April 17, 2007

Lewis on Non-Trivial Counterpossibles

Lewis offers a number of reasons for his, the now standard, reading of counterpossibles--viz., that they are all vacuously true. He does not take his reasons to be decisive. Let's revisit the reasons. Lewis writes,

There is some intuitive justification for the decision to make a 'would' counterfactual with an impossible antecedent come out vacuously true. Confronted by an antecedent that is not really an entertainable supposition, one may react by saying, with a shrug: If that were so, anything you like would be true! Further, it seems that a counterfactual in which the antecedent logically implies the consequent ought always to be true; and one sort of impossible antecedent, a self-contradictory one, logically implies any consequent. [Counterfactuals 24]

There are two reasons given here. The first is difficult to disagree with. An antecedent that is not really an entertainable supposition, invokes triviality. If Socrates were a potato, then, sure, anything goes! I'm down with that intuition. But this is not an objection to the position that I'm more inclined to, which says that counterpossibles are sometimes non-vacuously true (and sometimes false). Lewis' point suggests incorrectly that all counterpossibles involve an antecedent that is not entertainable. Counterlogicals, such as 'if excluded middle were invalid, then double-negation elimination would be invalid too", sometimes express non-trivial consequences of alternative logics. The sincere entertaining of such logics is something that we really can, and often do, do. That said, I share Lewis' view about non-entertainable suppositions. In sum, the position that allows for a non-vacuous reading of some counterpossibles should also allow for the vacuous reading of others.

The second reason that Lewis articulates above says that a counterfactual whose antecedent logically implies the consequent ought always to be true. That is, it ought to be that logically strict implication implies counterfactual implication. Perhaps there is a pre-semantic-theoretic intuition here. But counterpossibles are strange animals from any point of view. It shouldn't be too much of a concern if they run contrary to a pre-theoretic intuition. Moreover, putting too much weight on the intuition would be question-begging. Anyone who is convinced that there are non-trivial counterpossibles will not share this intuition about the relation between strict and counterfactual implication. Lewis seems to have ruled out in advance that there might be impossible antecedents that we sincerely entertain. For this reason, I don't think that he was considering the more interesting candidates for the non-trivial reading. The bottom line is that the Lewis intuition just isn't shared by those who take seriously the idea that some impossibilities may be sincerely entertained counterfactually.

Lewis offers an explanation for why some of us believe in non-trivial counterpossibles. He hypothesizes a mistaken need to explain why some counterpossible that we do want to assert are true and others that we do not want to assert are false. He writes,
I do not think, however, that we need to discriminate in truth value among such counterfactuals. Of course there are some we would assert and some we would not:
If there were a largest prime p, p!+1 would be prime.
If there were a largest prime p, p!+1 would be composite

are both sensible things to say, but

If there were a largest prime p, there would be six regular solids.
If there were a largest prime p, pigs would have wings.

are not. But what does that prove? We have to explain why things we do want to assert are true (or at least why we take them to be true, or at least why we take them to approximate to truth), but we do not have to explain why things we do not want to assert are false. We have plenty of cases in which we do not want to assert counterfactuals with impossible antecedents, but so far as I know we do not want to assert their negations either. Therefore, they do not have to be made false by a correct account of truth conditions; they can be truths which (for good conversational reasons) it would always be pointless to assert. [24-25]

I agree that a willingness to assert the former, but not the latter, examples is not a good reason to hypothesize distinct truth-values. Indeed, it appears to me that all four examples have the same truth-value. The context of the latter two presumably invokes the trivial world where anything goes, and the context of the former two presumably invokes non-trivial reasons for thinking that the consequents obtain in the counterfactual circumstances described. What Lewis leaves out are cases where, in fact, we do want to assert the negation of a counterpossible. For instance, it is false that if intuitionistic logic were the correct logic, then excluded middle would still be valid. This counterpossible is not merely something that we wouldn't want to assert; it is something we should want to deny! Lewis' pragmatic explanation of the desire for non-trivial truth conditions does not appear to generalize.


Aidan said...

There was some previous discussion of these issues from a while back over at Carrie's blog. The relevant thread is here.

Trent_Dougherty said...

Hey Joe, nice post.

We were actually discussing counterpossibles in class with Ed Wierenga last Thursday. I was defending the nontrivial truth of certain counterpossibles on grounds that they are explanatory and otherwise informative (Aquinas does so and recent defenses of this have appeard by both Freddoso and Zagzebski).

Turns out Ed has a paper called "Theism and Counterpossibles" (Phil Studies 89: 87-103, 1998) in which he says of any such case that it only appears non-trivial because "it is derivable from a nontrivial general truth or if it is justified by some such nontrivial truth" (97).

In an example he explains the apparent non-triviality of

"If the square root of two were rational there would be integers n and m such that at most one of n and m is even and the square root of two is equal to n divided by m."

is explained by the fact that it appears to be an instance of

"For every real number r, is r is rational, then, etc. "

It seems that similar considerations could explain the force of the conditionals you advert to at the end.

I take it that Ed's explanation goes like this:

"If intuitionism were the correct logic, then excluded middle would still be valid."

seems false because it seems non-trivially true that

"If intuitionism were the correct logic, then excluded middle would NOT still be valid."

[N.B. Doesn't seem like LCEM worries are relevant here.]

But *that* only seems non-trivially true because it seems a substantive instance of

"If *these* were your axioms, then you would be able to prove *this* conclusion."

What do you think?

Joe Salerno said...

Hi Trent. I haven't yet read the paper you mentioned, but my initial reaction is that there is a difference between talking about what follows from this or that formal system, and what would be the case if this or that formal system were correct. I'm pretty sure that many of MY counterlogicals thoughts are about the world, and not simply about the consequences of formal systems. I'm talking about what the world (and the space of possible worlds) would or wouldn't be like, given that this or that logic were the correct logic. I think those questions can be literally entertained. The pragmatic explanation that you alluded to appears to deny that such literal philosophizing is non-trivial. So I resist for now.

Trent_Dougherty said...

"I'm talking about what the world (and the space of possible worlds) would or wouldn't be like"

I think impossible worlds semantics are best for counterlogicals, but I imagine you'll want to resist that as well.

However, perhaps I can coax you into saying just a tiny bit about what you have against impossible worlds (and if you don't have something against them then perhaps the quoted statement is a bit quick).

Joe Salerno said...

Right. I'm actually favorable to an impossible worlds account of non-trivial counterpossibles. The quote suggests that if an alternative logic were correct then the space of possible worlds would be different. Granted, some of those "possible" worlds are in fact impossible.