**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 *w*_{1}
in which Curley is offered the bribe and takes it is closer to the actual world
than the world *w*_{2} in which Curley is offered the bribe and
refuses it, since the subjunctive conditional is evidently true at *w*_{1}
but false at *w*_{2}.

Observe that SCLEM has an immediate implausible
consequence on the Lewis semantics: it entails that one never has a pair of
distinct worlds *w*_{1} and *w*_{2} such that *w*_{1} and *w*_{2} 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 *w*_{1} is actual or *w*_{2} is actual, and let *q* be the
proposition that *w*_{1} 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 *w*_{1} is actual, and so *w*=*w*_{1} since only one world is actual at a
time. But there is a world equally close to the actual world as *w*_{1} at which *p* and ~*q* hold,
namely *w*_{2}, which
contradicts the fact that *w*=*w*_{1}
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*=*w*_{2}, and once again we obtain a
contradiction since *w*_{1}
is a world just as close to the actual world as *w*_{2}, but at *w*_{1} 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, *w*_{1},
*w*_{2}, *w*_{3}, ... with the property that for every *n*,
the world *w*_{n}_{+1} is closer to the actual world than *w** _{n}* 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

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*=*w*_{2m}
for some appropriate choice of *m*. But then *w*_{2m+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*=*w*_{2m+1}
for some non-negative integer value of *m*. But then *w*_{2m+2} is closer to the actual world
than *w* is. Since *p* and *q* hold at *w*_{2m+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 *w** _{n}* worlds
would be actual. The same kind of intuition that makes one want to infer from
the fact that

(3) If (*p*·®(*q*_{1} or *q*_{2} or ...)), then (*p*·®*q*_{1}) or (*p*·®*q*_{2}) or
....

To recover the ordinary SCLEM from this, let *q*_{1}
be *q*, and let *q*_{2}, *q*_{3},… all be ~*q*
and observe that from the ordinary LEM it follows that *p*·®(*q*_{1}
or *q*_{2} 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 w* _{n}* actual), then

(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 *w** _{m}* would be actual violates (4), let

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 *w*_{1},
*w*_{2}, ... 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 *q** _{n}* be the proposition that

"*n*
(*p*·®~*q** _{n}*).

It is tempting to infer from this that:

*p*·®"*n*(~*q** _{n}*)

But of course the latter claim is false, since "*n*(~*q** _{n}*) is equivalent to ~