I spent last weekend at the Rutgers Semantics Workshop, which was a great success. Thanks to Angela Harper and Jason Stanley for putting on such a good conference. I learned a few things there, some of which quite surprised me. What I’m going to write about here is something I learned from my colleague-to-soon-be Thony Gillies. (I’m translating a little what Thony said, so if there are errors in what follows, they are my responsibility. I also don’t claim originality for any of this – it’s just something I didn’t know, so I’m recording it here in case other people were similarly ignorant.)
In chapter 6 of Counterfactuals, when he is discussing the relation between semantic constraints on the similarity and accessibility relations, and axioms of the counterfactual logic, Lewis writes
There is no special characteristic axiom corresponding to the Limit Assumption. We can therefore say that if any combination of axioms corresponds to a combination of conditions without the Limit Assumption, then then same combination of axioms corresponds also to that combination of conditions with the Limit Assumption added.
Lewis is here talking about characteristic axioms for theories of comparative possibility. But since counterfactual conditionals can be defined in terms of comparative possibility, the result quickly generalises to the counterfactual conditionals. What I never knew was that this result turns crucially on the expressive limitations of the logics Lewis is working in. If we have quantifiers, and an actuality operator, there is an axiom that corresponds to the Limit Assumption. It’s a little hard to state in HTML, but I’ll try. I’ll use > for the counterfactual conditional, -> for the material implication, A for the universal quantifier, and $ for the actuality operator. Then consider the following axiom schema, where F and G are any predicates, and p is a sentence in which x does not occur free.
(L) $(Ax(Fx -> (p > Gx)) -> (p > Ax($Fx -> Gx)))
That is, if it’s actually the case that for any F, it would be G if p were true, then if p were true, all the actual Fs would be G. That sounds pretty plausible to me. Given the limit assumption, I believe it holds fairly trivially. (Assume it doesn’t hold. Then in the actual world the antecedent is true and the consequent false. So if p were true there would be some object that isn’t G despite being actually F. (Wo points out in comment 5 that I should say could here, not would.) By the Limit Assumption, that means that in some of the nearest p-worlds there are some actual Fs that are not Gs. Call some such object b. That means in the actual world, Fb, but it isn’t true in all the nearest p-worlds that Gb. So p > Gb is not actually true. That contradicts Ax(Fx -> (p > Gx)), which we assumed is actually true.) Surprisingly, (L) does not hold given the failure of the Limit Assumption. (Italicised words added due to correction in comment 4.)
Consider a model with a countable infinity of Fs in the actual world, with these Fs numbered 1, 2, 3, etc. And set the similarity ordering to be as follows. For two worlds w1, w2 that differ on the distribution of F-ness, w1 is more similar to actuality than w2 iff the lowest number individual that is F in just one of these two worlds is F in w1, and not F in w2. Now consider a substitution instance of (L) with G = F, and p is Finitely many actual Fs are F.
The first thing to note is that the Limit Assumption fails. For any world where exactly n of the actual Fs are F, the world where the first n+1 actual Fs are F is closer. So for any world that satisfies p, there is a closer world.
Let c be any number, and let o be the c’th actual F. Then consider the world where the first c actual Fs are F, and no other actual Fs are F. That’s a world where p & Fo is true obviously. Moreover, any closer world to actuality must still have Fo true, since the only way we can get closer to actuality is to make some later actual Fs back into Fs. So on Lewis’s semantics for counterfactuals, p > Fo is actually true. Since o was an arbitrary actual F, it follows that Ax(Fx -> (p > Gx)) is actually true.
But clearly it isn’t true that p > Ax($Fx -> Fx). Indeed, p is inconsistent with Ax($Fx -> Fx), since there are infinitely many actual Fs. So (L) fails in this model. Since (L) is true in all models where the Limit Assumption holds, it follows I think that (L) is a nice way to characterise the Limit Assumption.
If we have the Barcan and converse Barcan axioms in our logic, we can simplify (L) considerably. In that case, the axiom (LB) characterises the Limit Assumption, though I won’t prove this.
(LB) Ax(p > Fx) -> (p > AxFx)