## Back to the PSR

If we do grant that causes always yield explanations, we can do even better than a CP on the given assumptions - we can get the PSR, supposing that the arguments previously outlined have successfully established the following claim:

(42) Necessarily, all wholly contingent, positive states of affairs have causes.

This is not only a CP, but it seems to entail the necessary truth of a PSR for positive propositions, that is, propositions that report positive states of affairs. For if p is a proposition reporting a positive state of affairs S, we can let S* be the maximal, wholly contingent part of S. Recall that, necessarily, S obtains if and only if S* does, and by (42), S* has a cause C. Hence, we can explain the obtaining of S* as follows: S obtains because S* has a cause C

and because contingent states of affairs obtain if and only if their maximal, wholly contingent substates do. If one objects that the noncontingent substates of S have not been explained, we can instead say that their obtaining is explained by the necessity of their obtaining or that they are self-explanatory, or stipulate that we are talking about explaining things modulo necessary truths, or perhaps hope that there is some way in which ultimately even the necessary truths all have explanations in terms of self-explanatory necessary truths (such as that each thing is identical with itself).

But now the necessary truth of a PSR for positive contingent propositions entails the necessary truth of a PSR for negative contingent propositions, where a negative proposition is the denial of a positive one. For if p is a negative contingent proposition, we can explain why p holds as follows: p holds because (a) there is nothing to explain why not-p holds, and (b) not-p is a positive contingent proposition, while (c) necessarily all positive contingent propositions that hold have explanations. This explanatory scheme is a variant of the scheme: E did not take place because no cause of E took place (see the discussion of the dog that did not bark in Section 2.3.2.2).

Finally, it is plausible that once we have explained all the positive contingent propositions and all the negative ones, then all contingent propositions will thereby have been explained, since their truth-values should supervene, in an explanation-preserving way, on the truth-values of the positive and negative ones.

This argument has an interesting consequence. I have argued (Pruss 2004a) that if we reject the PSR because we think that it has some counterexamples, such as the BCCF according to the van Inwagen argument, we should instead accept the restricted PSR (R-PSR):

(R-PSR) Every proposition that possibly has an explanation actually has an explanation.

Now, since the R-PSR claims to be a metaphysical principle, we should take it to be a necessary truth. We are now, however, in a position to see that the necessity of the R-PSR actually entails the PSR, if the arguments of the preceding section are successful. The argument is easy. The previous section shows, independently of any CP, that every wholly contingent, positive state of affairs can have a cause. Hence:

(43) Necessarily, every proposition reporting a wholly contingent, positive state of affairs can have an explanation.

The R-PSR then entails that it does have an explanation. But the same argument that shows that (42) entails the PSR also shows that (43) does so as well.

There is good news and bad news here for the cosmological arguer. The bad news is that if there are counterexamples to the PSR, there will be counterexamples to the R-PSR, and so the R-PSR does not make possible a cosmological argument that would work even if the PSR were false. The good news is that those whose intuitions lead them to accept that whatever is explainable is explained need to also accept the PSR for all contingent propositions.

If, on the other hand, the arguments of the preceding section are not successful, then it seems to be possible to accept the R-PSR without being committed to the PSR. In Section 4.4, I will show how to run a cosmological argument for a First Cause based only on the R-PSR.

### 3.2.2.1. The Brouwer analogue

The greatest difficulties in the given modal argument for the CP are with (33). The first difficulty is that (33) cannot be a conceptual truth on Lewis's semantics for counterfactuals. According to David Lewis, p □—> q is true if and only if either p is necessarily false, or there is a p&q-world (i.e. a world where p&q holds) closer to the actual world than any p&~q-world (i.e. a world where p&~q holds).

Write Aw for a proposition true at w and only at w. We might take Aw to be the BCCF of w, or we might take Aw to be the proposition that w is actual. Let q = Aw0, where w0 is the actual world. Let wi be any other world, and let p = ~Awj. Then, q & p & M~p holds. Consider the consequent of (33). This says that there is a ~p-world w at which p O—> q and which is closer than any ~p-world at which ~(p Q—> q). In fact, there is only one ~p world, namely wx. Thus, the consequent of (33) says simply that p Q—> q holds at wx. Now, p Q—> q is equivalent to ~(p □—> ~q). The proposition p □—> ~q holds at wx if and only if there is a p&~q-world that is closer to w1 than any p&q-world is. Now, there is only one p&q-world, namely w0, and a p!k~q-world is just a world different from w0 and wx. Thus, p □—>~q holds at w1 if and only if there is a world different from w0 and w1 that is closer to w1 than w0 is. Thus, ~(p □—> ~q) holds if and only if no other world is closer to wx than w0.

What we have shown is that if (33) holds, then for any world w1 other than the actual world w0, the closest world to w1 is w0. But this is most unlikely. Moreover, (33) is presented as a conceptual truth. If it is such, then the said argument should work in all possible worlds. It follows that for every pair of worlds w and w1, no other world is closer to w than w1. This is equivalent to the claim that one never has a chain of three distinct worlds w1, w2 and w3 such that w2 is closer to w1 than w3 is. But surely there are such chains, and thus the consequence is absurd. Hence, the assumption that (33) is a conceptual truth leads to absurdity on Lewis's semantics.

However, all we need (33) for is the special case where q reports a wholly contingent, positive state of affairs and p reports the nonexistence of a state of affairs under a certain description (namely, under the description of being a cause of the state of affairs reported by q), and it might well be that in those cases (33) could still hold on Lewis's semantics. The given counterexample was generated using very special propositions - the proposition q was taken to be true at exactly one world and the proposition p was taken to be false at exactly one world. Ordinary language counterfactuals do not deal with such special propositions, and hence it might be that the intuitions supporting (33) do not require us to make (33) hold for these propositions, and hence these intuitions are not refuted in the relevant case by the counterexample.

This, however, is thin ice. One might perhaps more reasonably take (33) to entail a refutation of Lewis's semantics. In any case, Lewis's semantics are known to be flawed, especially when applied to propositions like the ones in the given counterexample. To see one flaw in them, suppose that w0 is the actual world, and we have an infinite sequence of worlds w1, w2, w3, . . . such that wn+1 is closer to the actual world than wn is. For instance, these worlds could be just like the actual world except in the level of the background radiation in the universe, with this level approaching closer and closer to the actual level as n goes to infinity. Let p be the infinite disjunction of the Awn for n > 0. Fix any n > 0. On Lewis's semantics, we then have:

For w„+1 is a p&~Aw„-world that is closer than any p&Aw„-world, since there is only one p&Awn-world, namely wn, and wn+1 is closer than it. This implies that it is true for every disjunct of p that were p true, that disjunct would be false! But, surely, there has to be some disjunct of p such that were p true, that disjunct might also be true.

Like the counterexample to (33), this counterexample deals with propositions specified as true at a small (in the case of p here, infinite, but still only countably infinite and hence much "smaller" than the collection of possible worlds, which is not only not countably infinite but is not even a set18) set of worlds. This shows that there is something wrong with Lewis's semantics, either in general or in handling such propositions (see also Pruss 2007).

To see even more clearly, although making use of a slightly stronger assumption about closeness series, that there is a commonality between a problem with Lewis's semantics and the Lewisian counterexample to (33), suppose the following principle of density: for any nonactual world w, there is a nonactual world w* closer to the actual world than w is. This should at least be an epistemic possibility: our semantics for counterfactuals should not rule it out. Let w0 be the actual world and put p = ~Aw0. Then, by the principle of density, on Lewis's semantics, there is no possible world w such that were p true, w might be actual, that is, such that p 0—> Aw. To see this, suppose that we are given a w. First, note that it is hopeless to start with the case where w is w0 since p and Aw0 are logically incompatible. Next, observe that if w is not w0, then we have p □—> ~Aw. For let w* be any world closer than w. Then, w* is a p&~Aw-world that is closer than any p&Aw world, there being only one of the latter, namely w. But if we have p □—> ~Aw and p is possible, then we do not have p O—> Aw.

But, intuitively, if p is possible, then there is some world which is such that it might be actual were p to hold. Lewis's semantics fails because of its incompatibility with this claim, on the aforementioned not implausible principle of density, which should not be ruled out of court by a semantics of possible worlds. Note further that the failure here is precisely a failure in the case of a might-conditional p O—> q with p of the form ~Awi and q of the form Aw2, which is precisely the kind of might-conditional that appeared in the analysis of the putative counterexample to (33). Lewis's semantics makes too few might-conditionals of this sort true, and it is precisely through failing to make a might-conditional of this sort true that it gave a counterexample to (33).

Thus, rather than having run my argument within Lewisian possible worlds semantics, it was run on an intuitive understanding of counterfactuals, which intuitions do support (33). It would be nice to have a complete satisfactory semantics for counterfactuals. Lewisian semantics are sometimes indeed helpful: they are an appropriate model in many cases. But as we have seen, they do not always work. Other forms of semantics meet with other difficulties. We may, at least for now, be stuck with a more intuitive approach.

If we want some more precision here, we might speak as follows. To evaluate p □—> q and p O—> q at a world w, we need to look at some set R(w,p,q) of "q-relevant p-worlds relative to w" and check whether q holds at none, some, or all of these. If q holds at all of them, then p □—> q and p 0—> q. If it holds at none of them, then neither conditional is true. If it holds at some but not all of them, then ~(pn—>q) and po—>q. The difficulty is

with specifying the q-relevant p-worlds. Proposition (33) then follows from the claim that the actual world is a q-relevant p-world relative to every world w which is a relevant ~p-world relative to the actual world. This is plausible, and somewhat analogous to the Brouwer Axiom. However, this does not let us embed the discussion in a precise semantics because we do not have an account of what R(w, p, q) is.

David Manley (2002) has come up with the following apparent counterexample to (33), which I modify slightly. Suppose our soccer team wins 20 to 0. Then, it is true that the team won overwhelmingly in the actual world w0. What would have happened had our team not won? Presumably, the score would have been rather different, say 20 to 20, or 0 to 5, or something like that. Suppose the score is one of these - that we are in a possible world w1 where our team has lost. Then, it is not true that were our team to have won, it would have won overwhelmingly. If our team in fact failed to win, as at w1, then worlds where the team wins overwhelmingly are much more distant from our world than worlds where it wins by a bit. Thus, it is true at w1 to say that were our team to have won, it would have won by a tiny amount. Putting this together, we conclude that were our team not to have won, then were it to have won, it would have won by a tiny amount. But this is incompatible with (33), which claims that were our team not to have won, then were it to have won, it might have won by a tiny amount.

But this account of the situation also relies on David Lewis's semantics, and again does so in a context in which Lewis's semantics fail. For, by this reasoning, if we are in a world where our team has not won, then we should say that were it to have won, it would have won by exactly one point. But this need not be true. Perhaps were it to get ahead by a point at some point in the game, then the other team would have become disheartened and lost by more. We can even more clearly see the problem in the Lewisian reasoning if we substitute a game very much like soccer except that its scores can take on any real value: perhaps instead of a flat one point for a goal, one gets a real-valued additive score depending on how close to the middle of a goal one hits. Then, by this reasoning, were our team not to have won, it would be true that were it to have won, it would have won by no more than 1/10 of a point. Worlds where one wins by no more than 1/10 of a point are closer than worlds where one wins by more than that. But this reasoning is perfectly general, and the "1/10" can be replaced by any positive number, no matter how tiny. But this is absurd. It is absurd to suppose that were our team not to have won, it would be true that were it to have won, it would have won by no more than 10-1,000 points.19