Counterfactual conditionals ("counterfactuals," for short) are statements about situations which we know did not occur -- as opposed to indicative conditionals, where the antecedent is, or at least might be, true. For example, if we know that it's not raining, it is natural to make some judgement about what would have happened, if it had rained:
Of course, someone might disagree, and hold the opposite counterfactual conditional:
But even if we disagree over which of two opposed counterfactuals is true -- that is, about which consequent would have been true, if the counterfactual antecedent were true -- even then we all agree that not both of the (opposed) consequents would have been true. That is, we agree in rejecting the absurd counterfactual (3):
(Also, of course, we may sometimes not be able even to develop an opinion about which counterfactual is more likely than another: if I don't know anything about the person in question, for example, I may be in no position at all to say whether he would have come to class in the rain or not. Still, I'd be confident that the guy wouldn't -- indeed couldn't -- both come to class and miss the class at the same time.)
But think about the truth-table for the conditional, from back in sentence logic:
P | Q | ~Q | P --> Q | P --> ~Q
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1) T | T | F | T | F
2) T | F | T | F | T
3) F | T | F | T | T
4) F | F | T | T | T
How does our traditional truth table handle counterfactual conditionals -- cases where the antecedent is false? Notice that in valuations where the antecedent is false -- valuations (3) and (4) -- the truth table says both conditionals are true. But from what we just said above, that can't be right: even if we disagree over which counterfactual conditional is true, we don't think both of them are. So it looks like the truth-table, while perhaps faithful to conditionals whose antecedents we don't know are false ("indicative conditionals"), doesn't do justice to counterfactuals. Here we have one of those cases where traditional logic seems to get tripped up.
An improved theory of counterfactuals needs to recognize several things. First, when we judge what would have happened if it had rained, we appeal to our knowledge -- perhaps knowledge about human nature, about this person's class attendance behavior in previous rainy situations, and so on. We don't just wipe the slate clean, and imagine an entirely new and different situation where it rains -- rather, we hang on to all this background knowledge, sentences in the Conversational Background, and make use of it, imagining a situation where much of what we know about the person remains the same. So making such a counterfactual judgment involves keeping information in the Conversational Background, but adding to it the hypothetical assumption that (contrary to the facts) it rained:
Here "R" is the counterfactual assumption ("It rained"), and the sentences in brackets are relevant bits of the Conversational Background. If adding the counterfactual antecedent to the conversational background were all there is to judging counterfactuals, we would have a nice simple pragmatic theory. But there's a problem with this little scheme: one of the many things we know, and have recorded in the Conversational Background, is that in fact it didn't rain -- "~R" is one of the sentences in the Conversational Background. So when we add "R" we get a blatant contradiction:
That's just wrong: we can perfectly well imagine that it had rained, without automatically imagining a contradictory, impossible -- situation. The situation where it rains isn't impossible -- it just didn't happen to occur. So we need to fix our method of adjusting the Conversational Background. Clearly, one thing we should do is remove the conflicting sentence "~R" from the Conversational Background -- a hypothetical situation where we believe it's raining isn't also a situation where we believe it's notraining. Likewise, certain other sentences, to the claim that it's not raining, should also be thrown out. For example, if we originally held that it didn't rain, we probably also believed that lots of little drops of water didn't fall through the air just then; so if we throw out the claim that it didn't rain, we should also throw out its consequences, like the claim that there weren't little drops of water falling through the air.
We then get the following rule for adjusting the background assumptions:
Following this rule keeps counterfactual situations from being immediately impossible, and that agrees with our basic intuitions about how counterfactuals work.
As it stands, however, the rule is still flawed, because it turns out there's more than one way of throwing out sentences to keep the Conversational Background consistent. This spells real trouble when following the same rule in different ways, leads us to two different, and opposed, counterfactuals. Don't think it could happen? Read on.
Suppose my lucky match is never struck in its whole 'lifetime'. I keep it stored safely in a box, until I feel its luck has run out, and then cut it up into tiny pieces and flush it down the toilet. And, to be perfectly clear, suppose it was a normal match, and was never wet (before it was flushed). Even though it was never struck, I could still consider what would have happened, if I had struck it on some occasion where there was a dry flint, sufficient oxygen, and the "Law of Matches" held. (In case you don't know, the Law of Matches says that normal dry matches struck on a dry flint with enough oxygen will ordinarily light.). Would my lucky match have lit if it had been struck in that situation? There are only two possible outcomes to such a counterfactual situation:
These two counterfactuals do not strike most people as equally believable. Rather, people usually lean strongly in favor of (4). How does our counterfactual rule (CR) lead us to the consequent of (4)? We have various bits of information in the Conversational Background: the match was not struck (~S), it was a normal match (N), it was dry (D), there was enough oxygen (O), there was a dry flint (F), the Law of Matches held (L), and so on.
Since we're picturing a scenario where the match was struck, CR tells us to add that sentence (S) to the Conversational Background, and then adjust the Conversational Background to maintain consistency -- which we can do by throwing out "~S" (and any consequences it has). From this revised Conversational Background it seems to follow pretty clearly that the match would light -- hence we get conditional (4). Nice, eh?
Except for this: as we noted above, there may be more than one way of adjusting the Conversational Background to accomodate the counterfactual antecedent. Consider another way we could apply the same rule (CR): we add "S" to the Conversational Background, making it contradictory overall; then we eliminate this contradiction by throwing out the sentences "~S" and"O". This would be a situation where the match was struck, it was dry, etc., but there was not enough oxygen; in that sort of situation, it seems more likely the match would not have lit -- so by the very same rule we can also settle on sentence (5) instead.
The problem is that our rule (CR) is not detailed and specific enough about which sentences get jettisonned from the Conversational Background -- just so long as it ends up consistent. So we can apply (CR) in different -- and incompatible -- ways. (CR) needs to be tightened up to avoid mis-applications like the one yielding sentence (5). Obviously, in this second case we threw out more than we absolutely had to in order to make the Conversational Background consistent: the sentence "O" wasn't causing a contradiction, and so it didn't need to be thrown out. But our rule (CR) didn't stop us from throwing out more than enough, just so long as we threw out at least enough. Oops! We need to revise (CR) in light of this observation, to make sure it only throws out the bare minimum number of sentences needed to maintain consistency:
This new and improved rule will block acceptance of counterfactual conditionals like (5) -- which again agrees with our intuitions about which counterfactual is right in that situation.