Artificial Intelligence Research Report AI-1991-02
Research Report AI-1991-02
Tense and Conditionals
Donald Nute
Artificial Intelligence Research Group
Boyd Graduate Studies Center
The University of Georgia
Athens, Georgia 30602 U.S.A.
NOTE: This paper uses many special characters
which are lost in the ASCII text version and may
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TENSE AND CONDITIONALS
Donald Nute
Department of Philosophy
Artificial Intelligence Programs
The University of Georgia
Athens, GA 30602
DNUTE@UGA.CC.UGA.EDU
Abstract
In this paper I explore the possibilities for developing a formal language
containing both tense and conditional operators and a model theory for such a
language.1 The criteria for success will be that we may provide formal
counterparts for a wide variety of English conditionals and that the truth
conditions for these formal counterparts will be appropriate for the English
conditionals which they represent.
Temporal relations play an essential role in determining the truth values of
many and perhaps most conditional assertions. This fact is recognized and
explored by many logicians including David Lewis (1979) and John Pollock
(1981), yet the attention which investigators of the logic of conditionals
have given to temporal relations has not in general included an explicit
consideration of the interaction of tense and conditional constructions. Two
exceptions are Thomason and Gupta (1980) and van Fraassen (1980) who do
develop an account of the logical and semantical properties of conditional
sentences based upon the occurrence of various tenses within those sentences.
This paper will include a critique of this account, particularly as it is
developed by Thomason and Gupta, and a "correction" of what I take to be some
of the major problems of this account. Beyond that, the paper will be an
exploration of issues which have not received very much attention by logicians
and philosophers of language.
The next eight sections assume that time can be represented as a linearly
ordered set of points or instants. Sections 1 - 3 provide background summaries
of techniques developed in tense logic and of techniques developed in condi-
tional logic, but no attempt is made in these sections to integrate tense and
conditional logic. In section 2, I also make a distinction between two kinds
1Part of the research for this paper was performed during the summer of
1981 while the author was a participant in a project in tense logic conducted
at the University of Stuttgart under the direction of Professor Christian
Rohrer and with the support of the Deutsche Forschungsgemeinshaft. This
research was also supported by a grant from the University of Georgia. I am
grateful to both institutions for their support. I also wish to thank Chris-
tian Rohrer, Franz Guenthner, Dov Gabbay, and Hans Kamp for their helpful
comments and criticisms.
Part of the material from this paper has been revised and published in
(Nute 1991).
1
Tense and Conditionals 2
of conditionals which I call material and intentional conditionals. These two
kinds of conditionals will require different analyses. A semantics for a
formal language containing both tense and conditional operators is developed
and criticized in sections 4 and 5, and an alternative language containing
special tensed conditional operators is developed, provided with a semantics,
and evaluated in sections 6 and 7. The discussion in sections 4 - 7 is
restricted to intensional conditionals. In section 8, we look at the affects
of tense on material conditionals and at some special problems which arise in
trying to distinguish intensional from material conditionals where the future
tense is concerned.
Sections 9 - 11 explore an interpretation of tense which is based on a
non-linear model for time, a non-deterministic, branching time. Such a
conception of time allow us to entertain the Aristotelian notion that contin-
gent future tense sentences may lack truth values. Some of the puzzling
features of such a semantics for tense are emphasized when we try to adapt
this semantics to a formal language containing both tense and conditional
operators. I offer a semantics employing what I call pseudo-branching time as
an alternative to the branching time of Thomason and Gupta, and I argue that
this semantics avoids certain objectionable metaphysical assumptions found in
the Thomason-Gupta account.
1. Tense Logic for Linear Time
We will rely on familiar techniques of tense logic in our investigation of
those special problems which arise when we mix tense and conditionality. Our
initial assumptions about the nature of time will be very limited. In our
first semantics for tensed language, we will represent time as a set of
moments or instants of time linearly ordered by an earlier-than relation. We
will not be concerned with such questions as whether time has a first or last
moment, whether time is dense or continuous, etc., nor with the problems
involved in expressing various answers to these questions within a formal
language containing tense operators. The interested reader may refer to
Burgess (1984) for a survey of tense logic, including discussion of these
issues.
We will begin our examination of the logic of tense by developing a formal
language within which we can hope to represent various ordinary English
sentences involving tense. Actually, we will simplify our task in this section
and the rest of this paper by confining our attention to sentential languages.
In this way we put off for the time being any problems which may arise when we
try to incorporate machinery for representing tense within a quantificational
language. We construct our formal language for tense logic by adding four
monadic sentence operators P, F, H and G to a language for classical senten-
tial logic which contains infinitely many sentence letters A, B, C, etc., and
the usual truth-functional operators ª, , , , and ð. Using p, q, r, etc.,
as sentence variables, we may read Pq as `It has been the case that q', Fq as
`It will be the case that q', Hq as `It has always been the case that q', and
Gq as `It will always be the case that q'. (Here and elsewhere I use formal
sentences autonomously to denote themselves. I believe that no confusion will
result from this.)
Tense and Conditionals 3
A model for our tensed language is an ordered triple <T,®,[]> satisfying the
following conditions:
1.1 T = í.
1.2 ® is a strict, total ordering of T; i.e., ® is a relation in T
which is connected in T, asymmetric, and transitive.
1.3 [] is a function which assigns to each sentence q of our formal
language a subset [q] of T.
1.4 [ªq] = T - [q], [q r] = [q] ï [r], and so on for the other
truth-functional connectives.
1.5 t î [Pq] iff there is a t1 such that t1 ® t and t1 î [q].
1.6 t î [Fq] iff there is a t1 such that t ® t1 and t1 î [q].
1.7 t î [Hq] iff for every t1 such that t1 ® t, t1 î [q].
1.8 t î [Gq] iff for every t1 such that t ® t1, t1 î [q].
Intuitively, T represents the set of all moments or times, ® represents the
earlier-than relation, and [q] represents the set of all times at which q is
true. The conditions 1.5 - 1.8 provide truth conditions for sentences contain-
ing one of our tense operators. Another way of developing our semantics would
be to interpret the sentences of our formal language as being true or false
over an interval of time rather than at individual times. An interval would be
a subset I of T such that for any times t,t1,t2 î T, if t,t1 î I, t ® t2, and
t2 ® t1, then t2 î I. This might be more appropriate for interpreting English
sentences like `He ran a mile', since it is obvious that there is no single
moment of time at which it is true that he runs a mile. An interval semantics
will still allow us to interpret a sentence q as being true at an individual
time t since we can say that q is true at t just in case q is true at the
interval whose only member is t. For further discussion of interval semantics
and its advantages, see Humberstone (1979). For present purposes, we will
simplify our task by avoiding examples
which might require the use of an interval semantics.
2. Indicative, Subjunctive, Material, and Intentional Condi-
tionals
The prime example of a conditional in English is a sentence which contains the
words `if' and `then'. Examples of sentences of this sort are
2.1 If Anthony's door is unlocked, then he will be back soon.
and
2.2 If Anthony had left for the weekend, then he would have locked his
door.
Tense and Conditionals 4
Of course, the word `then' could be omitted in either of these
sentences without any change in meaning. We could also reverse
the order of the antecedent (grammatically, the dependent clause)
and the consequent as in
2.3 Anthony would have locked his door if he had left for the weekend.
It is also possible to omit both `if' and `then' in conditionals like 2.2
which contain verbs in the subjunctive mood. We do this by changing the order
of the subject and verb in the antecedent of the conditional as in
2.4 Had he left for the weekend, Anthony would have locked his door.
We see that while the words `if' and `then' readily come to mind when we think
of English conditionals, there are really a number of constructions in English
which may be used to produce sentences of the sort we want to consider. The
important feature of the conditional sentence semantically is the presence of
an antecedent and a consequent, where the antecedent expresses some condition
which somehow mitigates the sense normally expressed by the consequent.
Certain constructions signal special kinds of conditionals which have their
own truth conditions. Examples are `might' conditionals like
2.5 If we had invited Frank, he might have come.
and `even if' conditionals like
2.6 Even if we had invited Frank, he wouldn't have come.
Note, however, that we can delete the word `even' in 2.6 without a change of
meaning. This means that a conditional can have the logical and semantical
properties of an `even if' conditional even though it does not contain the
word `even'. I will say nothing more about these kinds of conditionals
although they have some interesting properties. There is further distinction
between different kinds of conditionals, however, which I will discuss. Some
pairs of conditionals seem to have exactly the same structure except that the
verbs in one member of the pair are all in the indicative mood while the verbs
in the other member of the pair are all in the subjunctive mood. Furthermore,
it is often the case that one member of such a pair is true while the other
member is false. One such pair is
2.7 If Nute didn't write this paper, then someone else did.
2.8 If Nute hadn't written this paper, then someone else would have.
2.7 is true and 2.8 is false, yet the two conditionals have the same apparent
antecedent and consequent. Thus, 2.7 and 2.8 represent distinct ways in which
a condition contained in the antecedent of a conditional may mitigate the
sense of the consequent of the conditional. 2.7 and 2.8 represent different
kinds of conditionals having different truth conditions. Investigators have
for the most part associated the difference between 2.7 and 2.8 with the
Tense and Conditionals 5
difference in the mood of the verbs and hence distinguished `indicative'
conditionals like 2.7 from `subjunctive' conditionals like 2.8.
Examples like these certainly point to the existence of two different kinds of
conditionals in ordinary usage, but it may be a mistake to identify this
difference with the difference in the moods of the verbs. Consider, for
example
2.9 If President Reagan runs for another term, he will win.
2.10 If President Reagan were to run for another term, he would win.
The inclination of the native English speaker, I believe, will be to say that
these two conditionals must have the same truth value. Nor is this a peculiar-
ity of these two specific conditionals. It is difficult and perhaps impossible
to find two conditionals, one indicative and the other subjunctive, involving
the same future tense antecedent and consequent, which strike us as being as
clearly different in their truth conditions as are 2.7 and 2.8. The difference
which investigators draw between indicative and subjunctive conditionals might
not be a difference which is invariably signalled by the mood of the verbs
after all. It may be true that indicative and subjunctive conditionals in the
past and present tenses have different truth conditions, but distinguishing
future tense conditionals on the basis of mood is unreliable.
I suggest that the truth conditions for future tense conditionals are usually
very much like those for past and present tense subjunctive conditionals,
while past and present tense indicative conditionals have different truth
conditions. David Lewis (1973) and others have suggested that all indicative
conditionals have truth conditions very similar to those of the material
conditionals of classical sentential logic. This seems a likely analysis for
past and present tense indicative conditionals and it is the analysis which I
will adopt in this paper, with some modifications to be developed in section
8. With this in mind, I propose that we adopt a new nomenclature for these two
kinds of conditionals. I suggest that we call a conditional `material' if it
has the same truth conditions as the material conditionals of classical
sentential logic, i.e., if the conditional is true just in case its antecedent
is false or its consequent is true. I am suggesting that most and perhaps all
past and present tense indicative conditionals are material conditionals. On
the other hand, I propose that we call an English conditional `intensional' if
it is not material and instead has the same truth conditions which most
subjunctive conditionals have. The appropriateness of this label will become
clearer in the next section.
Any classification of conditionals which is based upon the moods or tenses of
the verbs occurring in the conditionals is an explicitly grammatical or
syntactic classification. The distinction between material and intensional
conditionals, on the other hand, is a semantic distinction. The long-standing
assumption which I am questioning is that there is a simple relationship
between these syntactic and semantic distinctions. Of course, there may be a
regular connection between the mood and tense of the verbs in the conditional
and the semantic category of the conditional even if this connection is not
the one I am questioning. For example, it is tempting to think that all future
Tense and Conditionals 6
tense conditionals are intensional conditionals. But I believe that this would
also be an oversimplification. I will attempt a better explanation of future
tense indicative conditionals in section 8.
3. The Logic of Intentional Conditionals
In recent years we have seen a number of proposals for interpreting inten-
sional conditionals. A review of these proposals is beyond the scope of this
paper, but the interested reader may wish to consult Nute (1984). The later
sections of this paper will rely upon one or the other of two model theories
for a formal language for conditionals, each of which uses the notions of a
possible world and of a selection function on the sentences of the language
and a set of possible worlds.
When the antecedent of an intensional conditional is false, we cannot deter-
mine the truth value of the conditional by considering the truth values of its
component antecedent and consequent. The simple fact is that English condi-
tionals are not always truth-functional, and it is those conditionals which
are not truth-functional that are intended by our term `intensional'. For
example, the conditional
3.1 If Reagan were bald, he could stick his elbow in his ear.
is clearly false even though both its antecedent and its consequent are false.
The corresponding material conditional, of course, is true. Robert Stalnaker
(1968) suggests that we evaluate such conditionals as 3.1 by performing a kind
of thought experiment in which we imagine, construct, or consider counterfac-
tual situations in which the antecedent of the conditional is true and
determine whether or not the consequent is also true in these situations. Each
of these situations represents a different way the world might have been, what
is often referred to as a possible world. So Stalnaker's procedure for
determining the truth value of an intensional conditional involves determining
whether the corresponding material conditional is true in certain possible
worlds where the antecedent of the intensional conditional is true. Many other
proposals have shared this basic approach. The differences in these different
proposals have concerned the way in which the appropriate worlds are to be
chosen.
Stalnaker's particular proposal, like many others, depends upon the idea that
it makes sense to talk about the relative similarity between different worlds.
For a given counterfactual antecedent q, one world in which q is true may be
more similar to the actual world than is some other world in which q is true.
Stalnaker proposes that for any antecedent q, if it is possible for q to be
true at all then there is some possible world at which q is true which is more
like the actual world than is any other possible world at which q is true. If
we call a world at which q is true a `q-world', then Stalnaker's assumption is
that for every sentence q, either q is impossible or there is some unique
q-world which is most similar or `closest' to the actual world.
Our formal language is obtained by augmenting the language of classical
sentential logic with a special dyadic operator >. We will use the subjunctive
Tense and Conditionals 7
mood in reading the conditional sentences of this language, e.g., we will read
`q > r' as `If it were the case that q, then it would be the case that r'.
Stalnaker's interpretation of such a language involves what we will call world
selection function models. A world selection function model for our condition-
al language is an ordered triple <W,f,[]> satisfying the following conditions:
3.2 W is a non-empty set.
3.3 f is a function which assigns to a sentence q and a member w of W
either the empty set or a member f(q,w) of W.
3.4 [] is a function which assigns to each sentence q of our condi-
tional language a subset [q] of W.
3.5 [ªq] = W - [q], [q r] = [q] ï [r], and so on for our other
truth-functional connectives.
3.6 If f(q,w) is not empty, then f(q,w) î [q].
3.7 w î [q > r] iff f(q,w) is empty or f(q,w) is contained in [r].
3.8 If w î [q], then f(q,w) = {w}.
3.9 If f(q,w) = í, then f(r,w) ï [q] = í.
3.10 If f(q,w) î [r] and f(r,w) î [q], then f(q,w) = f(r,w).
Where <W,f,[]> is a world selection function model, the intended interpreta-
tion of W is as a non-empty set of possible worlds, the intended interpreta-
tion of [] is as a function which tells us for each sentence q the set of
those worlds at which q is true, and the intended interpretation of f is as a
function which tells us for each sentence q and world w which world at which q
is true is most like w. The motivation for conditions 3.2 - 3.7 should be
obvious, and the motivation for 3.8 - 3.10 only slightly less obvious. The
class of world selection function models characterizes Stalnaker's favorite
conditional logic C2. Axiomatizations of C2 and discussions of the motivation
for and adequacy of Stalnaker's semantics can be found in several places,
including Stalnaker (1968). The reader should be warned that the present
formulation of the Stalnaker semantics differs from Stalnaker's original
formulation in certain ways. In particular, we assign the empty set as the
value of f(q,w) when there is no q-world at all similar to w. Stalnaker
posited an absurd world at which all sentences are true to play a similar
role.
One consequence of world selection function semantics which we must take note
of is Conditional Excluded Middle.
CEM: (q > r) (q > ªr)
If f(q,w) is empty, then clearly w î [q > r] by 3.7. On the other hand, if
f(q,w) = w1, then w1 î [r] or w1 î [ªr] by 3.5, and thus w î [q > r] or w î [q
> ªr] by 3.7. So CEM is true at every world in every world selection function
Tense and Conditionals 8
model. But CEM is not universally accepted as a logical truth. In fact, more
authors seem to have rejected CEM than have accepted it. Consider the follow-
ing two conditionals:
3.11 If Robert had wrecked his bicycle, he would have broken his arm.
3.12 If Robert had wrecked his bicycle, he would not have broken his
arm.
In most contexts where the antecedent of 3.11 and 3.12 is false, we would
likely say that both 3.11 and 3.12 are false. The simple fact is that if
Robert had wrecked his bicycle, he might or might not have broken his arm.
Despite the evidence against CEM, we consider Stalnaker's semantics here
because both Thomason and Gupta (1980) and van Fraassen (1980) use Stalnaker's
semantics as the foundation for their discussions of tense and conditionals.
We can avoid CEM if we allow our selection function to pick out a class of
possible worlds instead of an individual world. It seems reasonable that we
should consider more than one way things might be if the counterfactual
antecedent of a conditional were true. Consider, for example, a roll of a die
where an ace comes up. If we consider what would have happened if an ace had
not come up, we will surely consider at least five different worlds, one for
each of the other five values which might have come up on that roll of the
die. Contrary to Stalnaker's assumption, it would seem that there is no unique
closest world in which an ace is not rolled, but rather that there are several
worlds which are equally similar to the actual world. We will want to consider
each of these worlds in determining the truth value of a conditional like
3.13 If an ace had not come up, Clyde would have won his wager.
We would say 3.13 is true just in case Clyde wins his wager in all of these
equally close alternative worlds. Furthermore, we may consider a particular
world relevant to the truth value of a particular conditional even though that
world is not a closest world at which the antecedent of the conditional is
true. Suppose Mack has an ancient lawn-mower which will barely cut grass. On
high grass, the mower stalls. Now suppose Mack's lawn is just slightly too
short for the blades of the mower to hit the grass. Is the following condi-
tional true or false?
3.14 If Mack's grass were higher, his mower would cut it.
I believe that 3.14 is not true, even though the closest worlds in which
Mack's grass is higher, i.e., those worlds in which it is just barely long
enough for the blades of his mower to reach it, are worlds in which his mower
cuts the grass. But we would object to 3.14 on the grounds that if the grass
were any more than this bare minimum higher, then the mower would stall and
would not cut the grass. In many cases, we consider worlds which are close
enough to suit our purposes in evaluating conditionals without regard for
whether they are the very closest worlds in which the antecedent of the
conditional is true.
Tense and Conditionals 9
This approach to the analysis of intensional conditionals is captured in the
formal notion of a class selection function model. A class selection function
model for our formal language for conditionals is an ordered triple <W,f,[]>
satisfying conditions 3.2, 3.4, and the following:
3.15 f is a function which assigns to each sentence q and each w in W a
subset f(q,w) of W.
3.16 f(q,w) is contained in [q].
3.17 If f(q,w) = í, then f(r,w) ï [q] = í.
3.18 w î [q > r] iff f(q,w) is contained in [r].
3.19 If w î [q], then w î f(q,w).
3.20 If f(q,w) ï [r] is not empty, then f(q r,w) is contained in
f(q,w) ï [r].
3.21 If f(q,w) is contained in [r] and f(r,w) is contained in [q], then
f(q,w) = f(r,w).
This semantics characterizes the conditional logic CV which is axiomatized in
Lewis (1973) and elsewhere. It is the underlying semantics for intensional
conditionals assumed by the account of tense and conditionals to be developed
in this paper.
The notion of similarity of worlds which lies behind either of the two model
theories summarized in this section must remain vague. Given different
purposes and interests which speakers may have on different occasions, various
features of the world might be considered more important than others in
deciding which worlds are more similar to the actual world than others. The
intuitive interpretation of class selection function models offered in this
section introduces a further cause of vagueness since it allows the consider-
ation of worlds which are reasonably similar to the actual world even though
they are not most similar. This means that we not only have to decide on a
particular occasion which features of the world are most important for
determining similarity, but we also have to decide how similar a world has to
be for us to include it in our deliberations. (For a discussion of some of the
pragmatic features involved in shaping the selection function used on a
particular occasion, see Nute (1980).) Despite this variability of the
selection function, it is also widely accepted that any selection function we
use, no matter what are the circumstances in which it is used, must at least
have certain formal characteristics. The conditions proposed above for class
selection functions is one suggestion about the characteristics which any
suitable selection function must have.
4. Tense and Intentional Conditionals: the Language CT
An obvious first step in the analysis of the combined logic of tense and
conditionals is the development of a formal language CT which contains both
conditional and tense operators. Let CT be the language formed by augmenting
Tense and Conditionals 10
the language of classical sentential logic with a conditional operator > and
tense operators P, F, H, and G. CT is obviously the result of combining the
formal language for tense defined in section 1 with the formal language for
intensional conditionals defined in section 3.
A model for our language of tense and conditionals will be an ordered quintu-
ple <T,W,®,f,[]> satisfying the following conditions for all t,t1 î T, all
w,w1 î W, and all sentences q and r of CT:
4.1 T is a non-empty set.
4.2 W is a non-empty set.
4.3 T ï W = í.
4.4 ® is a strict total ordering for T.
4.5 f is a function which assigns to every sentence q î CT, time t î
T, and world w î W a subset f(q,t,w) of W.
4.6 [] is a function which assigns to every sentence q a subset [q] of
T x W.
4.7 [ªq] = (T x W) - [q], [q r] = [q] ï [r], and so on for the rest
of our truth-functional connectives.
4.8 If w1 î f(q,t,w), then <t,w1> î [q].
4.9 <t,w> î [q > r] iff for every w1 î f(q,t,w), <t,w1> î [r].
4.10 <t,w> î [Pq] iff there is a t1 such that t1 ® t and <t1,w> î [q].
4.11 <t,w> î [Fq] iff there is a t1 such that t ® t1 and <t1,w> î [q].
4.12 <t,w> î [Hq] iff for every t1 such that t1 ® t, <t1,w> î [q].
4.13 <t,w> î [Gq] iff for every t1 such that t ® t1, <t1,w> î [q].
4.14 If <t,w> î [q], then w î f(q,t,w).
4.15 If f(q,t,w) = í, then f(r,t,w) ï {w1:<t,w1> î [q]} = í.
4.16 If f(q,t,w) ï {w1:<t,w1> î [r]} is not empty, then f(q r,t,w) is
contained in f(q,t,w) ï {w1:<t,w1> î [r]}.
4.17 If f(q,t,w) is contained in {w1:<t,w1> î [r]} and f(r,t,w) is
contained in {w1:<t,w1> î [q]}, then f(q,t,w) = f(r,t,w).
These restrictions on our models for CT derive from the conditions on models
for tense in section 1 and from the conditions on class selection function
models for intensional conditionals in section 3. The connections should be
obvious.
Tense and Conditionals 11
While we have defined a formal language containing both tense and conditional
operators, and while we have developed a semantics for this language, our
semantics effectively segregates the two notions of tense and conditionality.
Notice that in the truth conditions 4.10 - 4.13 for tense operators the world
mentioned in any one of these conditions remains constant. On the other hand,
the time remains constant in the truth condition 4.9 for conditionals. In the
next section I will explore the expressive power of our formal language CT and
advance certain arguments to show a need to introduce operators whose truth
conditions will involve `simultaneous' change in time and world. These
operators will be used to represent genuine tensed intensional conditionals.
5. What CT Can't Do
A great many interesting sentences of English can be symbolized in CT in
obvious ways. For example,
5.1 If I had received an invitation, I would be at the party.
may be symbolized as Pq > r, and
5.2 If I had received an invitation, I would go to the party.
may be symbolized as Pq > Fr. But we run into difficulty when we consider the
English sentence
5.3 If I had received an invitation, I would have gone to the party.
We cannot capture the full meaning of 5.3 by symbolizing it as Pq > Pr, for
this would allow my attendance at the party to precede my receiving an
invitation. Surely the intent of 5.3 is that I would have gone to the party
after I received the invitation and not before. The time at which q would have
been true must be later than the time at which p would have been true for the
entire sentence to be true. Thus the time of the antecedent and the time of
the consequent are related to each other in the truth conditions for the
sentence in some essential way. How can we capture this when our tense
operators only relate the times of the antecedent and consequent to the time
of utterance and not to each other?
One possible solution to the problem is to try, in effect, to shift the time
of utterance of the conditional part of 5.3 to the time of either the anteced-
ent or the consequent and then to relate that time in an appropriate way to
the actual time of utterance. Two possibilities would be P(q > Fr) and P(Pq >
r). The first of these possibilities is proposed in Thomason and Gupta (1980).
If this suggestion is correct, the antecedent of the conditional is in the
present tense and the consequent is represented as being in the future from
the point of view of the time of the antecedent. If the second suggestion is
correct, it is the consequent which is represented as being in the present
tense and the antecedent is represented as being in the past from the point of
view of the time of the consequent. In both cases the time at which the
conditional is true is represented as being in the past from the point of view
of the time of utterance of 5.3. Either of these proposals captures the proper
temporal relationship between the times of the antecedent and the consequent,
Tense and Conditionals 12
but I fear neither adequately captures the sense of the English sentence with
which we began.
Both of the formal sentences suggested as possible symbolizations of 5.3 will
be true if 5.3 is true, but the converse may not be the case. Suppose I want
to go to the party very badly and that I even sit by the telephone and wait
for an invitation until the party is half over. I finally decide that the call
is not coming. I telephone a friend and we decide to meet at a restaurant.
After calling the friend, I would not go to the party even if I were to
receive a belated invitation. Suppose in fact that the phone rings as soon as
I hang up from talking to my friend, and that the call is the very invitation
for which I have been waiting. I certainly would not say, "I'm sorry I can't
come. If I had received an invitation, I would have come." This response
would sound very peculiar under the circumstances. Nevertheless, both of the
sentences of CT which we considered as symbolizations of this English sentence
would be true under these circumstances.
The problem with these proposals is that the embedded conditional need only be
true at some single moment in the past in order for the entire formal sentence
to be true, while 5.3 requires that the embedded conditional be true during
some stretch of past time. We might try to mend the situation by using the
past tense operator H in place of the operator P. Perhaps the correct repre-
sentation of 5.3 is H(q > Fr). But this will not work either. To see this,
let's consider a slightly different example. Suppose I received an invitation,
but the invitation fell behind my desk when my wife placed the mail in its
usual spot. Then I might well assert the following conditional:
5.4 If I had looked behind my desk, I would have gone to the party.
But surely it is not true that I would have gone to the party if I had looked
behind my desk the day before the invitation arrived, so H(q > Fr) is too
strong to be a correct symbolization of 5.4. In this case my intent in
uttering 5.4 is, of course, that I would have gone to the party if I had
looked behind my desk at any time after the invitation fell there. Perhaps
what we need to do is to introduce a new tense operator akin to H but relativ-
ized to a particular period of time, in this case the period of time beginning
at the moment when the invitation fell behind my desk. Using H* for this
operator, our symbolization of 5.4 will then be H*(q > Fr). One problem with
this proposal is that we cannot provide truth conditions for sentences
containing H* using the model theoretical devices which we have assembled so
far. The period of time associated with H* will change for different anteced-
ents. What we might do is add another function g to our models which will
assign to any sentence q, time t, and world w, an interval g(q,t,w) which is
open on the right and for which the right limit is t. We could then say that
H*q is true at t in w just in case q is true at every time t1 in w for every
t1 in g(q,t,w). If we do something like this, we introduce a second element of
vagueness in addition to the vagueness already inherent in our selection
function for interpreting the conditional operator. A problem with this
approach is that we can't really allow the set of times picked for q, t, and w
by g to extend all the way to the time of utterance in every case. Suppose,
for example, that the party was yesterday. Then it certainly isn't true that I
would have gone to the party if I had looked behind my desk this morning. If
Tense and Conditionals 13
we allow g(q,t,w) to be any set of times prior to t (or perhaps some such set
such that for any two times t1 and t2 in g(q,t,w), if t1 ® t3 and t3 ® t2, then
t3 is also in g(q,t,w)), our new operator H* looks less and less like the
familiar H. Furthermore, there seems to be no need for this operator in the
analysis of sentences which do not involve conditionals. It would be simpler
if we could get by with only one selection function f in our models and if it
were the only source of contextually dependent vagueness in our semantics.
This would also allow us to avoid the extra tense operator H*, although we may
still need to introduce new operators which combine elements of tense and
conditionality.
Another difficulty with the suggestion that we use an operator like H* in our
analysis is that this does not reflect very well the grammatical structure of
the English sentences which we are studying. In either H(q > Fr) or H*(q > Fr)
the scope of the conditional operator is smaller than the scope of the tense
operator H or H*. Yet when we look at an English sentence like 5.3, the scope
of the conditional operator appears to be the greatest possible. Other things
being equal (and it must be admitted that they often are not), we should
prefer formal representations of sentences of a natural language which most
closely copy the surface structure of the sentences of natural language that
are the objects of our analysis. In the present case, I see no way to repre-
sent the logical structure of certain English conditionals using separate
tense and conditional operators and still allow the conditional operator to
have greatest scope. I believe the tense and conditional constructions are
inextricably intertwined in these sentences to form tensed conditional
constructions which can not be analyzed into a part which is tensed and
another part which is conditional.
Similar problems arise for the suggestion that we represent our original
English sentence by either H(Pq > r) or H*(Pq > r), but an additional diffi-
culty confronts this proposal. The initial reaction to 5.3 may be that the
times of both antecedent and consequent are past times, but this is not a
necessary condition for the truth of 5.3. There is nothing peculiar about
saying, "I am not going to the party tomorrow, but I would have gone if I had
received an invitation." It is clear that this construction indicates the
time of the antecedent to be past, but the time of the consequent might be
past, present, or future. Both H(Pq > r) and H*(Pq > r) guarantee that the
time of the antecedent is past, but neither allows for the possibility that
the time of the consequent be either present or future. This makes these
symbolizations doubly unattractive.
We need some sort of tense operator which will be context dependent in a way
in which familiar tense operators are not. The times involved in the truth
conditions containing these operators will depend not only on the times of
utterance (or, perhaps, `projected' times of utterance in the case of embedded
operators), but also on the particular content of the sentences to which the
operators are attached. Since the need for such tense operators arises out of
a consideration of problems involved in adequately representing the semantical
structure of tensed conditional sentences of English, it is reasonable to
think that the needed operators themselves will be tensed conditional opera-
tors of some sort. Our next task will be to develop a formal language which
contains operators of this sort and a semantics for this language.
Tense and Conditionals 14
Let's review the combinations of tense and conditionals which we can represent
in CT. Where the times of both antecedent and consequent are only indicated as
being past, present, or future with respect to the time of utterance of the
sentence, we have no problem. The difficulty arises when the sentence indi-
cates something about the relation of the time of the antecedent to the time
of the consequent. Again, where the time of the antecedent is the same as the
time of utterance, there is no problem and we can represent the temporal
relations using our language CT. It is only when the time of the antecedent is
either past or future with respect to the time of utterance and the time of
the consequent is either past or future with respect to the time of the
antecedent that more sophisticated devices are needed than those provided in
CT.
There are four situations remaining for further analysis. In the first, the
time of the antecedent is earlier than the time of utterance and the time of
the consequent is at least as early as the time of the antecedent. We can call
such a conditional a past-past conditional. In the second, the time of the
antecedent is earlier than the time of utterance and the time of the conse-
quent is no earlier than the time of the consequent. These conditionals we can
call past-future conditionals. The other two new kinds of conditionals we will
call future-past conditionals and future-future conditionals. These are the
four varieties of tensed conditionals which we are unable to represent in CT.
In the next section we will develop a new formal language and model theory
which can accommodate these kinds of conditionals.
6. Tensed Intensional Conditionals: the Language TC
In the last section we discovered evidence that there are constructions in
English which combine tense and conditionality in such a way that the logical
structure of these constructions cannot be represented using combinations of
distinct tense and conditional operators. In this section we will develop a
new formal language which contains, in addition to all the symbols of CT, four
new tensed conditional operators which may be used to represent the four
tensed conditional constructions listed at the end of section 5. These
operators are >PP>, >PF>, >FP>, and >FF>. Each of these is a dyadic tensed
conditional operator, and the resulting, expanded language TC is not just a
language of tense and conditionals but also a language of tensed conditionals.
Thus we can represent in TC five different kinds of intensional conditionals
using our five distinct conditional operators.
Our new language TC requires a more complex model theory than that proposed
for CT. Models for TC will still be ordered quintuples <T,W,®,f,[]>, but our
selection function f will have some different properties and our truth
function [] will have additional restrictions resulting from the new formula-
tion of truth conditions for conditional sentences in TC. Since f will now be
used to interpret tensed conditionals, it will be necessary for f to pick out
for a sentence q, a time t, and a world w not just a set of worlds but rather
a set f(q,t,w) of ordered pairs <t1,w1> of times and worlds satisfying certain
conditions regarding similarity to t and w. Essentially, we have the following
new condition for all q, t, and w:
6.1 f(q,t,w) î [q].
Tense and Conditionals 15
Presumably the choice of pairs <t1,w1> in f(q,t,w) where t1 is earlier than t
will depend on and affect which past-past and past-future conditionals are
acceptable, the choice of pairs where t1 is later than t will depend on and
affect which future-past and future-future conditionals are acceptable, and
the choice of pairs <t,w1> in f(q,t,w) will depend on and affect which
conditionals of the familiar form q > r are acceptable. We could establish
separate selection functions for each of our conditional operators, but this
will not be necessary.
The truth conditions for our new kinds of conditionals should be fairly
obvious:
6.2 <t,w> î [q >PP> r] iff for every t1 and w1 such that <t1,w1> î
f(q,t,w) and t1 ® t, there is a t2 such that t2 ® t1 and <t2,w1> î
[r].
6.3 <t,w> î [q >PF> r] iff for every t1 and w1 such that <t1,w1> î
f(q,t,w) and t1 ® t, there is a t2 such that t1 ® t2 and <t2,w1> î
[r].
6.4 <t,w> î [q >FP> r] iff for every t1 and w1 such that <t1,w1> î
f(q,t,w) and t ® t1, there is a t2 such that t2 ® t1 and <t2,w1> î
[r].
6.5 <t,w> î [q >FF> r] iff for every t1 and w1 such that <t1,w1> î
f(q,t,w) and t ® t1, there is a t2 such that t1 ® t2 and <t2,w1> î
[r].
Of course, we must also amend our truth condition for untensed conditionals:
6.6 <t,w> î [q > r] iff for every w1 such that <t,w1> î f(q,t,w),
<t,w1> î [r].
While it is certainly possible to introduce tensed conditional operators
having the interpretations suggested here, it might be the case that there are
no constructions in English or any other natural language which correspond to
each of these operators. In fact, there are English intensional conditionals
corresponding to each of our tensed conditional operators. We have already
seen that a sentence like `If I had received an invitation, I would have gone
to the party' is a past-future conditional. An example of a past-past condi-
tional is `If I had been admitted to the party, I would have had to have
received an invitation'. `Were I to be invited, I would go to the party' is a
future-future conditional and `Were I to be admitted to the party, I would
have to have received an invitation' is a future-past conditional. The only
past-past and future-past conditionals which I can suggest in the subjunctive
mood involve the rather awkward phrases `would have had to have' and `would
have to have'. Both past-past and future-past conditionals are varieties of
back-tracking conditionals. (For discussions of these, see Lewis (1979) and
Pollock (1981).) True back-tracking intensional conditionals are relatively
rare, which may explain the fact that past-past intensional conditionals are
not provided with simpler forms of expression in English. Since we rarely have
an occasion in which it would be appropriate to assert such a conditional,
Tense and Conditionals 16
there is no great practical need to evolve more efficient constructions for
such conditionals. Of course, all the truth conditions for conditionals which
have been offered above are for intensional conditionals. I shall have
something more to say about tensed material conditionals later.
7. What TC Can Do
A major advantage which the analysis of the previous section enjoys over one
which employs relativized versions of the familiar tense operators H and G is
that only one selection function f appears in our models. Recall that if we
were to represent a past-future conditional as H*(q > Fr) where H* is a
relativized version of H, we would have to add a new item to our models, a
selection function which would serve as the basis for interpreting the new
operator H*. We would have to add a dual operator G* to our formal language to
represent future-past and future-future conditionals, and we would have to add
a selection function to our models to interpret this operator as well. Adding
either relativized tense operators or tensed conditional operators to our
formal language makes our language more complicated, but adding tensed
conditional operators rather than relativized tense operators results in
considerably less complication for our model theory. Furthermore, the very
grammatical structure of the English sentences we are considering indicates
that these sentences are conditionals and that the conditional constructions
in these sentences have greatest scope.
Despite the greater complexity of the corresponding model theory, there is a
reason why we might prefer to use H* and G* rather than tensed conditional
operators to represent the kinds of English conditionals we have been discuss-
ing. Consider the case of a tennis player, let's call him Franz, who suffers a
fall during the opening round at Wimbledon. Fortunately for Franz, he suffers
no serious injury and ultimately competes in the finals of the tournament.
Later we might assert:
7.1 If Franz had broken his leg, he wouldn't have played in the
finals.
After the tournament, Franz develops some soreness in his knees and consults a
physician. The physician orders x-rays of his knees and examines them in the
presence of Franz's coach. The coach asks the doctor if there is anything
wrong with the leg Franz broke. To this the doctor replies, "Franz never broke
his l eg." The doctor goes on to assert:
7.2 If Franz had broken his leg, there would be evidence of the break
in the x-rays.
Here we have two tensed conditionals involving the same antecedent condition,
`Franz breaks his leg'. These two sentences present a problem since the range
of times which may be considered in evaluating 7.1 is usually going to be far
smaller than the range of times which may be considered in evaluating 7.2. It
is obvious that for a given time t and world w our selection function f can
pick out only one set f(q,t,w) of times and worlds at which Franz broke his
leg, but we want to pick out quite different sets of pairs of times and worlds
for 7.1 and 7.2. Use of the operator H* provides one solution to this problem.
Tense and Conditionals 17
While the selection function associated with a conditional operator takes only
the antecedent of the conditional as argument, the selection function which we
would use to interpret H* in H*(q > Fr) would take q > Fr as argument and
hence, indirectly, both q and r. This would allow us to use different times in
interpreting the two English conditionals. While I would prefer not to accept
this proposal so long as there is no demonstrated need for H* in contexts
which do not involve conditionals, we must recognize its advantages.
Dov Gabbay (1972) has suggested another approach which may help us explain the
tennis player examples. For reasons which do not really involve considerations
of tense at all, Gabbay proposes that the set of worlds which we consider in
evaluating a conditional is always a function of both the antecedent of the
conditional and the consequent of the conditional. If we follow Gabbay, then f
becomes a function which assigns to sentences q and r, time t, and world w a
set f(q,r,t,w) of pairs <t1,w1> of times and of worlds similar to w such that
q is true at t1 in w1. By making f a function of both antecedent and conse-
quent, we are clearly able to distinguish between the truth conditions for the
two conditionals concerning the tennis player since these two conditionals
have different consequents. The difficulty with Gabbay's proposal is that it
would force upon us an extremely weak logic for conditionals, a logic so weak
that we could not even count among its theorems such theses as
7.3 ((q > r) (q > s)) (q > (r s))
(For a further discussion of Gabbay's semantics, see section 3.4 of Nute
(1980a).) While Gabbay's approach would allow us to solve the immediate
problem, I for one am not willing to pay the price of the very weak condition-
al logic which goes with it.
I think that a proper solution to our tennis player example lies not in a
revision of our formal language and its semantics but rather in a careful
consideration of the pragmatics of conditionals. It would be reasonable to say
of the tennis player, "If he had broken his leg, he would not have played in
the finals," and it would also be appropriate to say of the tennis player, "If
he had broken his leg, there would be evidence of the break in his x-rays."
But it would not be appropriate to utter both of these sentences on the same
occasion. Contrary to what Gabbay suggests, we do not need to provide differ-
ent truth conditions for these two sentences since both would not be uttered
in the same context. What we need is an account of the pragmatic principles
which prevent the utterance of both sentences on the same occasion. But we
need more than this. Each sentence is true when uttered in appropriate
circumstances and given that certain conditions hold. Given the clear meanings
of the two sentences on different occasions, how can we express exactly these
same two meanings on a single occasion?
As was mentioned earlier, the selection function we use to interpret condi-
tionals on one occasion may not be the same function we use on another
occasion. Furthermore, the particular function we use on a particular occasion
is never fully defined. It could even be said that there really is no function
which is being used on a particular occasion. Instead there is at best a
partial function which becomes defined for additional arguments as a conversa-
tion progresses. It is indeterminate which times and worlds will be picked out
Tense and Conditionals 18
for the antecedent `Franz breaks his leg' until a sentence with this anteced-
ent is actually used in a conversation. Once such a sentence is used and
accepted, the speaker and the hearer have tacitly arrived at an understanding
about the value of the selection function for this antecedent, an understand-
ing which will make the sentence which has been uttered and accepted turn out
to be true. These shared restrictions on the interpretation of conditionals
comprise a component of what David Lewis (1979a) has called conversational
score. For a further discussion of the role which conversational score plays
in the interpretation of conditionals, see Nute (1980). A consequence of this
view of the pragmatics of conditionals is that the selection functions
occurring in our models must represent semantic ideals which we only approach
in actual speech.
In the tennis player example, the value of our selection function for the
antecedent `Franz breaks his leg' will be determined by whichever of our two
English conditionals occurs first in the conversation. Thus the consequent of
the conditional does affect the selection of worlds to be considered in
evaluating a conditional, but in a far more subtle way than Gabbay suggests.
If the consequent were itself an argument for the selection function, it would
not seem abnormal to assert both sentences in whichever order we wished on a
single occasion and without further restriction. But this would be abnormal,
for Franz might very well have played in the finals of Wimbledon if he had
broken his leg several years before the tournament. The consequent does not
serve as an argument for the selection function; rather, it helps to determine
what the selection function itself may be. Whichever of the sentences is
accepted first, it then becomes necessary to modify the antecedent of the
other before it can be asserted on the same occasion. Thus we might say, "If
Franz had broken his leg, the mend would show on an x-ray; and if he had
broken his leg recently, he would not have played in the finals." We might
also say, "If Franz had broken his leg, he would not have played in the
finals; and if he had ever broken his leg, the mend would show on an x-ray."
In each case, the antecedent is modified in the second sentence by the
insertion of a qualifying temporal adverb like `recently' or `ever'. The
interpretation for the unqualified antecedent is different in the two cases
even though exactly the same English sentence serves as antecedent in the two
examples. Once an interpretation is tacitly accepted for the unqualified
antecedent, the antecedent of the other conditional must be modified so as to
expand or restrict the set of times selected for the unqualified antecedent to
produce the set of times appropriate to the qualified antecedent. This account
may not be as simple as an account built on operators like H*, or as an
account like Gabbay's which makes the selection function take both antecedents
and consequents as arguments, but it provides a better description of what
occurs in actual discourse.
Another possibility would be to eschew a formal language of tenses and
conditionals altogether. We could then attempt to provide a formal semantics
directly for the particular English constructions in which we are interested.
This is the approach of the Montague grammarians and there is much to be said
for it. It seems much simpler to go directly from natural language to models
for that language without the mediation of a formal language. But the present
approach has several advantages. First, it allows us to axiomatize the logic
of the regimented constructions which we use to represent the constructions of
Tense and Conditionals 19
the natural language if we choose, although such axiomatization is not a goal
of the present paper. Second, the simplicity of the formal language makes it
easier in many instances to see the consequences of various decisions concern-
ing our formal semantics and to see where to look in the natural language for
difficult cases to test our semantics. Third, consideration of the regimented,
formal language may result in a reform of ordinary usage. This third possibil-
ity may seem to be much less of a benefit to the linguist than it does to the
philosopher. The philosopher is attempting in many cases to clarify the
concepts underlying a particular linguistic usage and may decide that these
concepts are confused and require certain refinement or correction. Thus the
philosopher's analysis of language may ultimately result in the formation of
new linguistic intuitions as well as a better understanding of the linguistic
intuitions already shared by speakers of the language under study. Presumably
the linguist, or at least the descriptive linguist, is never interested in
changing usage in any way.
Our formal language TC allows us to represent true tensed conditionals of
various sorts, and our formal semantics for TC allows us to interpret these
conditionals. But both the language and the semantics suffer from various
limitations which we have noted. First, we may require an interval semantics
if we are to provide an adequate analysis of certain kinds of sentences
involving both tense and conditionals. Second, there may be additional tensed
conditional constructions which we can not represent even among those which do
not require an interval semantics for their interpretation. Third, we can not
explain puzzles like that of the tennis player example without augmenting our
formal semantics with a fairly detailed pragmatics for conditionals. None of
these limitations will be explored in greater detail in this paper. Neverthe-
less, I see none of these issues as a source of insurmountable difficulties
for the account which has been provided. Rather these issues show ways in
which the present account must be expanded before we can have a complete
account of tensed conditional constructions in English or any other natural
language.
8. Tense and Indicative Conditionals
I have proposed that we adopt new tensed conditional operators if we are to
provide a formal language capable of representing the logic of intensional
conditionals adequately. Throughout the discussion so far, I have assumed that
all English subjunctive conditionals are examples of intensional conditionals,
but I have also suggested early in this paper that English indicative condi-
tionals may also be used intensionally. Now it is time that we look at
indicative conditionals more closely and try to determine their logical and
semantical properties more precisely. The first question I will consider in
this section is whether there is a need for tensed material conditional
operators parallelling the need for tensed intensional conditional operators.
Next I will show why I believe that certain English indicative conditionals
are used intensionally and what there is about the circumstances of such use
which makes this practice reasonable.
Thomason and Gupta (1980) suggest that the distinction I have made between
intensional and material conditionals is really a difference in the scope of
the tense and conditional operators in the sentences affected. Look at the
Tense and Conditionals 20
following two conditionals taken from Thomason and Gupta (1980) and originally
due to Ernest Adams:
8.1 If Oswald didn't shoot Kennedy then Kennedy is alive today.
8.2 If Oswald hadn't shot Kennedy then Kennedy would be alive today.
Thomason and Gupta propose that 8.1 is of the form Pq > r while 8.2 is of the
form P(q > r), where the conditional operator > is provided with Stalnaker's
semantics and r is the eternal sentence `Kennedy is alive today'. Consider
also the following two sentences taken from Thomason and Gupta (1980) with
slight modification:
8.3 If Max missed the train then he took the bus.
8.4 If Max had missed the train then he would have taken the bus.
According to Thomason and Gupta, these conditionals are respectively of the
forms Pq > Pr and P(q > Fr). The treatment of these four examples is consis-
tent, the difference being that in the first pair r is taken to be an eternal
sentence while in the second pair r represents an ordinary atomic sentence
which is true at some times and false at others. I have already offered a
critique of this kind of account for 8.2 and 8.4. Now let's consider whether
this is an adequate account of conditionals in the indicative mood like 8.1
and 8.3.
Given the interpretation of most indicative conditionals as material condi-
tionals which I am adopting in this paper, I would of course not use the
conditional operator > in symbolizing 8.1 and 8.3. Instead, I would use the
truth-functional and symbolize these conditionals respectively as Pq r and
Pq Pr. I would agree with Thomason and Gupta that in the sentence 8.3 no
relation between the times of the antecedent and the consequent is guaranteed,
although such a relation is guaranteed by sentence 8.4. It is just as reason-
able to assert 8.3 in a case in which we wish to claim that Max's taking the
bus would explain his missing the train as it is in a case in which we wish to
assert that Max's taking the bus would be a result of his missing the train.
These examples suggest that we needn't worry about the relation of the time of
the antecedent to the time of the consequent in the case of material (indica-
tive) conditionals. This being the case, we would not have the reason for
inventing special tensed material conditional operators which motivated the
creation of our tensed intensional conditional operators. But this is not the
case. Although it may require the use of temporal adverbs to accomplish the
task, we can certainly construct material conditionals which guarantee
appropriate relations between their antecedents and their consequents. An
example of such a conditional is:
8.5 If Max missed the train, he subsequently took the bus.
Here it is clear that the bus-taking follows the train-missing. I think that
Thomason and Gupta would symbolize 8.5 as P(q > Fr), i.e., in the same way
that they suggest that 8.4 be symbolized. At least, this symbolization would
Tense and Conditionals 21
seem to be consistent with their symbolizations of other examples. I will not
press this suggestion with uncharitable vigor, however, since Thomason and
Gupta do not in fact consider the conditional 8.5 and since I myself find the
suggestion that 8.4 and 8.5 be symbolized in the same way very unattractive.
An obvious way to avoid this consequence is to replace > in the proposed
symbolization of 8.5 with , thereby symbolizing 8.5 as P(q Fr). This would
certainly indicate a difference in 8.4 and 8.5, but I still think that we
don't have 8.5 right.
Let's look at a modification of an earlier example:
8.6 If Jane received an invitation then she subsequently went to the
party.
We have difficulties if we represent this conditional as being of the form P(q
Fr). If this were a correct symbolization, then so also would be P(ªq Fr).
Now suppose Jane in fact did receive an invitation on Tuesday but did not
attend the party on Saturday. In this case we should say that 8.6 is false.
Still it is true that ªq Fr was true on Monday, so P(q Fr) is true now.
This cannot be a correct symbolization of 8.6.
A more promising candidate for the logical form of 8.6 is ªP(q ªFr). In
fact, this is almost correct. The only problem I can see with this suggestion
is that it would allow for the possibility that Jane received an invitation
yesterday and will go to the party tomorrow. The clear indication of 8.6, on
the other hand, seems to me to be that Jane went to the party, not that she is
going to the party. This possibility, that the time of the consequent is after
the time of the utterance, does not appear to be open in the case of 8.6 as it
is in the case of the intensional counterpart of 8.6, `If Jane had received an
invitation, she would have gone to the party'. To capture this additional
element of 8.6, I suggest the symbolization ªP(q ªFr) (Pq Pr). The
second part of this symbolization is essentially the same as that proposed for
8.1 and 8.3, taking into account the fact that the antecedent of 8.1 is
supposed to be an eternal sentence. The difference between a conditional like
8.6 and one like 8.4 is due to the occurrence of the temporal adverb `subse-
quently'. It is the presence of this adverb which forces us to append the
first conjunct in our symbolization of 8.6. I believe that 8.5 and 8.6 have
exactly the same logical form. The reason I changed examples in the discussion
is that the antecedent in 8.5 might well indicate a particular train leaving
at a particular time. Since there would then be one and only one time at which
Max could have missed the train, the possibility of there being some time at
which he either did not miss the train or did take the bus even though 8.5 was
false would not arise. But this peculiarity of 8.5 is due to the fact that the
train left at a specific time rather than to the tense or the conditionality
of the sentence.
A consideration of examples can hardly show that there is no need to augment
our formal language with special tensed material conditional operators, for no
matter how many examples we find which require no special operators there may
remain unexamined conditionals which require such treatment. Nevertheless, I
have been unable to discover any such examples. I therefore venture to propose
that the language TC, and indeed the language CT, is adequate for the repre-
Tense and Conditionals 22
sentation of all material conditionals whatever their tense structure or
temporal adverbs may be. It is well worth noting, though, that the logical
form of such English conditionals may be more complex than the account
included in a typical treatment of classical sentential logic would indicate.
Even without the complexities associated with intensional conditionals, the
combination of tense with conditionality is no trivial matter.
All of the examples considered in this section have concerned antecedents in
the past tense. A new problem arises when we consider future tense condition-
als. The problem is that in English we often do not distinguish between future
tense indicative conditionals and future tense subjunctive conditionals.
Consider the following examples.
8.7 If Joe strikes this match, it will light.
8.8 If Joe were to strike this match, it would light.
Under what conditions would we assert one rather than the other of these two
conditionals? We would be more likely to assert 8.8, I think, if we believed
that it is unlikely that Joe will strike the match, and we would be more
likely to assert 8.7 if we believed Joe might strike the match or if we were
trying to persuade Joe to strike the match. But is there a difference in the
truth conditions for the two sentences?
There is a temptation to say that 8.7 and 8.8 have exactly the same truth
conditions, and that both are intensional conditionals. The cause of this
temptation is that in deciding whether to accept 8.7 we have no option but to
perform the very same sort of thought experiment which we would perform in
evaluating 8.8. That is, we would imagine likely situations in which Joe
strikes the match and consider whether or not the match lights in all of those
situations. This is quite different from the position we find ourselves in
with regard to
8.9 If Joe struck the match, it lit.
Here we can investigate what actually happened to determine whether 8.9 is
true or false. Since the future is not open to investigation in the same way
the past is, we cannot use this method for evaluating 8.7. With no alterna-
tive, we form our opinion about the truth of 8.7 in much the same way we form
our opinion about 8.8. We might say that our epistemological situation with
regard to 8.7 is exactly the same as our epistemological situation with regard
to 8.8, while our epistemological situation with regard to pairs of past or
present tense indicative and subjunctive conditionals is quite different. This
explains why we do not distinguish as carefully between indicative and
subjunctive conditionals in the future tense.
While the epistemological distinction between indicative and subjunctive
conditionals in the past and present tenses collapses for indicative and
subjunctive conditionals in the future tense, this does not mean that the
difference in truth conditions also collapses. Just because we cannot now
employ different methods in estimating the truth values of indicative and
subjunctive future tense conditionals does not mean that these conditionals do
Tense and Conditionals 23
not in fact have different truth conditions. To better determine the facts in
this matter, let's consider the logic of future tense indicative conditionals
and see if it differs from the logic of future tense subjunctive conditionals.
A variety of logical principles which are acceptable for material conditionals
are not acceptable for intensional conditionals. Among these are left monoton-
icity, transitivity, and contraposition. Let's consider the principle of left
monotonicity as it applies to 8.7. Consider the conditional
8.10 If Joe dips the match in water and strikes it, it will light.
Not only does it seem plausible that someone would affirm 8.7 while denying
8.10, but it even seems likely. This suggests that 8.7 is being used as an
intensional rather than as a material conditional. Yet it is also plausible
that someone would insist that 8.10 is true because 8.7 is true, and conclude
from this that Joe will not both dip the match in water and strike it.
Inelegant though it may be, the honest conclusion to be drawn is that English
indicative conditionals in the future tense may be used either materially or
intensionally, and their intensional use is motivated by the fact that we
cannot maintain the same epistemological distinction between indicative and
subjunctive conditionals in the future tense that we maintain for past and
present tense conditionals.
To summarize briefly, I am suggesting that all English subjunctive condition-
als are probably intensional (I can find no persuasive counterexamples), that
all past and present tense English indicative conditionals are probably
material (again, I can find no persuasive counterexamples), and that future
tense English indicative conditionals may be used either materially or
intensionally. I further suggest that we need special tensed conditional
operators for symbolizing English subjunctive conditionals (and other inten-
sional conditionals), but that the resources of familiar tense logic are
sufficient for representing the logical form of past and present tense English
indicative conditionals (and other material conditionals).
9. Branching Time and Settledness
Our discussions so far have assumed that time is linear and that the
earlier-than relation is a strict ordering of the set of times. An alternative
account has it that the set of times together with the earlier-than relation
form a tree structure with branching toward the future. Such an account is
developed in Thomason (l970) and is employed in the investigation of tense and
conditionals in Thomason and Gupta (1980). The position of Thomason (1970) is
that contingent future tense sentences are often neither true nor false.
Thomason includes a `settledness' operator in his formal language for tense
logic and uses branching time together with van Fraassen's method of superval-
uations to provide an analysis of tense which admits of truth value gaps for
future tense sentences. When this theory is augmented with an account of
conditionals, some interesting problems arise to which Thomason and Gupta
provide no adequate solution.
The formal language for which Thomason and Gupta provide a model theory is the
language of classical sentential logic augmented by three tense operators P, F
and L, and a conditional operator >. There appears to be no reason why the
Tense and Conditionals 24
analysis could not be extended to a language containing the tense operators H
and G, but these devices are not included in order to keep the discussion
relatively simple. We may read Lq as `It is settled that q'. Let's call this
new formal language TGL. Thomason and Gupta actually provide two different
model theories for TGL, but I will only discuss the first and simpler of these
two semantics. The portion of the semantics which relates immediately to the
analysis of conditionals is adapted from Stalnaker's semantics and hence
validates the suspicious principle Conditional Excluded Middle, the CEM of
section 3. Thomason and Gupta prefer their more complicated second model
theory for TGL because it seems best equipped to preserve the thesis CEM
together with certain other theses to which they are committed. Since I reject
CEM in any case, I believe that the added complications of their second model
theory are unnecessary.
A Thomason-Gupta (TG-)model for TGL is an ordered quadruple <T,ó,s,[]> such
that
9.1 T is a non-empty set.
9.2 ó is a transitive relation on T such that if t1 ó t and t2 ó t,
then t1 ó t2 or t1 = t2 or t2 ó t1.
9.3 Ht is the set of all subsets h of T such that t î h, ó strictly
orders h, and there is no subset h1 of T having these two proper-
ties which properly contains h. In other words, Ht is the set of
maximal chains with respect to ó which contain t.
9.4 [] is a function which assigns to each sentence q a set of pairs
<t,h> where h is a member of Ht.
9.5 [ªq] = {<t,h>:h î Ht} - [q], and so on for the other truth-func-
tional connectives.
9.6 <t,h> î [Pq] iff h î Ht and there is a t1 in h such that t1 ó t
and <t1,h> î [q].
9.7 <t,h> î [Fq] iff h î Ht and there is a t1 in h such that t ó t1
and <t1,h) î [q].
9.8 s is a function which assigns to each sentence q, time t, and
member h of Ht either í or a pair <t1,h1> such that h1 î Ht1.
9.9 <t,h> î [q > r] iff h î Ht and either s(q,t,h) = í or s(q,t,h) î
[r].
9.10 <t,h> î [Lq] iff h î Ht and for all h1 in Ht, <t,h1> î [q].
T represents the set of times and ó is an earlier-than relation on T. But ó is
quite different from the earlier-than relation of our earlier models. In a
TG-model, distinct times might not be related by ó at all. The relation ó
imposes on T a tree-structure with branching toward the future. Ht represents
all those temporal branches which go trough a particular time t. It should be
Tense and Conditionals 25
noted that any two members of Ht will have the same members prior to t but
different members subsequent to t. The members of Ht may be called histories
which pass through t. The sentences of TGL are interpreted as being true or
false at a time in a history, and the function [] tells us for each sentence
the pairs of times and histories at which that sentence is true. The interpre-
tation of the truth-functional and familiar tense operators are the standard
ones. Additional restrictions on the function s will be modelled after the
restrictions for Stalnaker's semantics listed in section 3. Thomason and Gupta
also include a second, equivalence relation on their models. This
`co-presence' relation ÷ satisfies the conditions that (i) if t ÷ t1 then not
t ó t1, and (ii) if t ÷ t1 and t3 ÷ t2 and t ó t2 then not t2 ó t. Although
this co-presence relation plays a role in their second, more complicated model
theory, it is not mentioned in any of the truth conditions for their first
model theory and I have therefore omitted it for the sake of simplicity.
The feature of this theory which attracts our interest is the analysis of the
operator L. The intuitive picture corresponding to the formal semantics is
that at any given time the past and the present are completely determined
while there are several alternative paths which the future may take. Given
this semantics, we cannot in general say that a sentence of the form Fq is
either true or false at a time t. Instead, we can only say that Fq is true at
t from the perspective of some particular history which passes through t. Of
course, if Fq is true at t for every history to which t belongs, then we have
it that LFq is true at t regardless of the history we choose and hence Fq is
true at t simpliciter. Thus, the settledness operator turns out to be a kind
of truth operator within the formal language TGL, and if neither q nor ªq is
settled at t we say that there is a truth value gap for q at time t. Notice
that if q contains no occurrences of F and if q is true at t for some history
passing through t, then since time only begins to branch in the future q must
be true at t for every history passing through t. So if q contains no occur-
rences of the future tense operator F, then q and Lq are equivalent. Only
sentences containing occurrences of F can suffer a lack of truth-value.
While Thomason's intent is clearly to allow for truth-value gaps for contin-
gent future tense sentences, it does not appear that he or Gupta wishes to say
that all contingent future tense sentences lack truth value. It may be the
case that LFq is true at t even when Fq is contingent. For example, Thomason
and Gupta (1980) suggest that a sentence like
9.11 The local bus will not arrive at your place of business on time.
may be settled at some time t. Their view seems to be that the future may
remain undetermined in some respects while being determined in others. This
view seems at least plausible and I will not contest it. Model-theoretically,
this assumption must be accommodated by restricting the members of Ht to those
histories which are `lawful', that is to those histories all of which repre-
sent alternative fulfillments of the same set of physical laws. Otherwise it
is difficult to see how any future contingent statement could be settled
unless it contained some reference to the past of a sort which is lacking in
9.11. This in turn would mean that in a TG-model a time t could not belong to
two histories in which different sets of physical laws were operative. We
might, however, have two disjoint histories h and h1 and a one-to-one function
Tense and Conditionals 26
f from the times in h onto the times in h1 which preserves the earlier-than
relation ó, and we might have two times t î h and t1 î h1 such that for all
times t2 ó t and all sentences q, <t2,h> î [q] if and only if <f(t2),h1> î [q].
Then different laws might be operative in h and h1 even though h and h1 are
factually indistinguishable at least through the times t and t1. Perhaps this
is a case in which we should say that the times in h are co-present with their
corresponding times in h1. Either this or some other device will be essential
if we are to provide an adequate interpretation of the language TGL.
This theory encounters a problem when we turn our consideration from counter-
factual conditionals to the more exotic counterlegal conditionals. A counter-
legal conditional is one which proposes as its hypothesis a situation which
could only obtain if some physical law were violated, e.g.,
9.12 If the gravitational constant were to increase by 1% beginning
now, people would suffer more frequent fractures unless they
developed heavier bones.
I suggest that 9.12 is an example of a counterlegal conditional which is not
only comprehensible but also true. Furthermore, the antecedent of 9.12
certainly does not require any change in past history. Given a Stalnaker
semantics for conditionals, 9.12 should be true now for some possible history
h to which now belongs just in case the consequent of 9.12 is true now in the
history h1 at which the antecedent is true which is most similar to h. Given
familiar restrictions against changing the past gratuitously, h and h1 should
share the same past. But then something strange happens. If `now' in 9.12
denotes the same time t in h as `now' in 9.12 denotes in h1, then h î Ht and
h1 î Ht even though h and h1 are not subject to the same physical laws. Once
we allow this, we can no longer have contingent future tense sentences which
are settled because we have no physical laws common to all alternative futures
to guarantee their truth.
One way to attack this problem would be to use the co-presence relation
mentioned by Thomason and Gupta but deleted from the reformulation of their
model theory which I have provided. We might maintain that the times referred
to in h and in h1 by the word `now' in 9.12 are not the same time although
they are co-present times. They are what David Lewis might call temporal
counterparts. This brings into focus an interesting feature of the
Thomason-Gupta analysis. Depending on what happens, tomorrow may be one time
instead of another. Now we certainly say that there may be many different
tomorrows, but I don't think we intend by this that tomorrow could be one of
many different times. Instead, I think we mean that the one and only tomorrow
might turn out one of many different ways. While the notion of alternative
futures or of alternative histories is not counterintuitive, the idea that
these alternatives are made up of different times is not common. Of course one
might argue that two times at which different sentences are true must be
different times, but then every time would seem to be distinct from itself
since future tense sentences are in general true at a time only relative to a
particular history passing through that time. If we used this sort of argument
to try to justify the Thomason-Gupta theory, we would be forced to the
conclusion that any two histories must be completely disjoint. Even if this
conclusion is not accepted and we adopt a `co-presence' analysis of counter-
Tense and Conditionals 27
legals, we must still explain why a sentence like 9.12 should require us to
consider a co-present now while an ordinary counterfactual containing now in
its antecedent does not. It would be better to posit a single linear time (or
perhaps a single space-time) and to consider different events which might fill
it.
If we accept the Thomason-Gupta picture of alternative histories made up of
alternative times, there may be another way of handling the problem without
insisting that `now' in 9.12 denotes different times in the two histories h
and h1. Instead of a co-presence relation, we could introduce into our model
theory an accessibility relation R on the set of histories. This accessibility
relation would have to be relativized to times, so that in fact R would be a
function which assigned to each time t an equivalence relation R(t) on Ht. We
would then use R to interpret sentences of the form Lq. We could, that is,
replace 9.10 with
9.13 <t,h> î [Lq] iff for all h1 such that hR(t)h1, h1 î [q].
By doing this, we can explain counterlegals without recourse to co-presence
within the framework of a modified TG-semantics while at the same time
allowing for the possibility of settled contingent future tense sentences. If
we do this, we can no longer take sentences to be settled at a time t simpli-
citer, but only at a time t relative to some R(t) equivalence class of
histories. This may not be a bad thing, but it is much weaker than the
position taken in Thomason and Gupta (1980). While this repair of Thomason and
Gupta is technically possible, I prefer a model theory which is not motivated
by the view that there are not only alternate histories but also alternate
times. Alternate times might be necessary to interpret conditionals whose
antecedents require that time have a cyclical structure, etc., but for
ordinary conditionals, including most counterlegal conditionals, such devices
are not necessary. In the next section I will develop an alternative model
theory for TGL which is formally equivalent to the modified TG-model theory
presented here but which is not motivated by such assumptions about alternate
times.
10. Pseudo-branching Time and Settledness
Without the assumption that there are alternate times which stand in no
temporal relation to each other, we can produce much the same effect as that
which results from the Thomason-Gupta semantics by letting possible worlds
play a role similar to that of temporal branches or histories. As an alterna-
tive to TG-models, I suggest that we interpret the formal language TGL by
means of ordered hextuples <T,W,®,R,f,[]> satisfying the following conditions
for all t,t1 î T, all w,w1 î W, and all sentences q and r of TGL:
10.1 - 10.11 are the same as 4.1 - 4.11.
10.12 - 10.14 are the same as 4.15 - 4.17.
10.15 For each t î T let Ht = {<w,w1>: for all q and all t1 such that
t1 = t or t1 ® t, <t1,w> î [q] iff <t1,w1> î [q]}.
Tense and Conditionals 28
10.16 R is an equivalence relation on W.
10.17 <t,w> î [Lq] iff for all w1 such that wRw1 and <w,w1> î Ht, <t,w1>
î [q].
We see that two worlds w and w1 share the same past up to time t just in case
<w,w1> î Ht. Thus, Ht plays the same role in this semantics as it did in the
theory of TG-models. As in the case of TG-models, Ht can be defined in terms
of other items in our models and need not itself be an item in our models.
Intuitively, the relation R tells us which worlds in W share the same physical
laws. Then q is settled at time t in world w just in case q is true at time t
in every world w1 which shares the same physical laws as w and shares the same
past with w up to time t. We notice that if q contains no occurrences of the
future tense operator F, then [q] = [Lq] just as in the case of TG-models. In
fact, we can easily turn one of our present models into a modified TG-model of
the sort discussed at the end of the last section. Let's define a relation GT
on pairs of times and worlds such that <t,w>GT<t1,w1> iff t = t1 and <w,w1> î
Ht. Then let T+ be the set of GT-equivalence classes of time-world pairs. We
will let <t,w> be the GT-equivalence class of <t,w>. Next define a relation ó
on T+ such that <t,w> ó <t1,w1> iff t ® t1 and <w,w1> î Ht. Then <T+,ó,R,f,[]>
closely resembles a modified TG-model since worlds play much the same role in
our present models as do histories in TG-models. The primary difference in
these derived models and TG-models is that f is a class-selection function
rather than a Stalnakerian world-selection function. If we begin with a model
<T,W,®,R,f,[]> such that for any q, t, and w the set f(q,t,w) has at most one
member, and if we let s(q,t,w) = í if f(q,t,w) = 0 and let s(q,t,w) î f(q,t,w)
otherwise, then <T+,ó,R,s,[]> becomes a full-fledged TG-model with a
settledness-accessibility relation R.
We note that there exist modified TG-models which are not equivalent to models
of the sort just defined. This is because in a TG-model there may be a time t
between two times t1 and t2 such that no time co-present with t is between two
times co-present with t1 and t2 and related to each other by ó.
One important difference between the model theory developed in this section
and the original theory of TG-models is that we cannot speak simply of a
sentence being true or false at a time. We must, in fact, speak of sentences
being true or false at time-world pairs. But we can introduce the notion of
truth-value gaps in somewhat the same way Thomason does. Once again we
interpret our settledness operator L as a truth (or `supertruth') predicate in
our formal language. Where neither Lq nor ªLq is true at a time t and a world
w, we say that there is a truth value gap at t and w for q. Just as in
Thomason's original model theory, all purely past and present tense sentences
have a truth value at each time-world pair, but future tense sentences may
lack truth values at some time-world pairs. Interpreting TGL in this way, the
past and present are completely determined while the future is in general only
partially determined.
Something needs to be said about the role which the concept of an `actual
world' plays in the model theory developed in this section. I would distin-
guish the world from all possible worlds. By `the world' I mean that which
both you and I occupy, all that there is, or the totality of things. By
Tense and Conditionals 29
`possible world' I mean a way the world might have been or might be. I think
of a possible world as a pattern of properties and relations together with a
function which `fits' concrete individuals into niches in this pattern. (For
details, see Nute (1985).) The term `actual world' would on my view denote
the way the world actually is. Since the world is not the same thing as the
way the world is, the terms `the world' and `the actual world' designate two
different things. If we accept a model theory based on pseudo-branching time
and we assume that the future is not completely determined, then there is no
such thing as the way the world is. On this view, the world might yet be any
number of different ways, none of which it is yet. If we accept this view,
there is no `actual world'; there is only the evolving world and the many
different ways it might have been or might yet be. We could distinguish, then,
between all those merely possible worlds, those ways the world might have been
but clearly can now never be, and those possible worlds each of which accu-
rately describes the world to the extent it has so far been determined. These
latter we might call `actually possible' worlds. Where t is the present, the
actually possible worlds will be some R-equivalence class of the set of worlds
which accurately represent the world up to t. More exactly, the actual
possible worlds comprise the R-equivalence class of these `historically
possible' worlds all the members of which are governed by the actual physical
laws. We can say, then, that a sentence in TGL is true simpliciter iff it is
true now in every actually possible world.
I believe that no great problems will arise if we add the tense operators H
and G and the tensed conditional operators >PP>, >PF>, >FP> and >FF> to the
language TGL. We can adapt the model theory of this section to the interpreta-
tion of these operators in a straightforward fashion. Since we are allowing
truth value gaps we will not in general have [Hq] = [ªPªq] and [Gq] = [ªFªq]
as we did when our models were based on linear time. I will not provide the
details for such an expansion of our model theory.
11. Edelberg Inferences
There are two very interesting inference rules recorded by Thomason and Gupta
(1980) involving the settledness operator L. These are:
Edelberg 1: From Lªq and L(q > r) to infer q > L(q r).
Edelberg 2: From Lªq, q > Lq and L(q > r) to infer q > Lr.
In a footnote, Thomason and Gupta mention stronger versions of these two
inference principles:
Edelberg 3: From L(q > r) to infer q > L(q r).
Edelberg 4: From q > Lq and L(q > r) to infer q > Lr.
These principles, all of which Thomason and Gupta endorse, motivate the
second, more complicated model theory in their paper. Their goal was to
develop a semantics which validated the Edelberg inferences but which also
validated the Stalnakerian principle CEM. For the first model theory developed
by Thomason and Gupta, which closely resembles the theory of TG-models
Tense and Conditionals 30
developed in section 9 above, the Edelberg inferences can only be insured by
imposing a restriction which seems both ad hoc and incorrect. Thus, we have
the development of the more complicated, second model theory in Thomason and
Gupta (1980).
As Thomason and Gupta observe, the restriction required to guarantee that the
Edelberg inferences are validated by a class selection function semantics for
conditionals is not so counterintuitive as the restriction required for a
world selection function semantics. The only `advantage' to be gained by using
a world selection function semantics is that CEM turns out to be valid. Since
I consider CEM to be a disadvantage rather than an advantage, I believe the
extra complications of the second Thomason-Gupta model theory are unnecessary.
All that is required, then, is to spell out the conditions for satisfying the
Edelberg inferences in a class selection function semantics like that devel-
oped in section 10.
I suggest two further restrictions on our theory of conditionals in the
context of pseudo-branching time. These two restrictions are more than strong
enough to validate the Edelberg inferences. Both concern the notions of
historical and physical possibility built into our model theory.
Consider a time-world pair <t,w> and a sentence q. At which time-world pairs
should we look in evaluating at <t,w> a conditional with q as antecedent? We
want all of those time-world pairs which are reasonably similar to <t,w> at
which q is true. Suppose we have another world w1 such that w and w1 have
common physical laws and a common history up to at least time t. It is
completely reasonable to think that any time-world pair at which q is true
which is reasonably similar to <t,w> is also reasonably similar to <t,w1>. If
w and w1 both accurately describe the world up until now, we have no way of
choosing between them since they only differ in their descriptions of the
future which is yet to be determined. Any time-world pair reasonably similar
to either should certainly be included in our actual deliberations. Thus I
propose the following restriction for our model theory.
11.1 If <w,w1> î Ht and wRw1, then f(q,t,w) = f(q,t,w1).
This restriction together with 10.12 gives us the following quite reasonable
result.
11.2 If <w,w1> î Ht and wRw1 and w1 î [q], then <t,w1> î f(q,t,w).
In fact, we should get an even stronger result which cannot be stated precise-
ly. If w and w1 share the same laws and the same history up to t, and if t1 is
reasonably similar to t (which will depend upon context and upon the particu-
lar antecedent q), then we will also want <t1,w1> to be a member of f(q,t,w).
One interesting consequence of 11.2 is that we should expect the principle
CS: (q r) (q > r)
to be invalid. Where r is a contingent sentence which depends on the future in
a way that makes it indeterminate, we could certainly have a time t and two
Tense and Conditionals 31
worlds w and w1 which share the same laws and the same history up to t such
that q is true at both <t,w> and <t,w1> and r is true at <t,w>, but r is not
true at <t,w1>. Then given 11.2, q r is true at <t,w> but q > r is not. Thus
commitment to a theory of indeterminant time could provide additional reason
to reject the principle CS, a principle which has already received consider-
able criticism. It should be noticed, though, that a modified version of CS,
CSL: L(q r) (q > r)
escapes this particular criticism unscathed.
The second restriction I propose depends on the reasonableness of treating
similarly worlds which share laws and histories, in much the same way as did
the first restriction.
11.3 If <t1,w1> î f(q,t,w), <w1,w2> î Ht, w1Rw2, and <t1,w2> î [q], then
<t1,w2> î f(q,t,w).
The motivation for 11.3 should be clear. Again, we might endorse a stronger
principle which can only be stated informally: if <t1,w1> î f(q,t,w) and t2
is reasonably close to t (given q and the context) and w1Rw2 and either <w1,w2>
î Ht1 or <w1,w2> î Ht2, then <t2,w2> î f(q,t,w). I feel much less confident of
this principle than I do of the one corresponding to 11.2.
The restrictions 11.1 and 11.3 are sufficient to guarantee all four of the
Edelberg inferences. 11.1 also guarantees the following very strong thesis:
11.4 (q > r) L(q > r).
If we add our tensed conditional operators >PP>, >PF>, >FP> and >FF> to the
language TGL, we find that 11.1 is also strong enough to guarantee all of the
theses produced by replacing the ordinary conditional operator in 11.4 by one
of the tensed conditional operators. The only reason I can see for opposing
11.4 and its tensed counterparts is a commitment to CS, and such a commitment
seems to me to be a mistake. 11.4 will certainly hold where q and r concern
only the present and the past. Where q or r concern the future, we should
surely want to say that an intensional conditional is only true if it is true
regardless of the particular alternative future which is actualized. Finally,
11.1 and 11.3 together allow us to strengthen the Edelberg inferences in the
following ways:
Edelberg 5: From q > r to infer q > L(q r).
Edelberg 6: From q > Lq and q > r to infer q > Lr.
12. Loose Ends
We have explored a number of interesting issues involving the interaction of
tense and conditionality, but much remains to be done. One important task
which remains is the axiomatization of the logics characterized by the various
model theories developed in this paper. Efforts in this direction will likely
result in further refinement of the model theories themselves, and probably in
Tense and Conditionals 32
alternative refinements which will compete for acceptance. Another avenue for
further investigation, at which I have hinted repeatedly, is the adaptation of
the suggestions in this paper to an interval semantics for time. Still another
interesting problem which has been completely ignored in this paper is the
analysis of conditionals involving progressive tenses. There is also a need to
investigate the role which such temporal adverbs as `since' and `until' play
in the truth conditions of conditionals in which they occur. I think that the
progressive tenses always play an intensional role, and `since' and `until'
play an intensional role in conditional contexts which they do not always play
in other contexts. These additional intensional operators complicate the
analysis of conditionals in ways which will only be untangled through consid-
erable effort. Despite the large and growing literature in conditional logic,
the problems of tense are only just beginning to attract the attention of
conditional logicians. This paper, together with those by Thomason and Gupta
and by van Fraassen, are only a beginning.
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Gabbay, Dov, and Guenthner, Franz (eds.). 1984. Handbook of Philosophical
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