The Subjunctive Conditional Law of Excluded Middle
February 4, 2002
Alexander R. Pruss
Alvin Plantinga has argued that there are true counterfactuals of libertarian free will of the form “Were Curley to have been offered the bribe, he would have taken it.” Some, though not Plantinga himself, have attempted to help Plantinga argue for this conclusion on the basis of the subjunctive conditional law of excluded middle that says that for any two propositions p and q, one of “were p true, q would be true” and “were p true, not-q would be true” is true. Therefore, even if Curley is not offered the bribe, either he would take it were he offered it or he would not take it were he offered it.
I will write “p·®q” for the subjunctive conditional “were p true, then q would be true.” The subjunctive conditional law of excluded middle (SCLEM) is then:
"p"q ((p·®q) or (p·®~q)).
Note that SCLEM does not follow in any obvious way from the ordinary law of excluded middle (LEM) which only gives us the tautologous claims that (p·®q) or ~(p·®q) and that p·®(q or ~q). I will assume throughout this article that the ordinary LEM is true.
I will show that SCLEM is false if we assume a Lewisian similarity of worlds semantics for counterfactuals, which Plantinga actually is willing to allow. On this semantics:
(1) p·®q holds if and only if there is a world w at which both p and q hold but which is more similar to the actual world than any world at which p and ~q hold is.[1]
I will use Lewis semantics to construct a counterexample to SCLEM. The example will also work on a plausible weakening of the Lewis semantics. This paper will leave open the question of whether there are any true counterfactuals of libertarian free will, though any considerations in favor of this claim that stem from SCLEM will be discredited by the fact that SCLEM is false in general.
It is somewhat curious that Plantinga allows the use of Lewis semantics. The reason for this is that Plantinga will allow the truth values of subjunctive conditionals to be considered when comparing worlds. Thus, if it is actually true that were Curley offered the bribe, he would have accepted it, then, all other things being equal, worlds in which this subjunctive conditional is true are for this reason taken by Plantinga to be closer to the actual world than worlds in which this conditional is false. Thus, if the above subjunctive conditional is true, then the world w1 in which Curley is offered the bribe and takes it is closer to the actual world than the world w2 in which Curley is offered the bribe and refuses it, since the subjunctive conditional is evidently true at w1 but false at w2.
Observe that SCLEM has an immediate implausible consequence on the Lewis semantics: it entails that one never has a pair of distinct worlds w1 and w2 such that w1 and w2 are equally similar to the actual world, where by “equally similar” I mean simply that neither is more similar than the other is. To see this, let p be the disjunctive proposition that w1 is actual or w2 is actual, and let q be the proposition that w1 is actual. Then, by SCLEM, either p·®q or p·®~q, and both options lead to a contradiction. For suppose that p·®q. Then there is a world w at which both p and q hold and which is closer to the actual world than any world at which p and ~q hold. Since q holds at w, it follows that it is true at w that w1 is actual, and so w=w1 since only one world is actual at a time. But there is a world equally close to the actual world as w1 at which p and ~q hold, namely w2, which contradicts the fact that w=w1 is closer to the actual world than any world at which p and ~q hold. On the other hand, suppose that p·®~q. Then there is a world w at which both p and ~q hold and which is close to the actual world than any world at which p and q hold. But since both p and ~q hold at w, it follows that w=w2, and once again we obtain a contradiction since w1 is a world just as close to the actual world as w2, but at w1 both p and q hold.
Therefore, if both SCLEM and Lewis semantics hold, for any pair of worlds, one member of the pair is closer to the actual world than the other: there are no ties. This is obviously already highly implausible, but perhaps can be tolerated. But in fact we can do better by constructing an actual counterexample to the SCLEM.
To produce a counterexample, start by considering an infinite sequence of worlds, w1, w2, w3, ... with the property that for every n, the world wn+1 is closer to the actual world than wn is. Thus, we have an infinite sequence of worlds which get closer and closer to the actual world. For instance, we can imagine a whole sequence of worlds which differ only in respect of the color of some rose, and which color approximates more and more closely, but not perfectly, the color the rose has in the actual world. Let p be the proposition that there is a positive integer n such that wn is actual. Let q be the proposition that there is a positive integer m such that w2m is actual. Thus, p says that one of the wn is actual, and q says that one of the even-numbered wn is actual.
Now, by Lewis semantics, p·®q is false. To see this, suppose it were true. Then there would be a world w at which p and q hold and which is closer to the actual world than any world where p and ~q hold. Since w is a world at which q holds, it must be that w=w2m for some appropriate choice of m. But then w2m+1 is a world closer to the actual world at which p and ~q hold, contradicting the fact that there can be no world closer than w to the actual world at which p and ~q hold. Therefore, p·®q is false. However, by the same argument p·®~q is false as well. For let w be any world at which p and ~q hold. Then, w=w2m+1 for some non-negative integer value of m. But then w2m+2 is closer to the actual world than w is. Since p and q hold at w2m+2, it follows that w cannot have the property that it is closer to the actual world than any world at which p and q hold, and so p·®~q is false. Thus, both p·®q and p·®~q are false, in contradiction to the SCLEM.
The above counterexample to SCLEM makes two assumptions. The first is that we can in fact construct an infinite sequence of possible worlds such that things further down in the list are always closer to the actual world than things nearer to the start of the list. The second is that the Lewis semantics for counterfactuals are correct. The Lewisian assumption can in fact be weakened to the following plausible-looking assumption:
(2) If for every world w where p and q hold there is a world where p and ~q hold which is closer to the actual world than w is, then p·®q is false.
Note that this assumption is weaker than the full Lewis semantics, as it allows for the possibility that p·®q is true even though the closest world at which p and q hold is at the same distance from the actual world as the closest world at which p and ~q hold. There is good reason to allow for this possibility. For, pace Plantinga’s above-discussed measure of closeness of worlds, it is plausible on libertarian grounds that the world at which Curley in a libertarian-free way takes the counterfactual bribe is at exactly the same distance from the actual world as the world at which he does not in a libertarian-free way take it, and yet, unless we are to beg the question against the defender of counterfactuals of free will, we need to allow that perhaps there is a fact of the matter as to whether he would take it or he would not take it. Thus, (2) may be preferable to Lewis semantics for a defender of counterfactuals of free will. However, as can easily be seen, the above counterexample to SCLEM only used (2), and not the full Lewis semantics of (1).
Instead of questioning the Lewis semantics in its strong form (1) or in its weak form (2), the defender of SCLEM might instead question the construction. Perhaps in fact colors can only take on values from a discrete spectrum, and so we cannot have an infinite sequence of colors approximating the color of some rose better and better. But there are variant constructions. Consider any object in the world. We can imagine an infinite sequence of worlds where the object is at a slightly different position, and this position approximates the position of the object in the actual world more and more closely. Or if we prefer to work with wave-functions, we can take the wave-function of a particle in the actual world and consider an infinite sequence of worlds with corresponding wave-functions that approximate closer and closer to the actual wave-function of the particle. It would be really strange if the defender of SCLEM could claim to rule out these apparently physically-possible constructions on a priori grounds. Of course, the defender of SCLEM could insist that physics is wrong, and that in fact space and time are discrete, as are all observables. However even this extreme view is insufficient. For, SCLEM is presented as a law of logic. Therefore, the defender of SCLEM is committed to SCLEM holding in all worlds. Therefore, the defender of SCLEM would not only have to claim that space, time and all observables are in fact discrete, but that of logical necessity they must be, which is surely absurd.
Therefore, if one is to retain SCLEM, one will have to deny even the weaker form (2) of the Lewis semantics. But even after (2) is rejected, there is a difficulty. The intuition behind SCLEM is that were p true, then either q or ~q would be true: tertium non datur. From the fact that were p true it would be the case that q is true or that ~q is true, the defender of SCLEM concludes that either q would be true were p true or ~q would be true were p true. Formally, this inference seems invalid, since it pulls the disjunction outside the scope of the “were ... would” operator, but there is an intuitive attraction to this inference.
However, the same intuition would make one reason as follows. In the counterexample, were p true, then one, exactly one, of the wn worlds would be actual. The same kind of intuition that makes one want to infer from the fact that q or ~q would be true that q would be true or ~q would be true then pulls one to infer that there is one of the wn worlds which would be actual were q actual. In fact, we get this from the following extension of SCLEM in the case of infinite disjunctions:
(3) If (p·®(q1 or q2 or ...)), then (p·®q1) or (p·®q2) or ....
To recover the ordinary SCLEM from this, let q1 be q, and let q2, q3,… all be ~q and observe that from the ordinary LEM it follows that p·®(q1 or q2 or ...), and SCLEM follows from the consequent.
Any intuitive reason for believing SCLEM carries over to the more general principle, and this principle easily implies there is an m such that were p true (i.e., were one of the wn actual), then wm would be actual. But this violates the following plausible even weaker Lewisian principles:
(4) If there is a unique world w which is the closest world to the actual world from among the worlds at which p and q hold, and if there is a world closer than w to the actual world at which p and ~q hold, then it is not the case that p·®q.
(In fact, under the conditions described in (4), it is plausible that p·®~q.) To see that the claim that were p true then wm would be actual violates (4), let q now be the proposition that wm is actual. Then, there is only one world at which p and q hold, namely wm. But wm+1 is a world closer to the actual world than wm is at which p holds but q does not hold, and so by (4) it is not the case that p·®q, contrary to (3).
Therefore, if one accepts any of the Lewisian principles (1), (2) or (4), one will have to reject at least the reasoning behind SCLEM, and if one accepts (1) or (2), one will have to actually reject SCLEM itself.
Finally, as a closing remark, it is interesting to note that completely independently of SCLEM, the Lewisian semantics has a somewhat implausible consequence when applied to a variant of my counterexample. Let w1, w2, ... be a sequence of worlds getting closer to the actual world. Let p be the proposition that one of these worlds is actual. Now, let qn be the proposition that wn is actual. Then, by the Lewis semantics (1), it is true for any n that p·®~qn. For, wn+1 is a world closer to the actual world than any world at which p and qn hold, and yet p and ~qn hold at wn+1. Therefore:
"n (p·®~qn).
It is tempting to infer from this that:
p·®"n(~qn)
But of course the latter claim is false, since "n(~qn) is equivalent to ~p. Therefore, we have the curious situation that for every n it is true that were p true, then qn would be false, even though were p true, there would be an n such that qn is true.[2] Implausible as the claim "n (p·®~qn) might be, however, it is sufficiently far removed from our ordinary-language counterfactuals that we might tolerate it. Of course, the defender of SCLEM may take this as reason to save SCLEM by rejecting Lewis semantics in all its guises. But actually, this example is not reason to reject (2) or (4), but at most reason to reject (1), and (2) is sufficient to give us the counterexample to SCLEM.