definite operators in a dynamic framework 332
11.9 Analysis of tense in a typed dynamic system 334
11.10 Integrating (in)definite aspect 336
11.11 Conclusions 338

11.1 Introduction

Our focus in this chapter is the semantics of tense, one of the main devices
for encoding time in language. The grammatical category of tense is used
to locate a situation or event in time, typically in interaction with aspect.
When tense relates the speaker’s temporal focus, what is normally called the
reference time, to the speech time, we have deictic or absolute tense. Aspect,
on the other hand, is concerned with the internal temporal structure of
the event, for example, whether the event time is included or not in the
reference time.
Tense is typically marked by verbal morphology, normally an affix on the
matrix verb or on an auxiliary verb, but the semantic tense operator is not
interpreted at the verb. The operator can be located quite distant from the
verb at the level of logical form. Thus, we must distinguish between tense
semantics and tense morphology.
We will assume that our model contains a set of times I, and, furthermore,
that verbs have a temporal argument of the semantic type i (time intervals).

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 314 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

The latter assumption will be revised when we include aspect in the analysis.
Times are partially ordered by the relations ≺ ‘before’ and # ‘after’. Time
intervals are coherent sets of time points. Hence, they may overlap, stand in
the inclusion relation and so on (von Stechow, 2009b).
While everybody agrees that the meaning of past and future tenses is rela-
tional – with past placing the reference time before now and future after
now – there is a long-standing issue in the literature as to the exact status of
the reference time: is it quantificational or referential? We argue that this
question cannot be answered straightforwardly. In our view, tenses, such as
the simple past in English, are ambiguous between an indefinite (quantifi-
cational) and a definite (referential) interpretation of the reference time.
We will suggest that the data invite a dynamic account, according to
which indefinite terms are analysed as introducing a new discourse marker
(dm), whereas definite terms are considered to be anaphoric to an old dm
and are in fact entirely presupposed.
Furthermore, since tenses are relational, this last point also concerns the
other time involved in the tense relation, what is called the perspective time
in Kamp and Reyle (1993) and much subsequent work, but which we will
refer to as the temporal centre. For simple tenses in main clauses the tempo-
ral centre is either contextually given, that is anaphoric, or uniquely given
by the speech time, hence referential and not quantificational. In composite
tenses, such as the past perfect in Germanic and Romance languages, the
temporal centre is typically an anaphoric definite time, while the reference
time is shifted backwards.
Thus, following most contemporary work in the domain of tense and
aspect, we should carefully separate the reference time (aka the assertion
time) from the temporal centre (aka the perspective time, temporal anchor
for the sentence). In contrast, the classical notion of reference time going
back to Reichenbach (1947) suffers from the defect of not distinguishing
properly between these two temporal parameters.

11.2 Desiderata

The goal of this survey chapter is to indicate what kind of theory of tense is
needed for a coherent analysis of examples like in (1) and (2) below. Along
the way, we must address a wide range of questions concerning tense: its
morphological and syntactic expressions as well as its interpretation at both
the sentence and the text level.
(1) John left at 11. He didn’t turn off the stove.
(2) John left at 11. He had arrived yesterday.

Barbara Partee (1973a) remarked that the past in the second sentence of
(1) cannot be an existential quantifier. If it were an existential quantifier,

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 315

then we would have two options: it would have the meaning shown in either
(3), or (4).

(3) There is a time before now at which he didn’t turn off the stove.

This assertion is trivial, since (3) is a very weak statement and is likely to be
true. The second option follows:

(4) There is no time before now at which he turned off the stove.

This option is too strong to be true, and it is certainly not the meaning of
the sentence in the text. We can solve Partee’s puzzle if we interpret the past
in the second part of (1) as an anaphoric definite description. Then it would
have the meaning in (5):

(5) John leaves at [[a time1 before now] which1 is at 11]. He doesn’t turn
off the stove at [that time1 before now].

The key to the solution is that the anaphoric definite description, under-
lined in (5), is scopeless with respect to negation.
In the second sentence of Example (2), the composite past perfect involves
both an indefinite and a definite time, as we see from the paraphrase in (6),
where the anaphoric definite description is underlined.

(6) John leaves at [[a time1 before now] which1 is at 11]. He arrives at
[[a time2 [before that time1 before now]] which2 is on yesterday].

We will present a general theory that allows for different combinations of

(in)definite tense, simple and composite tenses, and temporal adverbials, as
in the examples above.
Importantly, both examples above involve an instance of cross-sentential
temporal anaphora, so a fully satisfactory analysis must arguably be dynamic.
The idea that tenses necessitate a dynamic semantics was originally put
forth by Hans Kamp and colleagues in the early 1980s (Kamp and Rohrer,
1983). The development of Discourse Representation Theory (DRT) (Kamp
and Reyle, 1993) was partly motivated by observations concerning tempo-
ral/narrative progression (and lack thereof) with French tenses.1 We con-
sider narrative progression to be a pragmatic phenomenon (be orderly!) and
we will not have much to say about this in our study of temporal seman-
tics. Hence, our argumentation for a dynamic semantics is to some extent
orthogonal to Kamp’s original insight. As pointed out by a reviewer, our
dynamic approach, which is based on the idea that a dynamic treatment
of definite/indefinite NPs could be replicated with tenses (and aspects – see
Section 11.10), is more in the spirit of Partee (1984b).

1
We thank an anonymous reviewer for reminding us of this point.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 316 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

However, we will start our discussion by presenting a standard analysis

of semantic tense, tense morphemes and temporal auxiliaries in a compo-
sitional, static framework. The idea is to clarify where exactly in the struc-
ture the various expressions of time fit in, and what actual contribution
they make. This will be the topic of Section 11.3. In Section 11.4 we will
have a closer look at the notion of referential tense. Next, in Section 11.5 we
will discuss the meaning of temporal adverbials and point out some tempo-
ral puzzles which temporal quantification gives rise to. In Section 11.6 we
briefly analyse the analytic perfect construction, which we consider to be a
sort of tense with either an extended now interpretation or a relative past
semantics. In both cases the perfect modifies or changes the reference time.
In Section 11.7 we motivate an analysis of tense in terms of (in)definiteness,
and we present the general tense architecture assumed here. Finally, in Sec-
tions 11.8 and 11.9, we formalise (in)definite tense in a dynamic framework.
Section 11.10 shows how the formal analysis can be extended to include
aspect. Some of the technical details concerning the dynamic framework
can be found in an appendix.

11.3 Tense in compositional semantics

Most contemporary approaches to the syntax–semantics interface assume a

structure as in (7):

(7) Architecture [TP TENSE[PerfP PERFECT[AspP ASPECT[vP]]]]

The following example motivates this particular three-way distinction

between tense, perfect and aspect.

(8) John has been working much lately.

[TP N[PerfP XN[AspP IMPERFECTIVE[John work]]]]

Although our examples will mostly come from English, the general archi-
tecture we discuss should in principle apply to other languages as well.
A complete inventory in (7) would comprise matrix deictic tenses, namely,
N, PAST* (i.e. PAST(N)), and FUT* (i.e., FUT(N)), at least two perfect relations,
XN (extended now – see Section 11.6) and PAST, a relative FUT, and a number
of aspects, such as PERFECTIVE, IMPERFECTIVE, but languages may differ in
which combinations of these they allow.
The nomenclature of verbal categories in (7), notably aspect, is most easily
analysed on the assumption that the verb comes with an event/state argu-
ment, see Rothstein, Chapter 12. We will see how events can be integrated
into our tense theory in Sections 11.6 and 11.10, but for most of the discus-
sion we will simplify the picture and leave aspect out. Here we assume that
the verb comes with a temporal argument, the time interval (event time) at
which the descriptive content of the VP obtains.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 317

However, what does it mean to say that a VP or a sentence is true ‘at an

interval’? A problem immediately arises with accomplishments (to build a
castle) which lack the sub-interval property and more naturally combine not
with at-adverbials, but with in-adverbials (to build a castle in two years). The
latter, however, do not specify the exact interval throughout which the event
obtains (see Ogihara, 2007, p. 400 and von Stechow, 2009b for discussion). In
the absence of durative adverbials (‘from 2 AM to 5 AM’), natural languages
thus typically remain vague as to the exact duration of the event. A way of
analysing such cases is to introduce a covert perfective aspect operator (see
Rothstein, Chapter 12, or footnote 5 below), which captures the temporal
configuration by letting the temporal trace of a VP-event be included in the
reference time.

11.3.1 Present tense

The present N denotes the speech time, a distinguished time s*. In our typed
framework, the semantic present has the simplest type i (mnemonic for
interval).

(9) Present: N ‘now’, type i

[[N]] = s* (‘speech time’)

We will not be able to discuss more fine-grained approaches to the

present.2 In fact, for many languages the present is rather a non-past. In lan-
guages like Japanese and Russian, Aktionsarten and aspect decide whether
the non-past is interpreted as denoting the speech time or a future time (e.g.
Russian perfectives with present tense morphology are coerced into a future
tense interpretation). Even a quick comparison of examples from English
and German suggests that there may be differences in the denotation of
present tense in ‘our languages’ too:

(10) a. (*) Mary comes tomorrow.

b. Mary will come tomorrow.
√
c. Maria kommt morgen.

The distribution of English present is more restricted than that of the Ger-
man present, since it cannot straightforwardly be combined with future
adverbials and is therefore less likely to denote a future time.

11.3.2 Quantificational past

English uses an auxiliary – an existential time shifter – in order to shift the
reference time to a time after now (will come in (10b)). Many languages, such
as French and German, can make use of auxiliaries (formally, a ‘present per-
fect’) also to shift the reference time to a time before now. For time reference
in the past, English normally uses the simple past:

2
See Altshuler and Schwarzschild (2013) for a recent discussion of alternative analyses.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 318 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

(11) Mary called.

(∃t)[t is before now & Mary call(t)]
There is a time t before now, such that Mary calls at t.

This statement of the truth-conditions in (11) is, not quite accurately, often
attributed to the logician Arthur Prior. Prior (1967) uses the temporal auxil-
iaries have and will in the paraphrases of the truth-conditions of simple past
and future: ‘It has been the case that’ and ‘It will be the case that’. In Richard
Montague’s PTQ (1973b), we find the syncategorematic symbol H (mnemonic
for has).

(12) [[Hα]]i = 1 iff (∃i ≺ i) [[α]] i = 1 (à la Montague)

In order to evaluate Hα at the evaluation time i, it is not enough to con-

sider the intension of α at i; we also have to consider some i ≺ i and evalu-
ate the intension of α there. As pointed out by Ogihara (2007, pp. 393; 396–
397), neither Prior nor Montague intended to represent the meaning of the
English past tense -ed per se.
In our static framework, P* is a deictic past, while P is the more general
version, i.e. a relative past. These are both functors in the object language:

(13) Deictic Past: P*, type (it,t)

[[P∗ ]] = λQit .(∃t )[t is before now & Q(t )]

(14) Relative Past: P, type i(it,t)

[[P]] = λt.λQit .(∃t )[t is before t & Q(t )]

‘is before’ will be written as ‘≺’. In matrix clauses it will not matter whether
we choose P or P*, since the temporal centre of P will be the present N. Of
importance, however, is our assumption, which is also in agreement with
recent ideas of Irene Heim’s, that the temporal centre (perspective time) is
the first argument of P. Other authors may assume a different type for P,
namely, (it,it). We will justify our choice below when we explicate our con-
ception of the syntax–semantics interface.

11.3.3 Have, will

The temporal auxiliaries contribute, in the simplest case, the same seman-
tics as the relative ‘Priorian’ tenses: they are time shifters, also called verbal
quantifiers, that shift the reference time backwards (or forwards):

(15) have (English), avoir (French): type i(it,t)

λt.λQit .(∃t )[t ≺ t & Q(t )]

A consequence of this analysis is that we somewhat incorrectly predict full

synonymy between the indefinite deictic past P* and the present perfect.3

3
This naïve parallel between deictic past and present perfect works better for German or French than for
English which has an XN-semantics in the present perfect, see Section 11.6.1 below.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 319

The structure differs, though, since [have t1 ] is a quantifier generated in the

head of the AuxP.

(16) John has called.

Jean a téléphoné.
[i N] [it λ1 [t [it,t have t1 ][it λ2 [John [call t2 ]]]]]
(∃t )[t ≺ s* & John calls at t ]

The auxiliary will is simply the mirror image of have:

(17) will: type i(it,t)

λt.λQit .(∃t )[t # t & Q(t )]

‘#’ stands for ‘is after’.

(18) John will call.

[i N] [it λ1 [t [it,t will t1 ][it λ2 [John [call t2 ]]]]]
(∃t )[t # s* & John calls at t ]

Note that English, unlike for instance French, does not have a synthetic/
morphological future tense as such.

11.3.4 Tense morphology and interpretation

Judging from the morphology, it would seem that tense is located at the
verb. However, most semanticists have treated tense as a sentential operator
and so, if we follow this practice, then the relation between tense morphol-
ogy and semantic tense must be more complicated.
From the examples above, we can see that the morphological past called,
the participle called (have subcategorises for a past participle) and the infini-
tive call (the auxiliary will subcategorises for an infinitive) all have a tenseless
interpretation in our system. Every verb has a temporal argument, which is
the first argument by convention:

(19) [[call]] = [[called]] = [[calls]] = λt λx. x calls at t

A problem arises from the fact that tense is not interpreted where it is
pronounced. Consider the past -ed and the meaning given in (13)/(14). Let us
assume that at deep structure (DS) -ed occupies the place of the temporal
argument of the verb. The semantics tells us that -ed is an existential quan-
tifier. Like all quantifiers -ed has to be QRed for type reasons. The derivation
of sentence (11) is therefore the following:

(20) DS: Mary call(-ed)

QR ⇒
LF: -ed λ1 Mary call(t1 )

We will adopt the theory of quantifier raising (QR) as described in Heim

and Kratzer (1998): quantifiers of type (at,t) are moved to a sentence-initial
position. They leave a trace of type a interpreted as a variable xa . The

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 320 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

movement index of the quantifier is interpreted as the binder λa . For

the example under discussion, this gives us the following structure and
meaning:

In order to mediate between semantics and morphology, we have to

assume that the QRed -ed is not pronounced at SS (surface structure), but
transmits a feature uP (uninterpretable past) to the bound variable ti , which
determines the pronunciation of the verb as a past form.
A more abstract but equivalent analysis assumes Heim’s theory of PRO (or
TPRO; ‘temporal’ PRO) as a semantically empty pronoun without type and
meaning. On this view, the DS of sentence (11) is as follows:

Given that PRO has no meaning, it cannot be interpreted in situ and has
to be moved. Since the verb requires a temporal argument, we obtain the
structure below.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 321

At LF, PRO is deleted by the principle of full interpretation. The movement

index i is interpreted as λi , and we derive the temporal abstract below.

This structure can be modified by the operator P*, and we get the
following:

The important point is that semantic operators like P* are not pronounced
but transmit features under binding, here the feature uP, which determines
the pronunciation of the verb call as a past form.

We write the origin of the feature P as iP (‘interpretable past’). The land-

ing site, that is, the morphological function of the feature, is written as uP
(‘uninterpretable past’).

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 322 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

When the main verb is non-finite, as in the perfect and future construc-
tions, the verb itself has no inherent tense feature, but it still inherits a fea-
ture via the auxiliary, which in the case of has called and will call is a present
tense feature. The temporal auxiliaries has and will are so-called verbal quan-
tifiers in the theory of Grønn and von Stechow (2010). They change the ref-
erence time of the main verb, but also come with their own morphology,
which must be licensed by a higher semantic tense N. When has and will
transmit their present feature to the non-finite verb, we have feature trans-
mission under semantic binding (Heim, 1994c, 2005). The inheritance of the fea-
ture from the verbal quantifier by the non-finite verb becomes important for
further feature transmission into embedded tenses (sequence of tense phe-
nomena), but it does not play any role in ordinary matrix sentences since
the feature is not pronounced on the non-finite verb.

11.4 Referential tense

The quantificational, indefinite analysis for the simple past predicts scopal
interaction with negation (and also with quantifiers and intensional oper-
ators). As we recall from above, Partee’s famous example (I didn’t turn off
the stove) challenged the standard Priorian and Montagovian analyses since
neither of the two possible scopal relations captures the meaning of the
sentence:

(21) a. (∃t ≺ s*) ¬ I turn off the stove at t

b. ¬(∃t ≺ s*) I turn off the stove at t

The first statement is trivially true and the second is too strong to be true.
Partee’s claim was that this example showed that the past could not be an
existential quantifier ‘there is a time before the speech time’. Instead, Par-
tee proposed an analogy between tenses and pronouns: ‘I will argue that
the tenses have a range of uses which parallels that of the pronouns, includ-
ing a contrast between deictic (demonstrative) and anaphoric use, and that
this range of uses argues in favour of representing the tenses in terms of
variables and not exclusively as sentence operators’ (Partee, 1973a, p. 601).
Partee presumably had in mind a referential past along the lines of what
was later formulated in Heim (1994a), with a presuppositional semantics as
follows:

(22) Referential Past

[[PASTi ]]g is only defined if g(i) ≺ s*.
If defined, [[PASTi ]]g = g(i).

We thus get the following LF for Partee’s example with PAST5 being a tem-
poral variable of type i:

(23) ¬ I turn off the stove(PAST5 )

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 323

Now, what is the meaning of the verb turn off above, a so-called achieve-
ment predicate? Turning off the stove is a very short event that takes only a
moment, so it is unlikely that PAST5 denotes a particular moment that the
speaker has in mind. Rather, PAST5 refers to a stretch of time, say the two
hours before my leaving. Suppose that this is the interval (10 AM, 12 AM).
We must say what I turn off the stove means when evaluated with respect to
this interval, which we call t5 . Obviously the action cannot take place at t5 ,
because the length of the event is not 2 hours. So the event must take place
in t5 .
This brings us back to aspect. We have to assume that the lexical entry of
the verb contains a covert aspectual operator, namely, the perfective. Accord-
ing to this analysis, the sentence actually means what is expressed in (24).

(24) ¬(∃t ⊆ g(5)) I turn off the stove at t,

where g(5) ≺ s* & g(5) = (10 AM, 12 AM)

Nevertheless, even though we have to slightly extend Partee’s analysis

by adding perfective aspect – the inclusion relation t ⊆ g(5), where t
is the event time – the Partee problem convinces us that tenses can be
anaphoric/definite.
In Partee’s original article, the slogan is that tenses are pronouns. The
present tense used deictically is like the indexical first person pronoun I,
while the past tense is an anaphoric or deictic pronoun – a free variable
denoting a particular time in the past. An interesting prediction of this
approach, noted by Partee herself, is that the English ‘future’, which is not
a semantic tense but makes use of the time-shifting verbal quantifier will,
should not be anaphoric in nature.
The referential approach was developed in a static framework by Abusch
(1994b), Heim (1994a), von Stechow (1995), Kratzer (1998a) and others. How-
ever, just as the indefinite tense analysis traditionally attributed to Prior
(1967) cannot be the whole story, there are also arguments against the pre-
suppositional semantics, i.e. the view that tenses are temporal variables
with a presupposition.
One problem is the question of how it would be possible, on the latter
approach, to explain obligatory backward-shifting of a past in an intensional
context under an attitude verb in non-sequence-of-tense (SOT) languages like
Russian (Grønn and von Stechow, 2010, 2012).

(25) Ona [ . . . ] sprosilaPAST,PF , spalPAST,IMPF li on. (Tolstoy, Anna Karenina)

She [ . . . ] asked him if he had slept. (translation from the RuN parallel
corpus)

The backward-shifting remains mysterious with the meaning of past

given in (22). We can also have backward-shifting in English with an embed-
ded simple past:

(26) John said that Mary left.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 324 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

Kratzer (1998a) assumes for such examples that the past tense is ambiguous
between a referential tense and a quantificational relative past, which she
calls perfect aspect. In Russian, an embedded synthetic future can shift the
event time as well:

(27) Ona byla nemnogo prostuzhena i skazalaPAST,PF, chto oni segodnja

pobudutFUT,PF doma. (Kurkov, Piknik na ldu)
She had a bit of a cold and said they would be staying in for the day.
(translation from the RuN parallel corpus)4

Advocates of a referential theory would have to say that the Russian future
is ambiguous between a referential tense and an aspect, say prospective. The
quantificational approach needs no such ambiguities.
There could be a way out of the dilemma if we were to develop a ‘mixed’
approach, analysing past/future tense as an existential quantifier over contex-
tually restricted times. For instance, the Partee example could be analysed as ‘at
some past time this morning’ (existential + referential). The anaphoric com-
ponent can thus be hidden in the domain restriction. We refer the reader
to Bäuerle (1978, 1979), Ogihara (2007), von Stechow (2009b), Altshuler and
Schwarzschild (2013) for further discussion of this possibility. Instead, we
will below advocate an ambiguity approach in the tense domain (indefinite
vs. definite tense). This is in our view the simplest analysis, where the dis-
tinction between quantificational and referential interpretations is directly
reflected at the level of logical form (LF).

11.5 Quantification and adverbials

11.5.1 Tense and quantification

We cannot do justice to all the quantificational puzzles that can arise in the
semantics of tense. Here we will only mention one example, the Bäuerle–von
Stechow Problem, as an illustration.

(28) a. Yesterday, Mary called exactly once.

b. Yesterday, Mary called exactly 3 times.

Comparison of the two sentences shows that once = at one time. In the fol-
lowing analysis, both yesterday(it) and exactly one time(it,it) restrict the past
reference time:

(29) [it,t P N] [it [it λ1 on yesterday(t1 )] [it at exactly one time λ2 Mary call(t2 )]]
(∃t)[t ≺ s* & t ⊆ yesterday & (∃!t )[t ⊆ t & Mary calls at t ]]
(∃!t )[α(t )] means ‘There is exactly one time satisfying α.’

The truth-conditions shown are obviously wrong, since they are compatible
with a scenario in which Mary also called at a time t included in yesterday,

4
www.hf.uio.no/ilos/english/research/projects/run/corpus/.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 325

but not included in t. This problem has caused a lot of trouble and led many
people away from the quantificational analysis of tense. Bäuerle and von
Stechow (1980) concluded that past has a predicate meaning λt.t ≺ s*. Mary
called means λt.t ≺ s* & Mary calls at t. The quantification is provided by
adverbials such as exactly once or, in the default case, existential generalisa-
tion. Several people have taken up this proposal, for example Musan (2002),
von Stechow (1995) and occasionally Ogihara.
In the following we will ignore this problem. For more puzzles related to
tense and quantification, see von Stechow (2002, 2009b).

11.5.2 Temporal adverbials

Temporal adverbials are of various kinds, some of which play a crucial role
in determining the reference time of the sentence, and sometimes also the
temporal centre (i.e. the perspective time at which the subject locates her-
self). For yesterday and PPs like on my birthday, sometimes referred to as posi-
tional or definite temporal adverbials, there are several alternative, though
ultimately perhaps equivalent, analyses in the literature. Here we will sim-
ply assume that these expressions are predicates of times:

(30) Predicate format of positional temporal adverbials, type (it):

[[on my birthday]] = λt.t is on my birthday

Positional adverbials always restrict tense or some other functor which pro-
vides the quantification:

(31) Mary called on my birthday.

[it,t P N] [it λ1 [[on my birthday t1 ] [Mary [call t1 ]]]]

The relevant version of Predicate Modification is as follows:

(32) Let α be a tree of type t with daughters β and γ of type t. Then

[[α]] = 1 iff [[β]] = 1 = [[γ ]]

The interaction between temporal adverbials and tenses is subject to cer-

tain pragmatic constraints. As observed by several authors, speech time
modification is usually disallowed:

(33) John leaves at six.

(#) N λ1 [[t1 at six] [John leave t1 ]]

The sentence can only have a so-called scheduled reading, that is, a special
kind of future reading, and hence the LF in (33) does not capture the mean-
ing of the sentence. Similarly, in (34), the adverbial modifies the past, not
the speech time:

(34) John left at six.

N λ1 P(t1 ) [[λ2 at six(t2 )] [λ3 John leave(t3 )]]

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 326 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

Klein invokes a pragmatic principle saying that a definite time (here: the
speech time) cannot be further specified (Klein, 1992). We will have more to
say about this in connection with the interaction between temporal adver-
bials and the past perfect. See (40) below.

11.6 The perfect

The analytic perfect construction in Germanic and Romance languages is

a rich source of puzzles and idiosyncrasies of various sorts. We will briefly
review some of these below.

11.6.1 Present perfect and XN

The so-called Present Perfect Puzzle refers to the fact that positional (defi-
nite) past adverbials cannot modify a present perfect in English (or Mainland
Scandinavian), but they can in German and French:

(35) a. Mary called yesterday.

b. (*) Mary has called yesterday.
√
c. Maria hat gestern angerufen. (German)
√
d. Marie a téléphoné hier. (French)

These facts follow from an extended now (XN) semantics for the
English/Scandinavian present perfect – provided that temporal adverbials
modify XN – and a relative past semantics for the German/French perfect
in combination with an intersective semantics for definite temporal adver-
bials, see Pancheva and von Stechow (2004) among others.

(36) Extended Now (McCoard, 1978), (Dowty, 1979); quantifier type i(it,t)
[[PERFXN ]] = λt.λQ.(∃t ) [XN(t’,t) & Q(t )],
where [[XN(t , t)]] = 1 iff t is a final subinterval of t .

Once more we have to assume an aspect operator5 in an eventive frame-

work – this time to solve the Present Perfect Puzzle. We will first show how
definite adverbials are compatible with the German variant of (35c). We are
assuming the same simple denotation for N in both English and German;
this will turn out to be correct here:

(37) Maria hat gestern angerufen (German)

N λ1 PAST(t1 ) [λ2 [[t2 on yesterday] [PF(t2 ) λ3 Mary
called(e3 )]]] (German)
(∃t)[t ≺ s* & t ⊆ yesterday & (∃e)[τ (e) ⊆ t & Mary call(e)]]

5
The standard semantics for perfective aspect, following Krifka (1992) and others:

[[PF]] = λQ.λt.(∃e) [τ (e) ⊆ t & Q(e)], where Q is a predicate of events, and τ (e) is the running time of e. In
our framework, the time argument will be the first: λt.λQ . . .

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 327

However, the same adverbial leads to a contradiction in an XN-configuration

as in English:

(38) N λ1 PERFXN (t1 ) [λ2 [[t2 on yesterday] [PF(t2 ) λ3 Mary called(e3 )]]]
(∃t)[XN(t,s*) & t ⊆ yesterday & (∃e)[τ (e) ⊆ t & Mary call(e)]]
⇒ s* ⊆ yesterday!

We conclude that the combination N + PAST, which we find in German

and French, is not available for the English present perfect (probably due
to the competition with the simple past). Furthermore, the time adverbial
yesterday cannot modify the event for type reasons. Note that if the theory
referred to event times instead of events, then modification of the event
time by yesterday would not be blocked.

11.6.2 Past perfect

The past perfect consists of two nested layers of ‘past’, the backward-shifting
lexically expressed by the auxiliary and the past operator licensing the past
tense morphology of the auxiliary.6 In interaction with the past perfect, tem-
poral adverbials may therefore create an ambiguity:

(39) Mary had left at six

a. [it,t P N] [it [it λ1 at six(t1 )] [it λ2 [t [it,t have(t2 )] [it λ3 Mary left(t3 )]]]]
(∃t ≺ s*) t at 6 o’clock & (∃t ≺ t) Mary leaves at t
b. [it,t P N] [it λ1 [t [it,t have(t1 )] [it [it λ2 at six(t2 )] [it λ3 Mary left(t3 )]]]]
(∃t ≺ s*)(∃t ≺ t) t at 6 o’clock & Mary leaves at t

The LF in (39a) is a past-time modification: the leaving is before six. (39b) is

a perfect-time or event-time modification, the leaving is at six.
The ambiguity of the past perfect was not attested for the present per-
fect in English. The present perfect in English is special since it has an XN-
semantics, while the past perfect mostly behaves as expected from a com-
positional point of view, on a par with the present perfect in German. The
relative past reading of had P-ed, unlike has P-ed, is compatible with definite
temporal adverbials, as we have just seen. Furthermore, the ban on modifi-
cation of the higher tense, as we saw in the present perfect, disappears in
the English past perfect, where the adverbial can modify either the higher
or lower time.
Consider, however, the following contrast:

(40) a. (#) At six John had left at five.

b. At six John had left an hour before.
c. (#) At seven John had left at five.
d. At seven John had left an hour before.

6
In addition, the past perfect may have an ‘XThen’ interpretation parallel to XN in the present perfect.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 328 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

Table 11.1 The interpretation of complex tenses

Present Perfect Past Perfect

English (*)N + PAST N + XN PAST(N) + PAST PAST(N) + XThen

German N + PAST N + XN PAST(N) + PAST PAST(N) + XThen

The first example in (40a) illustrates Klein’s prohibition of double temporal

specification (Klein, 1992). However, in the meta-language (40a) and (40b)
would be completely equivalent. An explanation for this phenomenon can
be found by comparing the truth-conditions of (40a) and (40c) – again the
truth-conditions are the same: [[(40a)]] = [[(40b)]] = [[(40c)]]. This is not so
for (40b) and (40d) with a topic time contrast, a minimal pair which clearly
express different propositions. So, when the highest adverbial does not play
any role semantically, as in (40a) and (40c), the construction is pragmatically
blocked.
The different distribution of combinations involving the perfect auxiliary
in English and German is listed in Table 11.1.
Thus, in German, every perfect is ambiguous. While German allows any
combination of a tense and a perfect relation, English does not have the
combination N + PAST.
Recall from the introduction that the past perfect, interpreted with the
admissible combination PAST + PAST, is typically used with an anaphoric
higher tense (the temporal centre for the lower tense). Thus, Example (39)
above will, when used in a discourse, contain the information t ≺ s* &
(∃t ≺ t). The underlined part is a presupposition, a standard case of cross-
sentential temporal anaphora. We cannot express this straightforwardly in
a static framework. We will turn to this issue in the following sections.

11.7 (In)definite tense

Both the quantificational approach to tense and the alternative referential

approach have their weaknesses. There is no simple way of treating temporal
anaphora in the quantificational approach.

(41) John came home at 12. Mary was asleep (then).

The referential approach can have the same temporal variable (or the same
deictic tense) in both sentences. A drawback of the referential approach,
though, is that it cannot account for backward-shifting in embedded con-
texts, for example in complement clauses. At least non-SOT languages like
Russian (see (25) above) and Japanese need quantificational tenses. For more
discussion of this question, see von Stechow and Grønn (2013a,b) and Sharvit
(2014).

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 329

We believe that any theory of tense that is able to overcome the disadvan-
tages of the two approaches mentioned must be a dynamic one. The follow-
ing system is an intensional version close to other dynamic systems, such
as the ones proposed in Muskens (1996) and Beaver (2001). The applications
will be focused on tense (and aspect). An alternative dynamic system with a
similar focus as ours is Bary (2009).
The main claim is that an indefinite tense introduces a new temporal dis-
course marker, while a definite tense is anaphoric to an old dm, presup-
posing its descriptive content, that is, the temporal relation. To implement
this idea, we will treat tenses and aspects as purely relational, with a covert
indefinite or definite article on top.
Many languages (e.g. Russian) have no overt [±def] marking in the nom-
inal domain. NPs in such languages are then ambiguous with respect to
[±def]. Since we find ambiguity in the nominal domain, we should not be
surprised to find it in other domains as well. By loosening the correspon-
dence between (in)definiteness and article morphology, we discover that the
ambiguity is found not only in the temporal domain, but also in the event
domain (e.g. aspect in Russian, see Section 11.10).

11.7.1 The [±def]-ambiguity of tense

One of the motivations for the development of dynamic approaches in the
1980s was the analysis of donkey anaphora in the nominal domain. We note
that similar cases can also involve tense, as in the following temporal donkey
sentence:
(42) When a farmer had a donkey, the farmer usually beat the donkey.
The past tense in the antecedent should be an indefinite term (like a farmer/a
donkey), but the past tense in the consequent must be a definite term (like
the farmer/the donkey). A similar observation was also made in Partee (1984b,
pp. 268ff.). The following paraphrase illustrates the intended interpretation
of donkey tense:
(43) Usually, when a farmer had a donkey at some past time, the farmer
beat the donkey at that past time.
By treating the simple past as ambiguous between an indefinite and a def-
inite tense, contra McCoard (1978) who claims that the simple past is only
definite, we can explain a lot of thorny data.

11.7.2 The [±def]-ambiguity and the perfect

The perfect is a shifter – a relative past – and therefore indefinite, though it
typically contains a definite temporal centre, cf. (2), repeated below:
(44) John left at 11. He had arrived yesterday.
In the style of Irene Heim’s dissertation (1982), we want to say, starting from
the first sentence, that a new temporal discourse marker 1i (i = indefinite)

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 330 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

is before now and that John leaves at t1 (bound variables are always definite),
which is at 11. For the second sentence, we intuitively want to pick up the
old discourse marker 1d (d = definite). This time is presupposed to be a past
time (before now), and it serves as the temporal centre (perspective time) in
a temporal relation with a new discourse marker 2i , such that t2 ≺ t1 and t2
is the time of John’s arrival yesterday.

11.7.3 A uniform type for ±def

Definites, like pronouns, are commonly analysed as referring terms with
contextually dependent anaphoric and deictic uses, while indefinites are
existential quantifiers. This view stipulates differences between definites
and indefinites along three dimensions: semantic type, uniqueness and pre-
suppositionality (Heim, 2011a). Since the (in)definite article is covert in the
tense domain, we want tenses to have a uniform type. In fact, we will pro-
pose below a uniform semantic type for definites and indefinites across all
domains.
We will adopt the idea from Heim’s dissertation, where definites, just like
pronouns, carry an old index at LF. However, our implementation will be dif-
ferent, since both indefinites and definites will have the format of dynamic
generalised quantifiers (see Muskens, 1996; Beaver, 2001). The restriction
of the quantifier in the temporal domain will contain at least the tempo-
ral relation contributed by tense proper (≺, #). In the case of definites, the
restriction will be entirely presupposed.

11.7.4 Tense architecture

In the system to be developed in what follows, we have deictic tenses and
relational tenses, that is, two-place predicates of times. These are constants
which have the same denotation in every model (we will adapt these mean-
ings to the dynamic framework below):

(45) Deictic tenses

F(N) = s*
F(PAST*) = {t | t ≺ s*}

(46) Relational tenses

F(PAST) = {<t, t > | t ≺ t } = F(have)
F(FUT) = {<t, t > | t # t } = F(will)
F(XN) = {<t, t > | t is a final subinterval of t} = F(has)

We distinguish between the temporal relation and the temporal cen-

tre. The T-centres are N (a deictic pronoun), Tpro (a definite dm), TPRO (a
semantically void pronoun that will create a temporal abstract). The T-centre
is obligatory, while the T-relation is optional. This is the theory we have
defended in several recent papers (Grønn and von Stechow, 2010, 2012; von
Stechow and Grønn, 2013a,b).

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 331

Figure 11.1 Tense architecture

To this system we add a covert indefinite or definite article. If the article

introduces a new discourse marker, then the tense will be a shifter; if the
discourse marker is old, the tense is anaphoric.
We thereby eventually arrive at the tense architecture shown in
Figure 11.1.
On this view, most tenses are not simply pronominal as in Partee
(1973a), but rather are relations between two times, of which only one is a
pro-form.

11.7.5 A note on temporal modification

In our system, tenses are definite or indefinite terms which have the format
of (dynamic) generalised quantifiers. In the nominal domain, the restriction
and nucleus of a generalised quantifier can easily be modified by intersec-
tive adjectives, relative clauses, and so on. We think that there are similar
devices in the temporal domain as well.
In von Stechow and Grønn (2013b) we give an intersective analysis of when
adjuncts as temporal relative clauses. Information structure determines
whether the when-clause is part of the restrictor or nucleus of the matrix
temporal quantifier, as in the following examples with indefinite tense:

(47) a. When I was young, I read War and Peace. (the topicalised
when-clause is a free relative clause in the restrictor of the
indefinite article)
b. I read War and Peace when I was young. (when-clause in the nucleus
of the indefinite article)

Which combinations of temporal modifiers and tenses are possible, is, of

course, an empirical question, but it is clear that the system needs a lot of
flexibility. For instance, the following past perfect (had left) should arguably
be decomposed into two indefinite past tenses:

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 332 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

(48) a. John’s wife had already left, when he came to the party.
b. There is a time at which John’s wife leaves (and this time is) before
a past time at which John comes to the party.

The temporal centre of the past perfect is not a definite anaphoric tense in
this case, but an indefinite past modified by a when-clause.

11.8 (In)definite operators in a dynamic framework

Our dynamic framework is presented in the appendix, written in telegram

style for those who are interested in the formalisation of the system. An
easier way out would be to make the points below informally in the DRT-
style of (Partee, 1984b).

11.8.1 Indefinites
Given the framework outlined in the appendix, we can introduce the indefi-
nite operator ind which serves not only in the formalisation of the indefinite
article in English but also in the covert indefinite article above tense and
aspect. The operator ind introduces d as a new dm and updates the context
with the restriction P(d) and the nucleus Q(d), formally:

(49) The indefinite operator ind, type d((d,cc),((d,cc),cc))

F(ind) = λd.λP.λQ.λc: d ∈ dom(c).{<f,w> | ∃g ∈ c: g ⊆ f & d ∈ dom(f) &
<f,w> ∈ (([c add d] + P(d)) + Q(d))}

This is the typed version of Heim’s semantics for indefinites. To understand

the definition, we have to introduce Beaver’s add-operator. What ‘c add d’
says is that we make the d-assignments in c a bit bigger: we enlarge their
domain by the discourse marker d, where d may have any value. We have
to carry out this trivial modification of the context, since c + P(d) might be
undefined if d is new.

(50) Beaver’s add-operator

c add d = {<f,w> | ∃g ∈ c: g ⊆d f & <g,w> ∈ c}
[i.e. f is an incrementation of g by adding d to the domain of g]

Let us consider the following example with an indefinite deictic forward

shifter in the temporal domain:

(51) It will rain.

ind5 WILL* RAIN
There is a time f(5) after now and it rains at f(5)

The symbol 5 stands for the dm 5di . This sentence doesn’t contain variables.
Formally, the context change potential (CCP) of the entire sentence is there-
fore as in (52):

(52) λc: 5 ∈ dom(c).{<f,w> | ∃ g ∈ c: g ⊆ f & 5 ∈ dom(f) & f(5) # N & f(5) ∈

F(rain)(w)}

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 333

11.8.2 Definites
The definite article picks up a familiar dm, that is, one that is in the domain
of the local context. Furthermore, the entire restriction must by entailed by
the context, i.e. it is presupposed. To make this precise, we first say what it
means for a CCP to be entailed by a context. The relevant notion is defined
by means of Beaver’s ↓-operator, which we explain below.
Heim defines entailment in this sense by saying that c entails p iff c+p=c.
The intuition is that an entailed CCP does not add any new information to
the context when the sentence is uttered, and hence the context remains
the same. However, this does not work for indefinites because indefinites
add new dms to the context, so after the update the context is different.
Beaver has found a method that works for indefinites. The crucial ingre-
dient is the ↓ -operator which is defined thus:7

(53) c + ↓ p = {<f,w> ∈ c | ∃g : f ⊆ g & <g,w> ∈ c + p}

(54) Entailment
c | p iff c + ↓ p = c.

Therefore the definite operator can be defined as in (55). Presuppositions are

written between the colon and full stop.

(55) The def-operator type d((d,cc), ((d,cc),cc))

F(def) = λd.λP.λQ.λc: c | P(d).{<f,w> | ∃g ∈ c : g ⊆ f & <f,w> ∈ c +
Q(d)}

So the definite article presupposes its entire restriction, and only the
nuclear scope can introduce new information. However, if P merely amounts
to the tense relation proper (e.g. ≺), then the presupposition can be rather
weak, since there may be several indefinite past tenses in the prior context.
As we noted above in Section 11.7.5, when-clauses and temporal PPs may be
added to the restriction of both ind and def. In the latter case, to avoid ambi-
guity, the speaker should make the restriction of the definite as informative
as possible, following the pragmatic principle: maximise presuppositions!
This will help the hearer to find the correct antecedent.
It should be noted that ind and def, despite their semantic type, are
not genuine quantifiers (like EVERY, which is not treated here), because
the result of applying either of them is still an open sentence. In truth-
conditional terms, the dm introduced/picked up by the article is a constant

7
The motivation behind the definition of entailment by means of the ↓ -operator is this:

Suppose we update a context c by uttering the sentence A1 farmer has a2 donkey. The semantics of the
indefinite article requires that we add a new dm for the indefinite articles, say 1 and 2.

The new context created by this utterance is c1 . This context should entail the CCP A farmer has a donkey,
because we have just said that. If we add something that has already been said, e.g. A3 farmer has a4
donkey, we don’t change the information contained in the context c1 . Beaver’s definition of entailment takes
care of this situation because the ↓ -operator is purely eliminative and doesn’t add any new dms to the
context.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 334 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

and ultimately a free variable in some sense. The result of an update is

always an open proposition.

11.9 Analysis of tense in a typed dynamic system

We now have to introduce temporal arguments for dynamic predicates. By

convention, the temporal argument will always be the first one after the
world argument, which is implicit.

(56) F(SLEEP) = λddi .λdde .λc: ddi , dde ∈ dom(c).{<f,w> ∈ c |

F(sleep)(w)(f(ddi ))(f(dde )) = 1}

We assume that verbs, adjectives, prepositions and tenses have temporal

arguments.

(57) Tenses
Present: N is a distinguished dm of type di.
PAST and FUT: type (di,(di, cc))
F(PAST) = λddi .λddi .λc: d, d ∈ dom(c).{<f,w> ∈ c | f(d ) ≺ f(d) }
F(FUT) = λddi .λddi .λc: d, d ∈ dom(c).{<f,w> ∈ c | f(d ) # f(d) }

Thus, tenses are purely relational and must be combined with either the
indefinite or the definite article. In the first case, they introduce a new ref-
erence time, while in the second case they are anaphoric. Let’s see how this
works on a concrete example (from German in order to avoid interference
with aspect):

(58) Alla schlief. Sie träumte.

ind1 PAST(N) [λ2 [def3 ALLA [λ5 SLEPT(t2 )(t5 )]]]
def1 PAST(N) [λ4 DREAMT(t4 )(SHE3 )]

The first PAST is indefinite. It introduces a new reference time at which Alla
was asleep (proper names like ‘Alla’ are treated as definites). The second PAST
is definite. The sentence does not mean that Alla was dreaming at some
time or other – it means that Alla was dreaming at the first time. Here is an
account of the CCP.

(59) c0 + ind1 PAST(N) λ2 def3 ALLA λ5 SLEPT(t2 )(t5 ) is only defined if 1 ∈

dom(c0 ), N,3 ∈ dom(c0 ). If defined, the result is {<f,w> | ∃g ∈ c0 : g ⊆1
f & f(1) ≺ f(N) & f(3) = Alla & f(3) sleeps at f(1) in w} = c1
c1 + def1 PAST(N) λ4 DREAMT(t4 )(SHE3 ) is only defined if
c1 | PAST(N)(1). This is given. So we have to calculate
c1 + DREAMT(1)(SHE3 ), which is only defined if 1,3 ∈ dom(c1 ), which
is the case.
So the result is {<f,w> ∈ c1 | f(3) dreams at f(1) in w}.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 335

Next, let us analyse an example involving the past perfect, which is typi-
cally used anaphorically.

(60) Mary arrived at 6 AM. John had left.

ind1 PAST(N) [λ2 [def3 MARY [λ9 [AT 6(t2 ) & ARRIVE(t2 )(t9 )]]]]
def1 PAST(N) [λ4 [ind5 HAVE(t4 ) [λ6 [def7 JOHN [λ8 LEFT(t6 )(t8 )]]]]]

The text means that Mary left at 6 AM, and John had left before that time. So
the past perfect introduces a new past (event) time, whose temporal centre
is anaphoric to a given past time. The HAVE in the second sentence is syn-
onymous with a relative PAST. Temporal adverbials (‘at 6 AM’) are analysed
as syntactic adjuncts which semantically combine with the VP through PM.

(61) [[AT 6 AM]] = λddi .λc: d ∈ dom(c).{<f,w> ∈ c | f(d) is at 6 AM}

The reader may check for herself that the effect of this text on a context c,
the domain of which contains 3de and 7de , is as follows:

(62) {<f,w> | ∃g ∈ c: g ⊆1,5 f & f(1) ≺ f(N) & f(1) is at 6 AM & f(3) = Mary &
f(3) arrives at f(1) in w & f(5) ≺ f(1) & f(7) = John & f(7) leaves at f(5)
in w}

Here is our analysis of the anaphoric PAST in Partee’s example:

(63) I left home at 11. I didn’t turn off the stove (at that time in the past).
ind1 PAST(N) [λ3 [AT 11(t3 ) & LEFT(t3 )(I)]] (= p)
1
NOT [def PAST(N) [λ2 TURN-OFF-STOVE(t2 )(I)]] (= q)

The standard dynamic semantics for NOT(π ,π ) is as follows:

(64) Negation. F(NOT) =

λpπ .λc: c + p is defined.{<f,w> ∈ c | ¬(∃g) f ⊆ g & <g,w> ∈ c + p}

Here is the calculation for the example.

(65) c0 + p = {<f,w> | ∃g ∈ c0 : g ⊆1 f & f(1) ≺ f(N) & f(1) at 11 & I leave at

f(1) in w} = c1
c1 + q = {<f,w> ∈ c1 | ¬(∃g) f ⊆ g & I turn-off-the-stove at g(1)}
= {<f,w> | ∃g ∈ c0 : g ⊆1 f & f(1) ≺ f(N) & f(1) at 11 & I leave at f(1) in
w & ¬ I turn-off-the-stove at f(1)}

This is the reading Partee had in mind. Since the entire definite tense is
presupposed, we get rid of the scope interaction with the negation without
giving up the relational semantics of the deictic PAST, namely that the ref-
erence time is before the speech time.
In the next section we will show that the application of the definite/
indefinite distinction to the temporal domain is a general phenomenon
which concerns not only times, but also events.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 336 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

11.10 Integrating (in)definite aspect

For simplicity we have chosen to mostly leave aspect and events out of the
analysis above. We will now show that we can introduce aspect without hav-
ing to modify the principles underlying the theory.
Let’s illustrate how aspect fits into the picture with a classical example of
event anaphora from Russian:

(66) Anaphoric use of the imperfective

V ètoj porternoj ja napisalPAST,PF pervoe ljubovnoe pis’mo Vere.
in this tavern I wrotePAST,PF first love letter to-Vera.
PAST,IPF
Pisal [karandashom]F
I-wrotePAST,IPF [with-pencil]F
‘In this tavern I wrote my first love letter to Vera. I wrote it with a
pencil.’ (Chekhov)

Grønn (2004) defends the view that the Russian imperfective is ambiguous.
Its core meaning is that the event time includes the reference time, but it
can also mean the same as the perfective, namely that the event time is
included in the reference time. The same thing happens when an imperfec-
tive mentions a previous event, as in (66).
Aspect localises the event time with respect to the reference time; see
Klein (1994). This means that we have to change the entries for verbs, so that
instead of the time argument, they have an event argument. The combina-
tion with tense is mediated by aspect, so the entry for WRITE is as shown in
(67), where we are using the notation e for dms of type dv.

(67) F(WRITE) = λedv .λdde .λdde .λc: e, d, d ∈ dom(c)

{<f,w> ∈ c | F(write)(f(e))(f(d))(f(d )) = 1}

The semantic aspects are now purely relational, just like tense:

(68) Dynamic aspects: type di(dv,cc)

a. Dynamic perfective:
F(PF) = λddi .λedv .λc: d, e ∈ dom(c).{<f,w> ∈ c | τ (f(e)) ⊆ f(d)}
b. Dynamic imperfective:
F(IPF) = λddi .λedv .λc: d, e ∈ dom(c). {<f,w> ∈ c | f(d) ⊆ τ (f(e))}

Compare these entries for aspect with the traditional ones in footnote 5
above.
As with tense, our new aspects are indefinite or definite. An indefinite
aspect introduces a new event, a definite aspect is anaphoric to an old event.
The ‘fake’ Russian imperfective in (66) is a morphological imperfective that
can be interpreted as an anaphoric perfective.
Usually, a semantic perfective marks its verb with the [pf] feature. How-
ever, when the aspect is definite, the aspect can be neutralised, and instead

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 337

the unmarked aspectual feature is assigned to the verb. As a result, we get a

semantic perfective and an unmarked imperfective verb form.
Here is an account of the Russian example, which for the sake of simplic-
ity is analysed as if it were English. The past tense originates as the first
argument of PF. It has to be QRed for type reasons.

(69) ind1 PAST(N) [λ6 [ind2 PF(t6 )[λ3 [A4 LETTER [λ5 WROTE(e3 )(x5 )(I)]]]]].
def1 PAST(N) [λ10 [def2 [[PF(t10 )][λ7 WROTE(e7 )(it4 )(I)]] [λ8 [A9 PENCIL [λ12
WITH(e8 )(x12 )]]]]]

It is revealing to see that everything that is indefinite in the first sentence

is definite in the second one. An Aspect Neutralisation Rule ensures that
the definite PF-operator can license the feature [u-ipf], even though it is
a semantic perfective.8 The second aspect operator could not be a seman-
tic imperfective because one and the same event cannot be both complete
and incomplete. It follows logically that the second aspect feature cannot be
semantically interpreted.
The meaning of the text is therefore as follows:

(70) c + (69) is only defined if 1, 2, 4, 9 ∈ dom(c). If defined:

c + (69) = {<f,w> | ∃g ∈ c: g ⊆ f & 1,2,4,9 ∈ dom(f) & f(1) ≺ f(N) &
τ (f(2)) ⊆ f(1) & letterw (f(4)) & writew (f(2))(f(4))(f(I)) & f(2) withw (f(9)) &
pencilw (f(9))}

This event anaphora example also illustrates the principle ‘maximise pre-
suppositions!’ by making the restriction of def as informative as possible.
We want to capture the fact that in the second sentence of (66) only the
adjunct ‘with a pencil’ is focused and provides new information – hence the
writing-predicate, not just the aspectual relation, should be backgrounded
and part of the presupposition. To achieve this we must make sure that the
definite operator takes the aspectual relation and the verbal predicate as its
first argument (the restriction). The two properties of events are then com-
bined by predicate modification. Only the adjunct ‘with a pencil’, which
is also a dynamic event property, is left in the nucleus of the definite
article.

8
In examples such as (66), we assume a checking relation from the semantic PF to the uninterpretable
imperfective (!) aspect feature of the event argument of the verb; see Grønn and von Stechow (2012) for
technical details.

Our Aspect Neutralisation Rule says that definite aspects are neutralised with respect to their morphological
feature. Whether a language adheres or not to this rule is an important factor in the division of labour
between morphological aspects in Slavic languages. In Russian, perfective morphology is typically used for
indefinite complete events (e.g. in narrative progression), while definite aspects – and imperfective
morphological aspect – are never used in cases of narrative progression. West Slavic languages, on the other
hand, apparently do not have this rule, and the unmarked imperfective is accordingly not used with definite
aspect but may occur in contexts of narrative progression.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 338 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

11.11 Conclusions

Do we need a dynamic framework for temporal anaphora? A static frame-

work would have to represent the arguments of tenses by free variables that
are existentially bound on the text level. We would still need the [±def]-
distinction to make sure that indefinite variables are new, while definite
ones are old. We would have to make sure syntactically that indefinite vari-
ables in the scope of negation or other operators are not used later in the
text. Furthermore, we would need a fourth kind of free variables in order
to represent deictic words, which would not be bound at the text level. It
is clear that the restrictions on co-indexing will be very difficult to formu-
late. Therefore Occam’s razor says that the dynamic framework is the better
framework for the analysis of temporality.

Appendix: A dynamic framework

The system consists of two parts, one static and one dynamic, with the latter
building on the former. The static part is here mainly used for making clear
the status of discourse markers that are not interpreted by the outer assign-
ment a, but by the inner assignment f. So our system has dms as constants
in the static part. There are also variables for dms that are interpreted by
the outer assignment a.
We introduce d (shorthand for de, di or dv) as the type of discourse mark-
ers (dm) and π as the type of context change potentials (CCP). Predicates of
type dπ are called dynamic predicates. With Beaver (2001) we assume that for
each dynamic predicate, e.g. [RAINdiπ ], there is a unique first order static pred-
icate (here of type it), i.e. we have a corresponding static expression written
in small letters: [rainit ].
We can also have dynamic λ-abstracts:

(71) [π [Ndi [diπ λν [RAIN ν]]]] (‘it rains’)

Apart from type differences, dynamic expressions look virtually the same as
their static counterparts, and we can use almost everything we know from
Montague Grammar. However, these expressions have a different interpreta-
tion, as we will see.

Syntax
For the syntax of the dynamic system we only need expressions of the types
d and π and the rule that combines two types a and b to (ab). d and π might
therefore be regarded as basic. But for the interpretation it is more perspic-
uous to decompose π. These are the types we use:

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 339

(72) Types
Basic: e (individuals), i (times), v (events), de (dms for ordinary
individuals), di (dms for times), dv (dms for events), t
(truth-value).9
Complex: If a and b are types, (ab) is a type. If a is a type (wa) and
(ga) are types.

Intuitively, w is the type of worlds and g is the type of dm-assignments. These

types are in some sense intensional. They are not basic but always form func-
tions. We use the following abbreviations: c := (g(wt)) (type of contexts, infor-
mation states), π := (cc) (type of CCPs).
The lexicon of the language consists of a set of constants and a set of
variables for any type a. The dms are constants. Thus the lexicon is ∪a∈Type
Cona ∪ ∪a∈Type Vara . We assume that there are no constants of type g or w in
the lexicon. Thus the language is intensional.

(73) Syntax
Con. If α is a constant of type a, α is an expression of type a.
Var. If x is a variable of type a, x is an expression of type a.
FA. If α is an expression of type ab and β is an expression of type a,
then α(β) is an expression of type b.
PM. If α and β are expressions of type (at) or (dπ), then [α β] is an
expression of the same type, where d stands for de, di or dv.
Abstraction. If x is a variable of type a and α is an expression of type
b, then [λx α] is an expression of type (ab).

The rule FA is the combination of a functor and an argument, that is, Func-
tional Application. PM is what Heim and Kratzer (1998) call Predicate Modi-
fication.

Semantics
The interpretation function for the language is based on a familiar inten-
sional model

(74) M = (E, I, V, DMe , DMi , DMv , W, N, . . . F)

where E, I, V are sets of individuals, times and events, respectively,
and DMe , DMi , DMv are sets of discourse markers for individuals,
times and events, respectively. N is a distinguished time (the speech
time). F interprets the constants.

Next we define the semantic domains for each type. Let DM be Dde ∪ Ddi ∪
Ddv .

9
Sometimes we ignore the sortal distinction between de, di and dv and simply write d for the basic type of
discourse markers.

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 340 AT L E G R Ø N N A N D A R N I M V O N S T E C H O W

(75) Semantic domains

De = E, Di = I, Dv = V, Dde = DMe , Ddi = DMi , Ddv = DMv , Dw = W
Dg = {f | f is a partial function from DM into E ∪ I ∪ V that respects
the types}
D(ab) = {f | f is a partial function from Da into Db }
F interprets the predicates by assigning them appropriate intensions.

All the functional domains are partial. It follows from the definition that
Dc = D(g(wt)) = sets of <f,w> pairs. Similarly, Dπ = Dcc i.e. the partial func-
tions from Dc into Dc.
The functions in Dg are called d-assignments. All the d-assignments in a con-
text c have the same domain. The variables used by these assignments are
denoted as dom(c). The worlds used in a context c are called the proposition
of c.

(76) Where c is a context:

a. f ∈ c := ∃w <f,w> ∈ c
b. dom(c) = ∪ f∈c dom(f)
c. Prop(c) = {w | ∃f <f,w> ∈ c}

(77) Truth
A context c is true in a world w if w ∈ Prop(c).

The interpretation function for our language depends on a model M and an

outer variable assignment a. We use the usual conventions, that is, a[x/u] is
that assignment a that is like a except for the variable x, which is mapped to
u. Variable assignments must not be confused with dm-assignments, which
may be thought of as Kamp’s embedding functions (here: inner variable
assignments). We use the +-notation of Heim (1982, 1983a), that is, if p is
a CCP and c is a context, then c + p stands for p(c). Since p is a partial func-
tion, the application might be undefined. The admissibility condition for p
is what is called the presupposition of p.
The semantics of the language is defined below:

(78) Definition of [[. . .]]M,a

Con. If α is a constant of type b, [[α]]M,a = F(α) [an intension].
Var. If x is a variable of type b, [[x]]M,a = a(x).
FA. If α is an expression of type ab and β is an expression of type a,
[[α(β )]]M,a = [[α]]M,a ([[β]]M,a ).
PM. If P and Q are expressions of type (dπ),
then [[[P Q]]]M,a = λd ∈ Dd .λc: d ∈ dom(c).{<f,w> | (∃g ∈ c) g ⊆ f &
<f,w> ∈ ((c + [[P]]M,a (d)) + [[Q]]M,a (d))}
[not an intersective ‘and’; but a dynamic ‘and’].
Abstraction. If x is a variable of type b and α is an expression of type
c, then [[[λx α]]]M,a = λu ∈ Db. [[α]]a[x/u] .

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012
 Tense 341

Dynamic predication
Let us illustrate the system by going through the evaluation of the sentence
he3 sleeps, ignoring tense.

(79) SLEEPde,π (3de )

Since the expression doesn’t contain a variable, the evaluation doesn’t

depend on the outer variable assignment a. Recall that the dm 3de is a con-
stant. Therefore:

(80) [[SLEEPde,π (3de )]]M

= [[SLEEPde,π ]]M ([[3de ]]M ) (by FA)
= F(SLEEPde,π )(F(3de ))

The result should be a CCP. In order to know which CCP it is, one has to
look at the interpretation of SLEEP by F. Recall that we assume with Beaver
that SLEEP uniquely determines a corresponding static predicate of type (et)
written in small letters, here sleep. The correspondence is given by the fol-
lowing meaning postulate (Beaver’s MP8):

(81) MP8 (Beaver, 2001, 178)

Let M be a model and let P be a dynamic predicate constant of type
(d, cc). Then there is static predicate constant V of type (et) such that
for every dm d: F(P)(d) = λc: d ∈ dom(c).{<f,w> ∈ c | f(d) ∈ F(V)(w)}

MP8 links dynamic predicates with static ones. Since F(sleepet ) is an inten-
sion, F(sleep)(w) is an extension. If d is a dm of type de, then f(d) will be an
individual that has to be in the set of the sleepers in w. Beaver’s postulate
MP8 can be generalised in an obvious way to n-place dynamic predicates.
Applied to the meaning of F(SLEEP), the postulate gives us the following:

(82) F(SLEEP(de,cc) ) = λd ∈ Dde .λc ∈ Dc : d ∈ dom(c).{<f,w> ∈ c | f(d) ∈

F(sleepet )(w)}

Suppose F(3de ) = 3de , then we have:

(83) F(SLEEPde,π )(F(3de )) = F(SLEEPde,π )(3de )

= λd ∈ Dde .λc ∈ Dc : d ∈ dom(c).{<f,w> ∈ c | f(d) ∈ F(sleepet )(w)}(3de )
(by F and MP8)
= λc ∈ Dc : 3de ∈ dom(c).{<f,w> ∈ c | f(3de ) ∈ F(sleepet )(w)}
(λ-conversion)

We add the definition of truth in a context.

(84) Let c be a context, w a world and p a CCP. Then p is true with respect
to w and c if w ∈ Prop(c + p). (i.e. iff f(3de ) sleeps in w.)

Downloaded from https://www.cambridge.org/core. The University of British Columbia Library, on 04 Jan 2019 at 09:03:02, subject to the Cambridge Core terms of
use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9781139236157.012