I was talking to Eliza Block the other day about ‘irrelevant’ counterfactuals, counterfactuals where the antecedent is intuitively irrelevant to the consequent, and by the end I was wondering why anyone should think there’s a serious problem here. In case I forget why this is so, I’ll write it down. (I can’t remember how many of these ideas were Eliza’s, or even which she agreed with, or which ideas are completely familiar, hence the somewhat cautious statement of where this all came from.)

So the problem cases are meant to be things like.

(1) Had I driven from Berlin to Moscow this summer, Churchill would have been Prime Minister of Britain during WWII.

Clearly there is something wrong in a Gricean sense with (1). But is there a semantic problem with (1)? I.e. is it false? If so Lewis’s theory of counterfactuals has a problem, since (1) is true on Lewis’s theory. It seems to me the answer to that is *no*, (1) is pretty clearly true. Here’s the argument for that.

First, (2) seems to be true.

(2) Had I driven from Berlin to Moscow this summer, I would have driven from the city where Hitler died to the city where Stalin died.

(I hope my history facts are right here – otherwise make relevant substitutions.)

Moreover, if (2) is true then (3), which just involves filling out the details of the relevant deaths in some detail, is true.

(3) Had I driven from Berlin to Moscow this summer, I would have driven from the city where Hitler died after the defeat of his army in WWII at the hands of the Allies, led in large part by British Prime Minister Winston Churchill, to the city where Stalin died.

Now we need a schematic principle. It is that the following inference is valid.

Had it been that p, it would have been that q

Necessarily, if q then r

Hence

Had it been that p, it would have been that r

That seems right, and so, it seems, is (4).

(4) Necessarily, **if** I drove from the city where Hitler died after the defeat of his army in WWII at the hands of the Allies, led in large part by British Prime Minister Winston Churchill, to the city where Stalin died, **then** Churchill was Prime Minister of Britain during WWII.

And from (3) and (4), (1) follows by the principle. It may take a bit of work to do it, but it seems for any old counterfactual that seems ‘irrelevant’ but turns out to be true on Lewis’s theory we can construct a similar argument.

The air of oddity to (1) is connected to the false suggestion, conveyed by the subjunctive in the consequent, that Churchill’s having been Prime Minister is somehow doubtful or contrary to fact. Though that’s the general rule governing the subjunctive, the requirement that antecedent and consequent match in mood overrules it. I think we normally get around the false implication by judiciously inserting ‘still’ (perhaps supplemented by ‘even’):

(2) Even if I had driven from Berlin to Moscow this summer, Churchill still would have been Prime Minister of Britain during WWII.

2 seems unproblematically true to me.

I don’t think you should be asking whether a sentence like (1) is true or false simpliciter, apart from any context in which we might say such a thing. Tell a story in which someone might have a reason to say such a thing, and then we can decide whether it is true or false (or neither) in that context. In those cases where the story you tell renders the conditional true in context, I think you’ll find that the antecedent and consequent do not stand in a relation of irrelevance. We can still ask whether “If p were the case then q would be the case” is simply true (as opposed to true in some context), but then that is another question that can only be assessed in light of some definite context.

Lewis’s theory validates the inference from “p and q” to “If it were the case that p, then it would be the case that q” for any p and q. If we accept the validity of that argument, does that mean that we allow conditionals to be true though the antecedent bears no interesting relation to the consequent? Not if what we mean in saying that an argument is valid is that in any context in which the premises are true the conclusion is true (which of course is not what Lewis meant). If we can think of some context in which p and q are both true together, then to that extent p and q are not entirely unrelated. This will be clearer if we relinquish the word “true” for describing the pertinent relation to a context and substitute something like … you know what, “assertible”.

An ordinary if-then statement, p—>q, is true if q is necessarily true. So “if it rains tomorrow, then 2+2=4” is true. This is unintuitive, since people usually use if-then statements when p is related to q in some way, but it’s the only consistent way to formally define logical implication (—>). Counterfactuals seem to me to be an extension of ordinary if-then statements to cases where p picks out possible worlds, as in “if it had rained yesterday, then 2+2 would equal 4”. So it’s not a surprise to me that counterfactuals follow the same rule as logical implication here.

The other counterintuitive rule for logical implication (—>) is that anything follows from contradiction. If p is necessarily false, then p—>q is true for any q. So “if 2+2=5, then it will rain tomorrow” is true, independent of tomorrow’s weather. It is not clear whether this rule follows through to counterfactuals. “If 2+2 were equal to 5, then it would have rained yesterday” sounds false to me, but I am not sure about it. The counterfactual requires looking at possible worlds where p (2+2=5) is true, but what if there are no such possible worlds? Does it matter for the theory of counterfactuals whether we consider counterfactuals with necessarily false “if” parts to be true or false?

It is interesting to note, as regards relevance and relevance logic, that Lewis wrote an argument against the

ex falsum sequitur quodlibetas a motivation for relevance logic, but did not touch, as far as I know, this kind of irrelevance that Brian discusses. The “quodlibet”, to use Lewis’s shortcut, is no relevant for relevance because any necessarily false (and necessarily true) proposition has as its subject matter all the possible worlds, therefore everything is already included in that subject matter, or as Lewis says, any subject matter is a mereological part of the subject matter of a contradiction, so anything follows.This is basically a slingshot argument, isn’t it? It’s perhaps made a bit easier by the use of weakening on the right, which lets you “bulk up” the consequent with supplementary detail, and then discharge all but the desired detail, but the whole thing could be run without weakening on the right:

Fa > Gb

Fa > (ix)(x=b ^ Gx) = b

Fa > (ix)(x=b ^ b =/= c) = b

Fa > b =/= c

Fa > (ix)(x=c ^ b=/=x) = c

Fa > (ix)(x=c ^ Hc) = c

Fa > Hc

(supposing Gb Hc, and b=/=c).

But it looks like the plausible reaction here is the same one generally has to slingshots — the required substitution principles look suspect. Think about how one would react to an attempt to run a similar argument on strict conditionals, rather than counterfactuals. One could start with:

(5) I drive from Berlin to Moscow \strictif I drive from Berlin to Moscow.

and then try to amplify to:

(6) I drive from Berlin to Moscow \strictif I drive from the city where Hitler died to the city where Stalin died.

But of course this is no good, because there are worlds in which Berlin is not the city in which Hitler died, so we can’t substitute using “Berlin = the city where Hitler died”. The identity isn’t stable over the relevant region of modal space (here, all of it).

So in the counterfactual case, we also need to know that the identities are stable across the relevant regions of modal space. But knowing that just is, in essence, knowing that C holds. There’s the usual question-begging vs. elucidatory issue still to grapple with, but it feels question-begging to me.