Wolfgang Schwarz

This has bugged me, too. Here's the thought I've been running with. Propositions come first, and they form a Boolean algebra. So the algebra is isomorphic to a field of sets, whose elements are the algebra's ultrafilters. But in general not *every* set of ultrafilters corresponds to a proposition from the original algebra. (Just the clopen sets in the Stone space.) Moral: not every set of worlds is really a proposition. Some of them are just artifacts of the world-representation. Some of those fake propositions got into your non-principal ultrafilter. How's that sound?

The main alternative is the way you went: start with a set of worlds W, consider propositions to be elements of the algebra 2^W, and then you find that not every ultrafilter is a world. Which ones are? Well, the principal ones. Why exactly do you need to say more?

The general point you're hitting on is that while philosophers often use the duality between Boolean algebras and sets of worlds, the duality is really between Boolean algebras and certain topological spaces. If you abstract to the underlying set, you lose the information about *which* sets correspond to propositions. And it just isn't true that an algebra of propositions generally corresponds to the field of *all* subsets of some set of worlds.

I'm inclined to think that this discrepancy has to do with not having strong enough closure principles on your sets of propositions (as you suggested.)

If S contained the complement of {w} for every w, then its intersection would be empty. So non-principle ultrafilters are not consistently closeable under arbitrary conjunctions. It feels to me like the non-principle ultrafilters are analogous to consistent but omega-inconsistent sets of sentences in the language of arithmetic. While they're consistent in some weak sense, they're not genuinely possible because they entail contradictions via valid (albeit infinitary) rules.

"I suppose this could be done by strengthening the closure condition, e.g. by saying that whenever T contains some propositions, then it also contains the (possibly infinite and uncountable) conjunction of those propositions. Would that work?"

I think this would work. Basically, suppose there was an ultrafilter which was also closed under arbitrary conjunctions. Note that it's principle iff it contains {w} for some w.

If it didn't contain {w} for any w, it would contain the complement of {w} for each w, since it's an ultrafilter. And so the intersection of the ultrafilter would be empty, so the ultrafilter would contain the empty set since it's closed under conjunctions. But since the emptyset entails every set, it entails p and ~p for some p, and so the ultrafilter contains p and ~p.

@Andrew: yes, that's pretty much how I was thinking about this as well!

@Jeff: you're right that if propositions come first, then one could simply exclude some sets of worlds from being propositions. But at what price are we forced to do this? The proposal to strengthen the intersection/closure condition would mean that we don't identify worlds with ultrafilters of the propositional algebra: only ultrafilters meeting the infinite closure condition would count.

Philosophically, this presupposes an understanding of infinitary conjunction or entailment on the level of propositions. That doesn't seem too bad. And then the infinite closure condition on worlds looks very plausible. (From his comment, I take it that Andrew shares this sentiment.)

I'm not so sure about what happens mathematically. Is there something like Stone's representation theorem to the effect that Boolean algebras meeting further conditions of infinite closure are generally isomorphic to full power set algebras?

"I'm not so sure about what happens mathematically. Is there something like Stone's representation theorem to the effect that Boolean algebras meeting further conditions of infinite closure are generally isomorphic to full power set algebras?"

No: there are atomless complete Boolean algebras (e.g. the regular open sets in R^n.) Anything isomorphic to a full power set algebra would be atomic.

However, if you add atomicity to your claim the answer is "yes". Also, if the plan is to start off with worlds, talk about propositions as sets of worlds, and get worlds back from the structure of the propositions you can do that. Let a *complete* ultrafilter be an ultrafilter which is closed under arbitrary conjunctions. Then every set is in one one correspondence with the set of complete ultrafilters over it's power set algebra (by the argument in my last post.)

Yes, that's right. Complete atomic Boolean algebras correspond to sets of their atoms, and obviously no other algebras correspond to power set algebras, as Andrew points out. And also, "complete ultrafilters" in such algebras correspond to atoms.

But I don't understand what problem the complete ultrafilters are solving. If you have a set of worlds, then you'll have a CABA of propositions, the sets of worlds. And if you have a CABA of proposition, then you have worlds—the sets of atoms, which as it happens correspond to complete ultrafilters. But if you have the atoms, why talk about the ultrafilters at all? What the ultrafilters were for, as I saw it, was giving you a way to talk about worlds even when you *don't* know your algebra of propositions is atomic. This is the real case, I think, because I see no antecedent reason to think that it would be atomic. (Completeness considerations don't motivate it, as Andrew also mentioned.) And that's why I wanted to go the way of saying not every set of worlds corresponds to a proposition.

Wo, you asked, "at what price are we forced to do this?" Did you have something in mind? I haven't come across any price I thought was serious, but maybe I'm missing something you're seeing.

I would like to allow both directions: starting with worlds and defining propositions as sets of worlds, or starting with propositions and defining worlds in terms of sets of propositions. (A third option is to define worlds directly as certain kinds of propositions, rather than as theories.) The "price" of saying that some sets of worlds aren't propositions is that the two directions don't line up. If we start with worlds, i.e. maximally specific ways things could be, then every (non-empty) set of worlds is a way things could be -- a way things are if the actual world is a member of the set. That's what a proposition is supposed to be: a way things could be, a cut through logical space. From that perspective, it is incomprehensible why some sets of worlds shouldn't qualify as propositions.

If I understand both of you correctly now, requiring completeness gets rid off the non-principal ultrafilters, but still doesn't ensure that the two directions line up. I agree that an atomicity constraint seems problematic if we start with propositions. Hm.

Oh, I missed a bunch of stuff.

So firstly: I would have thought that the requirement that a world should correspond to an (as it were) "omega-consistent" set of propositions is highly intuitive, and motivated completely independently of the question of whether the algebra of propositions is atomic or not.

Even if propositions are gunky there is something wrong with allowing worlds corresponding to consistent but omega-inconsistent sets of propositions (and in general, "kappa-inconsistent sets".) So I think the existence of non-principal ultrafilters makes a solid case against the project of regaining worlds in a non-atomic setting in terms of ultra-filters. (Which is something I hadn't properly realised until I read this post.)

Secondly: I think the next question is, what happens if we try to reconstruct worlds in a gunky algebra of propositions in terms of *complete* ultrafilters. Assume that the algebra has no atoms (so we're ruling out the case in which it has some atoms and some gunk.) Then, I think, there are *no* complete ultrafilters. I think this fact really puts the final nail in the coffin of the project of regaining worlds from the space of propositions if that space is not atomic.

I'll try and write out the argument later, but note that, in the special case of the regular open subsets of R^n, the conjunction of a set of sets is not their intersection, but the closure of the interior of their intersection. The intersection of a complete ultrafilter in this space will be a singleton set (which is not regular, and so not in the space.) However the closure of it's interior is in the space, and is the empty set. So in the space of regular open sets the conjunction of any set which contains any regular open set or its negation will be the empty-set -- i.e. it will be the inconsistent proposition. So there are no complete ultrafilters.

I think this means that if we take the space of propositions as primitive (as I think we should), and that space is gunky, then we should reject talk of worlds altogether as a façon de parler. (What the Stone representation theorem delivers as "worlds" is in fact a collection of unsatisfiable, but finitely satisfiable, sets of of propositions.)

"I see no antecedent reason to think that [the algebra of propositions] would be atomic."

Here's an argument that there is at least one atom: the set of true propositions is surely not just an ultrafilter but a complete ultrafilter, and therefore its conjunction is an atom.

The alternative is to say that the conjunction of all truths is inconsistent!

This is helpful.

Starting with the last thing. I had thought about an argument like Andrew's a while ago, but I thought it didn't generalize (and also I was in a sceptical mood about infinite conjunction anyway). But now that you bring it up again, I think I was just wrong about that: there is an interesting argument for atomicity. The main idea is to necessitate Andrew's argument.

+ Necessarily, the conjunction of all truths is true.

Or in other words,

+ Necessarily, some true proposition entails every true proposition.

By classical logic, such a proposition would in that case be an atom: Since exactly one of P and ¬P is true, exactly one of them is entailed by the conjunction. (We know the conjunction doesn't entail any false propositions, since it's true!)

And this implies that every consistent proposition is consistent with an atom: If P is possibly true, then whatever is necessarily true is compossible with P, so it is compossible with P that there is a true atomic proposition. I think doing this carefully relies on some principle about the rigidity of entailments, and maybe that propositions exist necessarily, but those aren't crazy thoughts.

Modulo those details, this seems pretty convincing, as long as you like infinite conjunction. At any rate, it doesn't seem likely to me that there's a stable position where you think worlds should be infinitarily consistent, but you *don't* think that the algebra of propositions is atomic. How does this sound so far?

(Another detail to check is that the idea of the conjunction of all truths can be expressed in a way that isn't vulnerable to paradoxes.

Here's a way to say the main thing we want, using propositional quantifiers:

+ Necessarily, for some P (P and for all Q (if Q then (P entails Q)))

So that ought to be ok.)

(I suspect that this kind of argument must be in Kit Fine's work somewhere.)

Ah nice!

"I think doing this carefully relies on some principle about the rigidity of entailments, and maybe that propositions exist necessarily, but those aren't crazy thoughts."

I wonder whether you do need the assumption about the rigidity of entailments. Assuming we're in some modal logic weaker than S5, then I guess there's two things you could mean by entailment in this context: strict implication, [](p->q), and containment between the set of possible and impossible worlds p and q correspond to (i.e. [](p->q)&[][](p->q)&[][][](p->q)...). In the second case you get rigidity pretty much for free, and in the first you can work around the contingency of entailment by necessitating your premise multiple times.

"+ Necessarily, for some P (P and for all Q (if Q then (P entails Q)))"

I don't know if you have the manuscript, but Williamson discusses the above principle in his book (p10 of chapter 5. He calls that "At". He also presents a couple of variants of it.)

I think Fine's appendix to Prior's "Worlds, Times and Selves" would probably be quite relevant; I can't remember if he discusses that exact argument though.

Great! Can I recap what happened, to make sure I get it?

If we have infinitary conjunction for propositions, then non-principal ultrafilters in the algebra of propositions are bad candidates for possible worlds, because the infinite conjunction of their members is inconsistent. So we better identify worlds with "complete" ultrafilters that are closed under infinite conjunction. This wouldn't work if the algebra of propositions were gunky, because then there wouldn't be any complete ultrafilters. Turning this observation around, we can see that the algebra of propositions can't be entirely gunky, because the set of true propositions is surely a complete ultrafilter. Moreover, this isn't an accident. It is plausibly necessary that there is some true proposition (namely the conjunction of all truths) that entails every true proposition. Now take any consistent proposition P. If P were true, then it would still be true that some truth entails all truths. So P is compossible with some such truth (an atom). Then the algebra of propositions is atomic, and we can identify the worlds with the atoms, without even going through the ultrafilter construction. Does that sound roughly correct?

I feel a bit uneasy about using necessity operators here, especially in a context where we quantify over worlds and propositions. But otherwise that does look pretty convincing.

I think Cresswell's "From Modal Discourse to Possible Worlds" is relevant here. I'm fond of the Aristotelian argument in that paper, which is closely related to the "this isn't an accident" argument above.

Hi Wo. Thanks for the summary -- yes, that's pretty much how I was seeing things.

It's clear that the mere combination of S5 and propositionally quantified logic does not entail Jeff's principle. I think this kind of brings out the substantial nature of PW semantics -- it validates principles that do not merely follow from the combination of propositional quantification and propositional modal logic.

A question I'd like to know is: is the principle Jeff stated complete for possible world semantics in the sense that if it were added to the axioms of S5 and propositionally quantified propositional logic it would be complete for the standard possible world semantics (i.e. a set of worlds with an equivalence relation over it, where the propositional quantifiers range over arbitrary subsets of the set of worlds.) My guess is yes.

The question whether it would be complete for arbitrary modal logics weaker or incomparable with S5 will I think depend on the issues Jeff was mentioning about the rigidity of entailment.

David: thanks for the reference!

@Andrew: I think you're right about completeness; Muskens mentions this in his article on Higher-Order Modal Logics in the Handbook of Modal Logic (p.626).

@Dave: Thanks, that does look relevant!