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)

That’s interesting. By the way, what do you mean by the Barcan axioms in this context? Do you mean something like (ExP > #) Ax(P > #)?

By the Barcan axioms, I was just meaning the familiar Barcan axioms for box, not some special axioms for >. The point was that with the Barcan axioms, we enforce a constant domain, and with a constant domain, we don’t need to do messy things like keep referring back to the actual Fs.

I see – I was just wondering whether you were taking box as a new primitive, allowing it to have a non-trivial accessibility relation, or defining it from > (getting you the thing I gave)?

I see now it doesn’t really matter for your purposes, so long as you have a constant domain over each island.

(1) I take it that you mean by “Surprisingly, (L) does not hold given the Limit Assumption” that, surprisingly, (L) does not hold, given the

failureof the Limit Assumption?(2) But that last thing is ambiguous. It could mean:

(a) given the failure of the Limit Assumption it follows that (L) does not hold.

or

(b) it is not the case that, given the failure of the Limit Assumption, (L) must hold (or, (L)

canfail to hold, given that the Limit Assumption fails).(3) What you show is (b). But for (L) to characterize the Limit Assumption, it seems to me you would need (a). Or, you would need: given that (L) holds, the Limit Assumption follows.

I agree with Michael. In fact, it seems that one can easily construct a model where L holds (is true at all worlds, or at the designated world), even though the Limit Assumption fails: suppose all worlds have just one individual b, who instantiates a real-valued magnitude M, with one world for every real number, and w1 is closer to w than w2 iff the M-value at w1 is closer to the M-value at w than the M-value at w2. Any proposition is then represented by a set of real numbers, and I think I’ve worked out that L is satisfied.

BTW, your proof that L holds given the Limit Assumption makes unnecessary use of Conditional Excluded Middle. The negation of ‘at all nearest p-worlds, all actual Fs are G’ is ‘at some nearest p-world, some actual F is G’, not ‘at all nearest p-worlds, some actual F is G’; but the former is enough for the proof.

Er, the last to ‘G’s in my post should be ‘~G’s. Sorry.

And by ‘to, I mean ‘two’. I give up now!

One more note: It looks like the argument used a bit more expressive resources than advertised, since the substitution instance used a “finitely many” quantifier. (I guess you just said “quantifiers”, but I took that to mean first-order quantifiers, which won’t suffice to express finiteness.)

Jeff, right if you think that ‘finitely many’ has to be cashed out quantificationally, then we’ll need more than first-order quantifiers. I was thinking that we could have propositional variables that picked out arbitrary sets of worlds in the model, and p was going to be an unstructured proposition.

(1) I take it that you mean by “Surprisingly, (L) does not hold given the Limit Assumption” that, surprisingly, (L) does not hold, given the failure of the Limit Assumption?Yep – my bad. I’ve corrected this now.

BTW, your proof that L holds given the Limit Assumption makes unnecessary use of Conditional Excluded Middle. The negation of ‘at all nearest p-worlds, all actual Fs are G’ is ‘at some nearest p-world, some actual F is G’, not ‘at all nearest p-worlds, some actual F is G’; but the former is enough for the proof.Agreed. I’ve made a small correction in the text to reflect this.

Michael, Wo

I think I was using ‘characterised’ in a different way. This is probably because I was using it in a non-standard way.

All I meant was that every model that satisfies the Limit Assumption makes (L) true, and that some models that don’t satisfy the limit assumption make (L) false.

My idea, and this could be wrong, was that (L) stood in the something same relationship to the limit assumption as Lp -> p stands to reflexivity. (By L here I mean box, and by reflexivity I mean the R-relation in a standard Kripke model being reflexive.) There are non-reflexive models where Lp -> p holds. Indeed, there are large classes of such models. But without Lp -> p you can’t guarantee reflexivity, and with Lp -> p you can.

Now there is a stronger claim you can make about Lp -> p and reflexivity. For any non-reflexive frame, there is a model on that frame where Lp -> p fails. What would be a good extension of the argument sketched here is a proof that every frame where the limit assumption fails could be extended to a model where (L) fails. And I don’t have a proof of that yet.

Indeed, I’m not entirely sure how one individuates frames in a quantified system. Do facts about identity of objects across worlds count as something that’s fixed by V, or fixed by the frames? I don’t know enough about quantified modal logic to say anything about this.

Jeff, the “Finitely many actual Fs are F” isn’t really a substitution instance of p, but a specification of the interpretation of p in the countermodel. So you don’t have to be able to express it in the language.

Wo’s counterexample seems to be pretty robust with respect to what you want to call a frame.

For these purposes, I assume that frame’s must

at leasttell us what the worlds and similarity relation are like or else it just wouldn’t make sense to say a given frame obeys or doesn’t obey the limit assumption”.But once you have that, the frame (or, any frame, if there’s more than one) consisting of the real numbers as worlds, with the natural closeness relation validates (L).

Andrew,

Right, if we let it be a feature of the frame how many things there are in each world, then there will be Limit Assumption violating worlds in which (L) holds. That’s because (L) will hold automatically in any frame where there are finitely many things in each world. So it will clearly hold given there is just one thing in each world.

I see what you’re saying.

I think you’d also have to stipulate that frames don’t include the designated world either, or else you could have cases where “x is closer to w that y” is a well order when w is the designated world, even though there are w where it’s not a well order.

(e.g. if the set of worlds is omega+1, the ordering is the standard ordering on the ordinals and the designated world is 0. (L) is true, even with an infinite domain, but it clearly violates the limit assumption, since there is no closest world to omega which is finite.)