Semantics: Handout 10
Tense and Modal Operators
Yılmaz Kılıçaslan
December 10, 2009
1 Introduction
Up to this point we have expanded our semantic machinery along an extensional direction. In
this lecture we will expand it in another way, this time in the direction of an intensional
language -- a language whose expressions are provided a definition of denotation not just for a
single state of affairs but for many possible states of affairs. We will begin this expansion by
considering the special case of tense operators and their interpretation and afterwards we will
look at some other phenomena, namely the modalities of necessity and possibility, that will
enable us to formalize “possible states of affairs” in more general terms.
2 Priorian Tense Operators
Up until now we have been assigning a truth value to a formula relative to a model “once and
for all”, thus ignoring the fact that many, if not most, sentences of natural languages may be
now true, now false, as circumstances in the world change. Thus to understand a sentence
such as ‘John is asleep’, it must be made clear to us implicitly or explicitly at what time it is
intended to “apply”, or else it gives us little useful information about the world. For a present
tense sentence it suffices (generally) to know when the sentence was uttered, since present
tense sentences (in one of their most common uses in English) are understood to describe the
state of the world that obtains concurrent with the time of utterance. It will be most practical
to begin by regarding all the existing formulas of our formal languages as “present tense”
formulas and state their truth conditions relative to a moment of time.
We first relativized the denotation (i.e. truth value) of formulas to an arbitrary model, then to
a model and assignment of values to variables; now we will relativize their denotations to a
model, value assignment, and to a moment in time (out of a given set of times). This given
moment in time on which we base our definition can be intuitively regarded as analogous to
the time of utterance of a natural language sentence, though of course we are not attempting to
represent the process of uttering and of comprehending sentences, nor of any kinds of speech
acts, in our semantic theory proper.
Accordingly, we will expand the definition of a model to include not only a domain of
individuals A and an interpretation function F, but also a non-empty set I with a linear
ordering imposed upon it (to be regarded intuitively as the set of moments in time of the
model). We will use the symbol < for this ordering. That is, < is simply a linear ordering of
the set I (i.e. a relation on I satisfying the conditions i i, (i < j  j < k)  i < k, i < j  i = j 
j < i for all i, j, and k in I). The expression i < i’ may be read as “i is earlier than i’ ” or
equivalently, “i’ is later than i.” Since we want to allow the truth values of formulas to come
out different for different times, we must obviously allow the denotations of the non-logical
constants to be different for various times. Thus the function F assigning a denotation to each
non-logical constant “once and for all” will now become a function of two arguments, a
constant and a member of I. For any constant  and any i in I, F(i, ) is understood to be the
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denotation of  at the time i. Of course, the denotation of certain non-logical constants may be
allowed to be the same for all times in the model – in natural languages, names typically
denote the same indvidual throughout time, but the denotation of predicates such as ‘is asleep’
changes constantly.
Formally, we will define a temporal model for a language L as an ordered quadruple A, I, <,
F such that A and I are any non-empty sets, < is a linear ordering on I, and F is a function
assigning to each pair consisting of a member of I and a non-logical constant of L an
appropriate denotation (out of the set of possible denotations we allow for each category of
non-logical constants in L). In the semantic rules we will uniformly replace each definition of
[]M, g with the definition []M, i, g – the denotation of  relative to the model M, time i, and
value assignment g.”
The interesting thing about the temporal interpretations of natural languages is that all such
languages have means for forming sentences that can be uttered at one time but are
nevertheless about a situation that obtains at a different time, a time either earlier or later than
the time of utterance. Perhaps the most obvious of these temporal indicators in a language like
English – and the kind of temporal indicator which most logical studies have taken as a
paradigm – are the tenses. The most common kind of tense logic is based on the assumption
that the present tense can be taken as the starting point for sentences in all tenses. That is, a
“present tense” formula is just a tenseless formula simpliciter, and other tenses – specifically,
past, future, past perfect and future perfect – can be best explained as the application of a
tense operator to a tenseless (i.e. present tense) formula. Thus to form a tensed language from
a language we have seen before, such as L2, we need to add only two operators:
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a future tense operator which we will symbolize as F and
-
a past tense operator which we will symbolize as P.
It was Arthur Prior who introduced these operators (as we will see below, meaning “at some
Future time”, and “at some Past time” respectively).
As for syntactic rules, we will simply say that if  is any formula, then F and P are
formulas. These are customarily read as “it will be the case that ” and “it was the case that
”, respectively. Thus, we add the following rules of syntax to the temporally interpreted L2 to
give the tensed first-order language L2t:
Syn B.10.
If  is a formula, then F is a formula.
Syn B.11.
If  is a formula, then P is a formula.
3 A Priorian Interpretation of Tense Operators
The semantic plausibility of this way of constructing pasts and futures out of presents comes
from the fact that it is intuitively plausible to say that a sentence ‘John was asleep’ is true
relative to a time t (i.e. if it were uttered at t) just in case the present tense sentence ‘John is
asleep’ is true relative to some other time t’, namely, relative to a time earlier than t. Likewise,
a future tense sentence ‘John will be asleep’ seems to be true relative to a time t just in case
the present tense sentence ‘John is asleep’ is true relative to a different t’ , but this time a t’
later than t. Thus on this view the past and future tenses are construed as a means for
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“shifting” the ordinary truth conditions of a present tense (i.e. tenseless) sentence to a
different point in time. This view leads to the following semantic rules, which we add to L2:
Sem B.10.
If  is a formula, then [F]M, i, g = 1 iff there is some i’ in I such that i < i’ and
[]M, i’, g = 1; otherwise, [F]M, i, g = 0.
Sem B.11.
If  is a formula, then [P]M, i, g = 1 iff there is some i’ in I such that i’ < i and
[]M, i’, g = 1; otherwise, [P]M, i, g = 0.
Since syntactic rules B.10 and B.11 make a formula from another formula, we can produce
formulas with more than one tense operator by using these rules recursively. This will enable
us to produce expressions that correspond more or less naturally to some of the compound
tenses of English. A formula PP corresponds to an English sentence in the past perfect tense,
and a formula FP corresponds to a future perfect sentence. The first type of formula will, by
the semantic rules, “direct” us to some time preceding the time of utterance at which P is
true, and this in turn “directs” us to some time preceding even this at which  is true. The
second formula similarly directs us to some future time at which P is true, and this in turn
directs us back to some time preceding this future time at which  is true.
4 A Reichenbachian Approach to Tense
Note that the above approach to the interpretation of tense operators does not offer a natural
way of treating the present perfect tense in English. Or, from a different point of view, what
we with have with a P preceding a bare formula is an interpretation with a present perfect
tense and we are unable to express the simple past tense in that case. The underlying problem
here is this: The Priorian approach fails to take into account the importance of reference to
specific times in the semantics of tense. As Areces and Blackburn (2005) point out, “[t]he
sentence ‘Dov smiled’ does not mean that at some completely unspecified past time Dov did
in fact smile (which is the meaning the tense logical representation [Psmile(dov)] gives it).
Rather, it means that at some particular, contextually determined, past time Dov did in fact
smile. Prior’s tense logical representations are interesting, and correct as far as they go, but
they do not go far enough” (p. 2). In order to find a way out of the problem, consider first the
following example:
(1)
a. I took a cab back to the hotel.
b. The cab driver was Latvian.
As Michaelis (2006) notes, “[i]f a speaker makes the assertion in (1b) following that in (1a),
no sensible hearer will respond by asking whether the cab driver is still Latvian now. This is
presumably because the cab driver’s Latvian identity is highly unlikely to desist following the
cab ride” (p. 221). So, a question comes to mind as to why the speaker of (1b) has chosen to
‘locate’ the cab driver’s Latvian identity in the past. A plausible answer to this question can
be deduced from Reichenbach’s (1947) account of tenses. He viewed temporal reference as
central to the semantics of tense in natural language. According to him, tenses do not express
the relationship between the time of utterance and the time of the state of affairs described.
Rather, tenses express the relationship between utterance time and a reference time (R).
Michaelis (2006) adds that “[i]n (1a), R is a specific past time both the speaker and hearer can
identify, while in (1b) R is the time established by (1a): the time of the cab ride. What (1b)
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shows us is that when a speaker makes a past-tense stative assertion, she or he may vouch
only for that portion of the state’s tenure that coincides with the mutually relevant interval”
(p. 221).
To speak more precisely, Reichenbach distinguished tenses in terms of a sequence of three
time points, namely what he called the point of speech (S), the point of event (E), and the
point of reference (R). Below is a tabulation of Reichenbach’s analyses of the tense forms of
English:1
Table 1. Reichenbach’s referential analysis of tense
TENSE
Past perfect
Past
Future in the past
Future in the past
Future in the past
Present perfect
Present
Prospective
Future perfect
Future perfect
Future perfect
Future
Future in the future
REPRESENTATION
E<R<S
E=R<S
R<E<S
R<S=E
R<S<E
E<S=R
S=R=E
S=R<E
S<E<R
S=E<R
E<S<R
S<R=E
S<R<E
ENGLISH EXAMPLE
I had spoken
I spoke
I would speak
I would speak
I would speak
I have spoken
I speak
I am going to speak
I will have spoken
I will have spoken
I will have spoken
I will speak
(Latin: abiturus ero)
We end this section with a final remark. Areces and Blackburn (2005) note that much of what
Reichenbach says is compatible with Prior’s views:
For a start, point of speech is the time at which the sentence is uttered, and this
concept is fundamental to the [Priorian] tense logic: it’s simply the particular time
at which we chose to evaluate a formula in a given model. The point of event is
the time at which the eventuality the sentence is talking about takes place. This
might be the same time as the point of speech, or to its past, or to its future. This
concept also fits naturally with Prior’s tense logic. If  is the representation of
some eventuality, then evaluating  at some time amounts to identifying point of
event with point of speech. Prefixing P to form P locates the point of event to the
past of the point of speech. Prefixing F to form F locates the point of event to the
future of the point of speech.
However, Reichenbach’s key innovation was the point of reference, and here
we encounter something that orthodox tense logic cannot handle. (p. 3)
Actually, in Reichenbach’s representations, the time points are separated either by a line or by a comma. The
line indicates that the left hand point lies to the past of the right hand point. The comma indicates that the two
points are not ordered with respect to one another but identical.
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5 Modal Operators: Necessity and Possibility
The truth value of sentences like
(2)
Barack Obama has been elected President of the U.S.
can vary as a function of the time with respect to which we evaluate their truth. Obviously
their truth value can vary with respect to factual contingencies also. It is easy to imagine a
different outcome of the 2008 presidential election which would have made (2) false rather
than true. We now take up some considerations relating to this kind of variation in truth value.
For the present we set aside the dependence of denotation on time in order both to simplify
the discussion and to bring out more clearly some formal parallels between variation of
denotation with the passage of time, on the one hand, and with change of factual
circumstances, on the other. Later we will treat the two together.
In addition to sentences like (2), some phrases of other categories have varying denotations as
a function of factual contingencies. Consider, for instance, those in (3).
(3)
a. The president of the U.S.
b. holds the winning poker hand.
We have no difficulty in imagining that, at the time John Kennedy was assassinated, the
situation might have been slightly different in that he would have recovered from his wounds
(or the assassin’s shots would have missed). In that case (3a) would have referred to John
Kennedy when it in fact referred to Lyndon Johnson. Similar observations can be made about
(3b).
In contrast to these expressions, those in (4) could not change denotation if some fact about
the world were different from what actually is.2
(4)
a. The President of the U.S. is the President of the U.S.
b. The set that is included in every set.
c. is a number that is not identical to itself.
A sentence like (4a) is a necessary truth in the sense that it could not possibly have been false.
This sense of necessity is recognizable from the preceding sentence as the dual of possibility,
in the sense of logical possibility. That is, for a sentence , it is necessary that  is equivalent
to it is not possible that it is not the case that . Some necessarily true sentences are not, like
(4a), true solely because of the logical words they contain, but in part because of nonlogical
words whose denotations necessarily stand in certain relationships to one another. For
example,
(5)
All bachelors are unmarried men
is necessarily true because of the fact that it is logically impossible for the denotation of
bachelor not to be a subset of both the denotation of man and that of unmarried.
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We exclude from consideration here changes of facts concerning what English phrases and sentences mean.
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It is customary to divide the set of all true statements into the contingently true (those that are
true as a matter of fact, though they might have been otherwise had the world been different,
e.g. Paris is the capital of France) and the necessarily true (those that could not be denied
without contradiction or departure from the normal meanings of words of the language, such
as mathematical and logical truths and analytic statements). Likewise, the false statements are
divided into the contingently false (e.g. Vincennes is the capital of France) and the
necessarily false (e.g. 2 + 2 = 5).
Now we can construct a formal language with the pairs of concepts possible/necessary treated
as sentence operators by adding two new syntactic rules to L2 to give the tensed first-order
language L2m:
Syn B.10.
If  is a formula, then □ is a formula.
Syn B.11.
If  is a formula, then ◊ is a formula.
The question now is how to interpret the necessity and possibility operators. Leibnitz is
credited with the idea that a necessary truth is a statement that is not merely true in the actual
world, but is true in “all possible worlds.” Following up on Leibnitz’ idea in constructing a
formal interpretation for modal operators parallel to the one for tense operators given above,
we will take it is necessary that  (i.e. □) as true in the actual world if and only if  is true in
all possible worlds. If possibility is to be the dual of necessity, then this commits us to the
truth of it is possible that  (i.e. ◊) in the actual world whenever  is true in at least one
possible world.
Just as we relativized the temporal definitions of denotation to a moment in time which was
intuitively regarded as “now”, the moment of utterance, we will relativize the definition of
denotation in L2m to an arbitrary member of the set of possible worlds which we will regard as
the “actual” world. Thus the definitions themselves do not single out any particular world as
the actual ones; we may alternately choose one or the other of them as the “actual” world, all
other worlds becoming possible but not actual worlds relative to this choice. Thus the formal
definitions needed are these:
A model for the modal language L2m is an ordered triple A, I, F such that A and I are any
non-empty sets and F is a function assigning to each pair i, α consisting of a member of I
and a non-logical constant of L2m an appropriate denotation (out of the set of possible
denotations we allow for each category of non-logical constants in L2m).
The semantic rules of L2m will be the same as for L2 except for uniformly defining [α]M, i, g
instead of [α]M, g. To these rules we add the new rules:
Sem B.10. If  is a formula, then [□]M, i, g = 1 iff for all i’ in I []M, i’, g = 1.
Sem B.11. If  is a formula, then [◊]M, i, g = 1 iff for some i’ in I []M, i’, g = 1.
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6 Languages Containing both Tense and Modal Operators: Coordinate
Semantics
We have now seen two ways of interpreting operators relative to a model consisting of a
domain of individuals A and a set of “other things” I; we thought of these first as times, then
as “possible worlds”. Now we will consider a language that contains both tense and modal
operators. For the interpretation of this language we will need a model with two sets besides
the set of individuals. We will designate these as the set W (which is to be thought of as the
set of worlds) and the set T (which is to be thought of as the set of times); < will be the
ordering of the set T. Thus we can think of our “semantic space” as expanded to two
dimensions. We have, as it were, differing worlds as we move along one axis and differing
times as we move on the other axis. Any particular point on this plane can be metaphorically
thought of as being a pair of coordinates w, t for some w in W and t in T; that is, a point
whose location is determined by which world it is in on the one hand, and by what time it is
on the other hand. We will call such a pair an index. Any one of these can represent “the
actual world now”, and our definitions of denotation will now be relativized to a choice of
some arbitrary index <w, t>. Thus our semantic rules will now give a definition of [α]M, w, t, g
for each expression α (i.e. the denotation of α relative to a model M, possible world w, time t,
and assignment of values to variables g). Below are the syntactic and semantic rules which we
will add to L2 to give a new first-order modal tensed language, which we will call L2mt:
Syn B.10.
If  is a formula, the □ is a formula.
Syn B.11.
If  is a formula, then ◊ is a formula.
Syn B.11.
If  is a formula, then F is a formula.
Syn B.12.
If  is a formula, then P is a formula.
Sem B.10. If  is a formula, then [□]M, w, t, g = 1 iff []M, w’, t’, g = 1 for all w’ in W and t’ in T.
Sem B.11. If  is a formula, then [◊]M, w, t, g = 1 iff []M,w’,t’,g = 1 for some w’ in W and t’ in T.
Sem B.12. If  is a formula, then [F]M, w, t, g = 1 iff []M, w, t’, g = 1 for some t’ in T
such that t < t’.
Sem B.13. If  is a formula, then [P]M, w, t, g = 1 iff []M, w, t’, g = 1 for some t’ in T
such that t’ < t.
PS: The content of this handout is mostly adopted from Dowty et al (1981).
References
Areces, C. and Blackburn, P. (2005) Reichenbach, Prior and Montague: A semantic gettogether. In S. Artemov, H. Barringer, A. d'Avila Garcez, L. Lamb, and J. Woods, editors,
We Will Show Them: Essays in Honour of Dov Gabbay, College Publications.
Dowty, D.R., Wall, R. E. and Peters, S. (1981) Introduction to Montague Semantics.
Dordrecht, The Netherlands: Kluwer Academic Publishers.
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Michaelis, L. A. (2006) Time and Tense. In B. Aarts and A. MacMahon, (eds.), The
Handbook of English Linguistics. Oxford: Blackwell. 220-234.
Reichenbach, H. (1947) Elements of Symbolic Logic. Random House, New York, 1947.
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