P and possibly not-P?

Robert Bass
Shatzel Hall
Department of Philosophy
Bowling Green State University
Bowling Green, OH 43402
rhbass@hushmail.com  

 

 

 

In a well-known passage from De Interpretatione, Chapter 9, Aristotle says that

With regard to what is and what has been it is necessary for the affirmation or the negation to be true or false.  And with universals taken universally it is always necessary for one to be true and the other false, and with particulars too, as we have said .... But with particulars that are going to be it is different....

[I]t is necessary for there to be or not to be a sea-battle tomorrow; but it is not necessary for a sea-battle to take place tomorrow, nor for one not to take place. So, since statements are true according to how the actual things are, it is clear that wherever these are such as to allow of contraries as chance has it, the same necessarily holds for the contradictories also.  This happens with things that are not always so or are not always not so.  With these it is necessary for one or the other of the contradictories to be true or false – not, however, this one or that one, but as chance has it; or for one to be true rather than the other, yet not already true or false.

Clearly, then, it is not necessary that of every affirmation and opposite negation that one should be true and the other false.  For what holds for things that are does not hold for things that are not but may possibly be or not be; with these it is as we have said. (18a 29-33, 19a 30- 19b 4)

 

This appears to say that for some statements about the future, where the occurrence or non-occurrence of the event is “as chance would have it” – which I shall take to mean that its occurrence is or depends upon some chance event(s)[1] – that neither the statements nor their denials are either true or false.  Just what Aristotle’s argument was here is not completely transparent, and there have been differing attempts to reconstruct and assess it.[2] It may be that we haven’t got sufficient materials at hand to be sure what the argument was.  However that may be, I think that Aristotle’s instincts were sound, even if, perhaps, his argument was not.  That is, I think it is correct to hold that statements about future events, where the occurrence or non-occurrence of those events depends on chance events,[3] are neither true nor false, but indeterminate.[4]  I shall try to do a little ground-clearing to get the issue in focus, make some terminological distinctions, and present an argument for my conclusion together with addressing some objections.

 

I. Ground-Clearing

Three points are important in order to be clear about what questions are at issue.  First, they must be distinguished from questions about knowledge.  Second, something must be said about the sense in which the events in question are or depend upon chance events.  (I shall only assume, rather than argue, that there are chance events.)  Third, something has to be said about the relation between truths and times.

Let’s begin with distinguishing the problem from questions about knowledge.  One guise in which people have considered the issue is by way of arguments about the relation between God’s omniscience and human freedom or responsibility.  In some of these arguments – which is where I first encountered the problem – the issue is identical.  However, if the problem is posed in this way, it is easy to lose sight of the fact that the issue is one about the truth or falsehood of statements about the future rather than about whether an omniscient being knows it.  For the issue I’m interested in, whether any being knows or can know whether the statements in question are true is irrelevant. If I am mistaken, there may still be problems reconciling omniscience with human freedom; if I am correct, no assumptions about omniscience are needed.  Nor, relatedly, am I interested (for present purposes) in questions about human knowledge.  I do not take denying that some statements about the future are either true or false to be equivalent to a denial that we are in a position to know of their truth or falsehood.  Of course, if they are not true or false, they cannot be known to be, but the converse does not hold: it may be that they cannot be known to be true or false but nonetheless are.

Second, I need to say something about the sense in which the future events in question may be attributed to chance.  I shall take a deterministic explanation to be one that explains the occurrence of an event, including all its features, in terms of a sufficient cause.  I shall assume that when there is a sufficient cause for the occurrence of an event, then the occurrence of that event can be deduced from (true) statements about antecedent conditions and about any relevant natural laws.[5]  Then, a chance event will be one for which there is no deterministic explanation, that is, one that lacks a sufficient cause.[6]  Given any relevant laws of nature and any set of antecedent conditions, there is at least one alternative possible outcome.[7]

Third, remember that in Aristotle’s version of  the argument quoted earlier, there was a reference to what was “not already true.” What is important to note here is that it is assumed that some statements are true at some times, but not at others.  In particular, a statement may be true tomorrow that is not today (which does not mean that it is false today).  Some philosophers have thought that notion problematic: truth must be timeless.  I confess I have little sympathy with that (perhaps inadequate exposure to Plato!), but, even if it is correct, it will not affect my argument.  For one way of understanding what it means to say that truth is timeless is just to say that a timeless truth is one that holds at all times.  Suppose the truth in question is that there is a sea-battle on a certain date.  If, under that interpretation, that is timelessly true, then it is true on the day before that date.  That’s the conception of “truth at a time” that I need for my argument.  Suppose, on the other hand, that the timelessness of truth is understood in some other way, so that the inference from “p is timelessly true” to “p is true at time t₀” is disallowed.  (I am not at all clear what this other way amounts to.  I’m just considering it for the sake of the argument.)  In that case, “there is a sea-battle at t₁” does not imply that it is true at any earlier time that there is a sea-battle at t₁.  But if it is not true at any earlier time (or, for the same reason, false either, where “false” means “true that not”), that amounts to acceptance of my thesis (actually, of a broader thesis) in the sense I attach to it.  At the earlier time, if the event being spoken of is or depends on some chance event, it is neither true nor false that it will occur at the later time.

 

II.  An Initial Formulation

An initial and largely intuitive formulation of the argument might go as follows:

“There will be a sea-battle tomorrow” is false if the battle doesn’t occur. Its denial, “there will not be a sea-battle tomorrow” is false if the battle does occur. But suppose there’s nothing about the current situation that settles whether there will be a sea battle or not. Whether there will be or not depends on chance events that haven’t occurred yet. So, it’s possible – really possible, given everything that’s true now with all the relevant laws of nature and causal factors taken into account – both that there will and that there won’t be a sea-battle. Given that, how can you also say that it’s true now (or false now) that one of them will occur? What does it mean to say that it’s true that it will happen but also possible that it will not?

 

 

Unfortunately, the simple formulation is open to simple objections.  It appears to be relying upon some claim to the effect that, if it is true that there will be a sea-battle tomorrow, then it’s not possible that there will not be one.  But surely, this will not do as it stands. ‘Not possibly not-p’ is equivalent to ‘necessarily p.’ It is certainly not true for the general case, however, that ‘p’ implies ‘necessarily p.’ If there is a sound argument here, it can’t rely upon something like that.

Susan Haack examines a version of the argument and, more formally, diagnoses the same modal fallacy:

The argument is invalid, I think, because it depends upon a modal fallacy, the fallacy of arguing from:

Necessarily (if it is now true ... that I shall be in Warsaw at noon on 21 December next year, then I shall ... be in Warsaw at noon on 21 December next year)

 

which is, of course, true, to:

 

If it is now true ... that I shall be in Warsaw at noon on 21 December next year, then necessarily I shall ... be in Warsaw at noon on 21 December next year

 

i.e., of arguing from:

 

L (A ÿ B)

 

to:

 

A ÿ L B[8]  

 

Again, if a sound argument is to be extracted, it will have to avoid this.

 

III.  Towards A Re-Formulation

A bit of additional terminology needs to be introduced.  When we are speaking of truths (at times) that hold of events, we are not speaking of whether an event is logically possible or not; we are concerned with whether, given what has already happened and any other relevant factors, including laws of nature, it is settled that it will occur.  From this point forward, I shall use ‘~p’ to mean ‘it is settled that p.’ Given antecedent conditions and laws of nature, it is not possible that ~p.  If it is not settled that p, then of course it is possible that ~p.

In addition, if we allow for chance events, then not all future events are settled.  Consider the case in which whether or not there will be a sea-battle tomorrow strictly depends upon a chance event that will occur in one of two possible ways, A or B, six hours before the sea-battle will (or won’t) occur.  If A occurs, then the sea-battle will occur as well; if B occurs, then the sea-battle will not. If the chance event has not yet occurred, then it is not now settled whether there will be a sea-battle tomorrow.  But once the chance event has occurred, it is settled (one way or the other).  Thus, since an event can be unsettled at one time and settled at a later time, in order to be clear, temporal indices will have to be attached to the modal operators.  For the above story, let ‘S’ be ‘the sea-battle occurs,’ ‘t₁’ be the time now, ‘t₂’ be the time of the occurrence of the chance event, and ‘t₃’ the time of the occurrence or non-occurrence of the sea-battle.  Suppose further that A occurs at t₂.  Then, �t₁ ~St₃, but ~t₂ St₃.

An intuitively appealing way to represent the kinds of relations between events that were just described is to view the temporal sequence of events as forming a branching structure.  That is, we will begin with the actual world, a, at a time, t₁, and connect it with an accessibility relation, where the accessibility relation is defined in terms of what is true at t₁, including all laws of nature, to all the future possible worlds consistent with that set of conditions.  For any claim, A, at t_n  (where n >1) that is true in any of the possible worlds accessible from at₁, then �t₁ At_n.  If there is any A at t_n that is true in all possible worlds accessible from at₁, then ~t₁ At_n.  And if there is any A at t_n that is false in all possible worlds accessible from a t₁, then ~�t₁ At_n.  That is, At_n is respectively possible, settled, and impossible under the conditions described.

Now we can return to the case of tomorrow’s sea-battle.  If it is to be both true at at₁ that the sea-battle occurs at at₃ and also possible at at₁ that the sea-battle does not occur at at₃, then there will have to be some possible world accessible from at₁ in which the sea-battle does not occur at at₃.  But, of course, there are no such possible worlds.  Such a world would be one in which it is both true and false that the sea-battle occurs at t₃.  With simple adjustments, a parallel argument can be constructed with respect to the claim that it is both false at at₁  that the sea-battle occurs at at₃ and that it is also possible at at₁ that the sea-battle does occur at at₃.

Note that in the present exposition, I have said nothing so far about the alleged intervening chance event at t₂.  As chance events were earlier characterized, there could not be one.  I said that “[g]iven any relevant laws of nature and any set of antecedent conditions, there is at least one alternative possible outcome.” In this case, however, there is an antecedent condition, namely, the truth about whether the sea-battle occurs, that precludes the alternative possible outcome.  If it is true at at₁ that the sea-battle will occur at  a t₃, and the sea-battle will not occur at at₃, unless A occurs at at₂, then A must occur at at₂.[9]  Conversely, if there is a chance event at at₂ upon which the occurrence or non-occurrence of the sea-battle strictly depends, then it is not true at at₁ that the sea-battle will occur at at₃.  And again, the parallel holds: if there is a chance event at at₂ upon which the occurrence or non-occurrence of the sea-battle strictly depends, then it is not false at at₁ that the sea-battle will occur at at₃.  So, if there are events that are “as chance has it,” then some statements about the future are neither true nor false.

This seems a simple argument and I anticipate objection.  But first note that I have tried to avoid the earlier objections.  I have not argued that there is any generally acceptable inference from a premise p to ~p.  Rather, I have argued that in the special case of truths about the future, the accessibility relation is such that the inference goes through for that case.  This should also suffice to answer Haack’s objection since she did not offer any argument that there could not be an accessibility relation with the feature to which she rightly objected for the general case.

I see two possible objections.  First, it might be objected that it is a mistake, in defining the accessibility relation, to include as a condition upon it the truth about whether there will be a sea-battle.  That should not count as one of the facts (or natural laws) obtaining at t₁.  But I fail to see why.  Admittedly, its being true when (it is claimed that) things could turn out differently is rather strange.  In one sense, that is my whole point: I claim that it is not only strange but incoherent.  But if it is insisted that it really is true at the earlier time, why may it not be used to infer that there are no accessible worlds in which it turns out to be false, and therefore that (in this special case) the inference from p to ~p goes through?

Second, it might be objected that the claim that it might be true at an earlier time that a sea battle will occur and possible that it will not has been misunderstood.  The claim should not have been understood to hold that it is possible that in the actual world the sea battle will not occur.  I am not sure what to make of this.  It sounds as though it is conceding the point rather than providing an objection.  If all that is meant by saying that it is possible that the sea-battle will not occur (though true that it will) is that there is some non-actual world in which it does not occur, I think I can agree and still insist that it is settled for the actual world.

On the whole, I think the argument is sound.  And if it is, then we will have to admit either that some statements about the future are neither true nor false or else deny that there are any chance events.

 

References

Haack, Susan.  Philosophy of Logics.  Cambridge: Cambridge University Press, 1978.

Sorabji, Richard.  Necessity, Cause, and Blame: Perspectives on Aristotle’s Theory. Ithaca: Cornell University Press, 1980.

 

 

 

 

Comments? I’d love to hear.