
























                                      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

            be lost in the PostScript version.









                                      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 ,  ,  ,  , and  . 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 = .


                  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   [q] =  T - [q],  [q   r]  = [q]   [r], and so  on for the  other

                        truth-functional connectives.


                  1.5   t  [Pq] iff there is a t1 such that t1  t and t1  [q].


                  1.6   t  [Fq] iff there is a t1 such that t  t1 and t1  [q].


                  1.7   t  [Hq] iff for every t1 such that t1  t, t1  [q].


                  1.8   t  [Gq] iff for every t1 such that t  t1, t1  [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  T, if t,t1  I, t  t2, and

            t2   t1, then t2  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   [q] =  W - [q],  [q    r] = [q]   [r], and  so on for  our other

                        truth-functional connectives.


                  3.6   If f(q,w) is not empty, then f(q,w)  [q].


                  3.7   w  [q > r] iff f(q,w) is empty or f(q,w) is contained in [r].


                  3.8   If w  [q], then f(q,w) = {w}.


                  3.9   If f(q,w) = , then f(r,w)  [q] = .


                  3.10  If f(q,w)  [r] and f(r,w)  [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 > r)


            If f(q,w) is empty,  then clearly w   [q > r] by 3.7.  On the other hand,  if

            f(q,w) = w1, then w1  [r] or w1  [r] by 3.5, and thus w  [q > r]  or w  [q

            > r] 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) = , then f(r,w)  [q] = .


                  3.18  w  [q > r] iff f(q,w) is contained in [r].


                  3.19  If w  [q], then w  f(q,w).


                  3.20  If f(q,w)   [r] is not  empty, then  f(q   r,w)  is contained  in

                        f(q,w)  [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   T, all

            w,w1  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  W = .


                  4.4    is a strict total ordering for T.


                  4.5   f is  a function which assigns to every sentence  q  CT, time t 

                        T, and world w  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   [q] = (T x W) - [q], [q   r] = [q]  [r], and so on for the  rest

                        of our truth-functional connectives.


                  4.8   If w1  f(q,t,w), then <t,w1>  [q].


                  4.9   <t,w>  [q > r] iff for every w1  f(q,t,w), <t,w1>  [r].


                  4.10  <t,w>  [Pq] iff there is a t1 such that t1  t and <t1,w>  [q].


                  4.11  <t,w>  [Fq] iff there is a t1 such that t  t1 and <t1,w>  [q].


                  4.12  <t,w>  [Hq] iff  for every t1 such that t1  t, <t1,w>  [q].


                  4.13  <t,w>  [Gq] iff for every t1 such that t  t1, <t1,w>  [q].


                  4.14  If <t,w>  [q], then w  f(q,t,w).


                  4.15  If f(q,t,w) = , then f(r,t,w)  {w1:<t,w1>  [q]} = .


                  4.16  If f(q,t,w)  {w1:<t,w1>  [r]} is not  empty, then f(q   r,t,w) is

                        contained in f(q,t,w)  {w1:<t,w1>  [r]}.


                  4.17  If  f(q,t,w) is  contained in  {w1:<t,w1>    [r]} and  f(r,t,w) is

                        contained in {w1:<t,w1>  [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)  [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>   [q >PP> r]  iff for every  t1 and w1  such that  <t1,w1> 

                        f(q,t,w) and t1  t,  there is a t2 such that t2  t1  and <t2,w1> 

                        [r].


                  6.3   <t,w>   [q >PF>  r] iff for  every t1 and  w1 such that  <t1,w1> 

                        f(q,t,w) and t1  t,  there is a t2 such that t1  t2  and <t2,w1> 

                        [r].


                  6.4   <t,w>   [q  >FP> r] iff  for every t1  and w1 such  that <t1,w1> 

                        f(q,t,w) and t  t1,  there is a t2 such that t2  t1  and <t2,w1> 

                        [r].


                  6.5   <t,w>  [q  >FF> r]  iff for every  t1 and w1  such that <t1,w1>  

                        f(q,t,w) and t  t1,  there is a t2 such that t1  t2  and <t2,w1> 

                        [r].


            Of course, we must also amend our truth condition for untensed conditionals:


                  6.6   <t,w>   [q  > r]  iff for  every w1 such  that <t,w1>   f(q,t,w),

                        <t,w1>  [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(q   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  q   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 P(q    Fr). 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  P(q   Fr)    (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,,s,[]> such

            that


                  9.1   T is a non-empty set.


                  9.2    is a  transitive relation on T  such that if t1   t and t2   t,

                        then t1  t2 or t1 = t2 or t2  t1.


                  9.3   Ht is the set of all  subsets h of T such  that t  h,   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  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   [q] = {<t,h>:h  Ht}  - [q], and so on for  the other truth-func-

                        tional connectives.


                  9.6   <t,h>   [Pq] iff h   Ht and there is  a t1 in h such  that t1  t

                        and <t1,h>  [q].


                  9.7   <t,h>  [Fq]  iff h  Ht and  there is a  t1 in h such that  t  t1

                        and <t1,h)  [q].


                  9.8   s is  a function which  assigns to  each sentence q,  time t,  and

                        member h of Ht either  or a pair <t1,h1> such that h1  Ht1.


                  9.9   <t,h>  [q >  r] iff h  Ht and either s(q,t,h)  =  or s(q,t,h) 

                        [r].


                  9.10  <t,h>  [Lq] iff h  Ht and for all h1 in Ht, <t,h1>  [q].


            T represents the set of times and  is an earlier-than relation on T. But  is

            quite different  from the earlier-than  relation of our  earlier models. In  a

            TG-model, distinct  times might  not be related  by   at all. The  relation 

            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  t1,  and (ii) if t   t1 and t3  t2  and t  t2  then not t2   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  q 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 , and we might  have two times t   h and t1   h1 such that for  all

            times t2  t and all sentences q, <t2,h>  [q] if and only if <f(t2),h1>  [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  Ht  and

            h1  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>  [Lq] iff for all h1 such that hR(t)h1, h1  [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  T, all w,w1  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   T let Ht = {<w,w1>:   for all q and  all t1 such that

                        t1 = t or t1  t, <t1,w>  [q] iff <t1,w1>  [q]}.









            Tense and Conditionals                                                      28


                10.16   R is an equivalence relation on W.


                10.17   <t,w>  [Lq] iff for all w1 such that wRw1 and <w,w1>  Ht,  <t,w1>

                         [q].


            We see that two worlds w and  w1 share the same past up to time t just in case

            <w,w1>  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> 

            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  

            on T+ such that <t,w>  <t1,w1>  iff t  t1 and <w,w1>  Ht. Then <T+,,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) =  if f(q,t,w) = 0 and let s(q,t,w)  f(q,t,w)

            otherwise,  then   <T+,,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 .


            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 Lq 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] = [Pq] and [Gq] =  [Fq]

            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 Lq and L(q > r) to infer q > L(q   r).


                  Edelberg 2:  From Lq, 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>  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>  Ht and wRw1 and w1  [q], then <t,w1>  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>  f(q,t,w), <w1,w2>  Ht, w1Rw2, and <t1,w2>  [q], then

                        <t1,w2>  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>   f(q,t,w) and t2

            is reasonably close to t (given q and the context) and w1Rw2 and either <w1,w2>

              Ht1 or <w1,w2>  Ht2, then <t2,w2>   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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            Tense and Conditionals                                                      33


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