For a long time I have wondered, with an uneasy feeling that there was something I couldn't see, about the relationship between two-dimensional semantics and my approach to analysing subjunctive necessity *de dicto*. As I flagged in the previous post, this has become even more urgent in light of my new, relational account involving the notion of a counterfactual invariance (CI) decider.

I think I've finally made a breakthrough here, and found a clear connection. There is more to say, but here it is briefly.

Recall that my account states that a proposition is necessary (i.e. necessarily true or necessarily false) iff it has a true positive counterfactual invariance (CI) decider.

(P is a positive counterfactual invariance decider for Q iff Q does not vary across genuine counterfactual scenario descriptions for which P is held true.)

A close analogue of this account can be stated in terms of two-dimensional semantics: a proposition Q is necessary iff there is a true proposition P such that for every scenario S in which P is true, Q's two-dimensional intension maps, for all W, <S, W> to the same truth-value.

And I think I can maintain, as CI deciderhood is plausibly *a priori *tractable and arguably a semantic matter, so too is the question whether, given some propositions P and Q, P is such that for every scenario S in which P is true, Q's two-dimensional intension maps, for all W, <S, W> to the same truth-value.

This makes clear one major way in which my analysis goes beyond the normal two-dimensional account of subjunctive necessity in terms of secondary (or C) intension - and this way can then be translated into two-dimensional terms. And looking at necessity this way, as opposed to with just the usual two-dimensional account of subjunctive necessity, gives us a finer grained picture of the role played by what Kripke called '*a priori* philosophical analysis' in our knowledge of necessity. You don't have to know which scenario is actual to know that a proposition is necessary - you just need to know that you're in one of some range of scenarios such that, if they were actual, the proposition would be necessary. And such a range can be characterized by a proposition which you can know *a priori* to be a CI decider for the necessary proposition in question.
Last Thursday I gave a talk at Sydney University's philosophy department about Kipper's bombshell, my old account of necessity, and my new account involving counterfactual invariance deciders. I was asked many good questions and got a lot out of it.

In preparing the talk, I came to realise that I may have been too quick to assume that 'Air is airy' disproves my old account, according to which a proposition is necessarily true iff it is in the deductive closure of the set of propositions which are both true and inherently counterfactually invariant. Because 'There is nothing more to being air than being airy' is plausibly true and ICI, and it does - at least on a rich enough notion of impication - imply 'Air is airy'.

Now, if that's right, what follows? Are my new ideas about abandoning, in the analysis of necessity, the property of ICI for a relation of deciderhood, to be thrown out? I don't think so. Even if I was pushed towards them by the possibly wrong idea that my old account can't be defended from 'Air is airy', they still seem to give us an account which seems better. The old account now seems clumsy, so to speak. Maybe it can be understood in a way - with a rich notion of implication - so that it doesn't go wrong on 'Air is airy'. But this still seems like a kind of lucky break, and it's not clear to me that there aren't more threatening examples in the offing. The new account, on which a proposition is necessary iff it has a true positive counterfactual invariance decider, seems to reveal the notion's workings more faithfully, and seems less hostage to as-yet-unconsidered examples.

(Also note that, with the new account, you *can* use 'There's nothing more to being air than being airy' as your decider, but it seems like you can also use something like 'Air has no underlying nature' or 'Air is not a natural kind', and these do *not* seem to imply 'Air is airy' - they do not seem to contain that information. And since it seems you can plug *these* into the new account and conclude that 'Air is airy' is necessary, but cannot conclude the same on the same basis with the old account, that the new account is superior here, in enabling us to conclude necessity on a sometimes slenderer basis than we can using the old account.)

In the talk I gave, there were a number of questions and examples suggested which could look like they may disprove my account, but I was able to respond to all of them straightforwardly and to my account's credit. (With some elements of the new account, it's hard to see immediately why they're there and are as they are, but working through some examples clarifies things.) I also fielded a question (thanks to N.J.J. Smith) about how my account goes beyond what we already find in Kripke. There too I was able to give what I think is a satisfactory answer: the account isolates a plausibly *a priori* tractable, maybe broadly semantic, aspect to necessity. Kripke's work doesn't do this. He says a proposition is necessary if it holds in all the ways things could have been, and one of his main points is that we don't in general know *a priori* what these ways are. True, he also allows that we know by '*a priori* philosophical analysis' (this occurs in 'Identity and Necessity') that 'Hesperus is Phosphorus' is necessarily true if true at all, but that isn't true of all examples. You might thus wonder, with respect to examples that *don't* work that way, what part '*a priori *philosophical analysis' might play in our knowledge of *their* modal status. My account gives us an answer to this.

But another sort of question arose in the talk was how my account relates to two-dimensional semantics, and I was less satisfied with what I had to say on that. The true CI deciding proposition(s) in my account seem to play a role close to the role played by what world is actual in two-dimensional semantics. I worry that some in the audience were beginning to suspect that I've just laboriously re-arrived at two-dimensionalism along a somewhat different path. (And I'm getting a bit suspicious myself.)

So, I think that now, the most pressing task is to clarify the relationship of my new account to two-dimensional semantics, rather than to defend it further from counterexample. (This has always been a background concern, even with my old account, but now it has become urgent.) The notions in my account come up in a different way, and most formulations of two-dimensionalism seem to bring up difficulties which I may be able to avoid. My account seems more minimal and focused on its topic, and thus potentially more instructive.

Such anyway is my hunch, but it remains to make this clear.
*I wrote the following dialogue as an antidote to the dogmatism I felt myself falling into when trying to write a paper about *a priori* propositions*. *The characters A and B are present-day analytic philosophers. Roughly, A represents the part of me which wanted to write the paper I was working on, and B represents the part which made trouble for the project.*

A: I've got a view about *a priori* propositions I'd like to discuss with you. I don't think you're going to like it.

B: Intriguing! I'll try to put up a good fight.

A: Good. Still, you won't just defend the opposite view no matter what, will you? I'm certainly going into this ready to modify my view, if not to completely relinquish it.

B: Sure. No, I won't just set myself up as an opponent debater. Let's try to give each other as much ground as our philosophical consciences allow, and see if we can agree on some things.

A: OK, great. So, here's the view: what is special about *a priori *propositions, which enables them to be known independently of experience, is that they have their truth values essentially. They do not reach outside themselves to get their truth values, but carry them within as part of their nature.

B: OK. Interesting use of the notion of essence. I'm used to associating views which tie *a priori* propositions' truth or falsity closely to meaning with more deflationary attitudes, not with philosophers who make positive use of metaphysical notions like that of essence.

A: Exactly. That's one of the exciting things about my view, I think. It brings out the fact that that sort of tight connection between meaning and truth value can be posited without embracing any problematic conventionalist or deflationary attitudes about essence or meaning.

B: I think you have a point there. A meaning-based view of *a priori* truth doesn't need to be deflationary or conventionalist. Still, I think it's wrong. Your view overlooks the fact that *a priori* propositions, or many of them at least, are about something, and we often have to inquire into that something to know them. When mathematicians discover new truths, they don't sit and try to get insight into the essences of the propositions they are wondering about. They try to get insight into the things that the propositions are about, like numbers, or sets, or graphs.

A: That is true, but does not affect what I am saying. Look, the *a priori *truths of mathematics either have their truth essentially, or accidentally. And if they really had to reach outside themselves for their truth, then they would only be true accidentally. And in that case it should be possible to depict those very propositions reaching out but getting the opposite truth value. But you can't even begin to imagine a situation where someone has expressed what is actually an *a priori *truth, but which in that situation is a false proposition. And it's not like the case of propositions whose instantiation vouchsafes their truth, like 'Language exists'. Instead, their truth is of their very essence. Now, we all agree that an *a priori* truth can have its actual truth value, but what would it look like for it to have the other one? The onus is on you to flesh out an answer here, and it seems to me that nothing you could say on this point would satisfy.

B: I do not dispute that I couldn't really flesh out a description of a situation where the same *a priori* proposition gets the opposite truth value, but I don't think I have to be able to. I can still maintain that these *a priori* truths do not have their truth off their own bat, due to meaning alone. The source of their truth lies in what they are about. *However*, unlike with empirical truths, what they are about is rigid and unmoving - necessarily the way it is. So it is no real objection that I cannot depict a situation in which their source of truth or falsity yields them a different truth value, since that is just because their source is necessarily the way it is. That doesn't make their source any less of a source.

A: So you are saying that the meanings of these *a priori *propositions are out there in a rigid, unmoving space of possible meanings, and that they get their truth or falsity from an equally rigid, unmoving space of mathematical objects. But since all this stuff is rigid, unmoving, and necessarily the way it is, it seems to me that your talk of sourcing is just empty talk. The very idea of sourcing seems dubious here. Granted, you may *seem* to have an advantage in the fact that our *knowledge* of these truths must have some source. But the sourcing you are talking about is all going on in Plato's Heaven. It does nothing to explain how we get the knowledge. So you might as well not posit it.

B: You are trying to cast aspersions on my talk of sourcing, but I want to suggest that what you are saying is, on examination, more dubious than what I am saying. You are no nominalist, no denier of the independent existence of mathematical objects. Right?

A: Sure. I mean, I think when people object to claims like 'Mathematical objects exist independently', they are perhaps bothered by something that really should bother them. But I do think that understood properly, such claims do make a sound and correct point.

B: OK, fine. And so, it seems to me that if you are saying that *a priori* truths about these objects have their truth essentially and off their own bat, you are positing a kind of *harmony* between the meanings and what they carry inside them on the one hand, and the mathematical objects on the other. But this harmony seems dubious. It cries out for explanation. Why should it exist? Coming around to the proper view, that the propositions are about the mathematical objects, and therefore the mathematical objects' being the way they are is the source of these propositions' truth values, the difficulty disappears.

A: I don't see how the harmony you complain about is particularly strange or objectionable. Don't parts of mathematics mirror and reflect each other in weird and wonderful ways? Since we accept that, it seems that it's not particularly costly to acknowledge that the meanings of mathematical truths are also part of this crystalline structure. Crucially, it seems less dubious than your sourcing talk - more of a piece with things we already acknowledge. And it seems to me that your view overdoes the analogy between mathematical and empirical truths, leading to confusion.

B: Do you see any positive value in your view? Or is it all about stopping that over-assimilation?

A: Well, perhaps my view helps with the problem of how we get mathematical knowledge. It seems to me an easier problem to say how we get in touch with meanings, than to say how we get in touch with things like numbers and sets. Our talk and thought *instantiates* meanings, I want to say, even if the meanings themselves are abstract, like numbers and sets.

B: But there are also "instantiation relationships", arguably more straightforward, between, say, numbers and piles of apples.

A: Hmm. Well, I don't know, I'll have to think more about that - but perhaps stopping the over-assimilation is enough. What value do you see in your view, anyway?

B: When I think about what is fundamentally wrong with your view, apart from my complaints about it being mysterious and ill-motivated, it seems to me that, in your effort to block the over-assimilation of mathematical and empirical propositions, you bring about *another* over-assimilation. Namely, between mathematical propositions which can be hard to discover the truth about, and what you might call paradigmatically analytic propositions - propositions where it really does seem that the way to know the truth about them is just to have insight into their meanings. *Those* propositions may perhaps be said to have their truth values essentially, since they don't seem to say anything substantial about anything, whether their subject matter be empirical or mathematical. And your view wrongly depicts substantial mathematical propositions as being like them. My view has the virtue of avoiding that over-assimilation. It may be that the over-assimilation you worry about is also a problem, but it should be combated in a different way.

A: Well, I am - or at least have been, up to having this conversation - inclined to think the corresponding thing about the over-assimilation that *you* are worried about. Positing a mysterious sourcing relationship between mathematical propositions and mathematical objects seems like a crude expedient. But I must acknowledge that the over-assimilation that bothers you is also a problem.

B: OK. So, it seems we can both agree that our respective views may have some power to prevent a certain over-assimilation, a different one in each case. And perhaps we can also agree that each of our respective views, when adopted, may increase the danger of falling into the over-assimilation targeted by the opposite view.

A: Hmm. I suppose we can both agree about that.

B: Now, isn't this worrying? I mean, where does it leave us? We have a question: Do mathematical propositions have their truth values essentially, intrinsically, inherently, off their own bat - or do they not? And it seems like our opposing answers have opposing strengths and opposing weaknesses. I feel the weakness of your view much more acutely, but I can't deny that your feeling that my view might be a somewhat crude expedient makes some sense as well.

A: I'm glad you're staying true to your intention of not just defending your view tooth and nail. Now it's starting to look like both our views have some merit, but that these merits crowd each other out. I am beginning to think that perhaps both our views can be said to suffer from crudeness on that score. We are both inclined to use a certain picture to ward off the over-assimilation which has most bothered us. And the pictures conflict, or at least seem to. Now, could it be that if our views were made clearer, these pictures could be seen to apply in different ways, so that there is no inconsistency in using one in its way, and the other in *its* way? The task then would be to clarify the difference between these two ways of using what appear to be conflicting pictures.

B: That is sounding more and more reasonable to me as a diagnosis of what's going on in this case. How Wittgensteinian! And to be honest, the Wittgensteinian-ness of this view worries me a bit, since this sort of approach, to this sort of problem, seems like it will turn many people off right away. If we are to try to resolve our difficulties this way, and if we expect the resolution to be given a fair hearing, I suppose we will also have to be careful to defend our resolution from objections which lump it together with features of Wittgenstein's views which people don't like.

A: I agree that is a worry. And it may be even worse than you are suggesting. What if the things people don't like and have turned their back on include this very power to resolve our difficulties!

B: Well, I see what you're saying. People are invested in a certain way of doing things, and in defending views of a certain type. And those ways of doing things may come naturally, at least to people with a certain background (including us), so that one slides back into them. But I think we may just have to try to give the naysayers about this method plenty of credit, and allow that there are serious problems with the sort of resolution we're talking about now. After all, why wouldn't there be? It could be that it's very promising, and still ultimately our best hope, but that there are serious difficulties with it which, in our desire to resolve our present issue, we aren't currently alive to.

A: I suppose I'm on board with what you're saying. As exciting and powerful as this approach may seem now, we must beware of coming off as if we think there's a silver bullet, a simple solution we've already got here. And I think that comes out more clearly when we come back from talking about pictures and consider the question, framed in terms of 'essence' or 'intrinsic' or what have you. Something about the idea of pictures makes us quite willing to allow different applications. Ambiguities, if you like. But it seems as though people, ourselves included, may be inclined to take a certain attitude to words like 'essence' and 'intrinsic', such that the word analogue of the move where we say 'These pictures appear to conflict, but if you look at their application, you see it's only an apparent conflict' seems less appealing. There is a feeling that with such words that for each there is a big, important, single job that they should be doing.

B: I think you're right. But again, I think you may be overplaying people's resistance. Yes, there will be people who just get turned off at the suggestion that such words should be understood as having various quite important roles to play. But probably, with many of the sort of people you have in mind, you must admit that they *are *willing to countenance such things as long as you keep things relatively clear and definite. I mean, if you start banging on about how complex and multifaceted it all is with these words, then yes, that will turn people off, because it sounds defeatist. It sounds like shirking hard and maybe very interesting work. But these sorts of people - and let's face it we're among them a lot of the time when we aren't just talking but trying to write papers - are quite willing to distinguish certain senses of weighty-seeming words, using little subscripts for example. So we shouldn't be too discouraged.

A: Yes, I suppose that's right. So, we should be ready to float the idea that our different pictures each having a role to play, but that just giving the picture and saying 'That's how things are' is a bit crude until we clarify and distinguish the application of the picture in each case. And we should be ready to try to take exactly this approach when it comes to our difficulties as posed in philosophical jargon, but be on guard against defeatist or wishy-washy sounding attitudes. I confess I'm worried about the extent to which this is possible. I mean, maybe once we try, we will find that the distinctions we might want to make by putting little subscripts on words like 'essence' tend to fall apart in our hands, or that possibilities multiply very quickly. But on the other hand, I must admit we haven't seriously tried yet. And maybe there is some progress to be made in that way, even if it does give out and get confusing again in a way similar to our original disagreement. So we should keep working on this.

B: Agreed.

A: I think I'm pretty worn out for now, though. And I suspect there are further problems with your view that I haven't brought out.

B: Same here, on both counts.

A: I hope we can find what it takes to continue soon.

B: So do I.
*The previous posts were quite raw and had me wrestling with new data. In this post, I try to be clearer and more accessible, and give a first outline of a new account of necessity that has emerged from my research on these topics. *

My old account of necessity was:

A proposition is necessarily true iff it is, or is implied by, a proposition which is both inherently counterfactually invariant (ICI) and true.

A proposition P is ICI iff P's negation does not appear in any (genuine) counterfactual scenario description for which P is held true.

(I.e. if you hold P true, then you won't produce (genuine) CSDs in that capacity (of holding P to be true) according to which not-P.)

(You might wonder about what exactly a CSD is and what it takes for one to be genuine, but this will not be our focus here.)

This account nicely handles an example like 'Hesperus is Phosphorus or my hat is on the table'. This proposition isn't itself ICI - after all, you can hold it true by holding it true that my hat is on the table but Hesperus is not Phosphorus, and in that case you'd be prepared to produce CSDs in which it's false. But it *is* implied by a true ICI proposition, namely 'Hesperus is Phosphorus'.

The account also handles more complicated cases where there is no *component *ICI proposition (as there happens to be in the last example). It is enough that a true ICI proposition *implies *the necessary truth we are interested in.

But this account recently fell, due initially to an example from Jens Kipper (discussed in recent posts here). The example is 'Air is airy'. The point of this sentence is that it denotes something which isn't a natural kind - i.e. has no particular underlying nature - and predicates of it its superficial properties. Since, as it turns out, air isn't a natural kind, 'Air is airy' is necessarily true; there couldn't have been non-airy air, since, as it turns out, what is is to be air is just to be airy. If on the other hand air had turned out to have an underlying nature, like water does, we would regard 'Air is airy' as contingent, like we do 'Water is watery'; there could have been non-watery water, i.e. H20 in a situation where it isn't watery.

The problem for my old account is that 'Air is airy' is necessarily true, but it is neither ICI nor is it implied by an ICI true proposition.

(After the Kipper example, I have also come upon an example due to Strohminger and Yli-Vakkuri: 'Dylan is at least as tall as Zimmerman'. Since Dylan is Zimmerman, this is necessary. But it isn't ICI, since you could hold it true while holding that Dylan and Zimmerman are distinct. With this example, you could try to save my account by maintaining that - in a rich sense of 'implies' - this troublesome example is implied by 'Dylan is Zimmerman' (which is true and ICI), so my account gives the right answer after all, provided we have the rich sense of 'implies' on board. But I see little point in this, as this trick doesn't help with 'Air is airy'.)

What I think all this shows is that, in our analysis of necessity, we need, not the notion of implication, but more specialised relevant relationships between propositions. In particular, we need to consider when the truth of a proposition P would make a proposition Q necessary. Or, for a more penetrating analysis, when P would make Q counterfactually invariant.

Let's say that P is a *positive counterfactual invariance decider *for Q iff Q does not vary across genuine CSDs for which P is held true.

(A proposition P *varies* across a bunch of CSDs iff it is true according to some of them but not according to others.)

So, for example, 'Hesperus is Phosphorus' is its own positive CI decider; if you hold it true, then, in that capacity of holding it true, you won't produce any genuine CSDs according to which Hesperus is *not* Phosphorus. ('Hesperus is not Phosphorus' is also a positive CI decider for 'Hesperus is Phosphorus', although it happens to not be true.) But really, these are vacuous cases; since 'Hesperus is Phosphorus' and 'Hesperus is not Phosphorus' are *inherently* counterfactually invariant, *any* proposition you like counts (on the above definitions, which may not be optimal) as a positive CI decider for these.

(UPDATE 26/10/2017: This last claim is false. Some random proposition like 'Snow is white' actually doesn't count as a positive CI decider for 'Hesperus is Phosphorus', since you might hold it true but not hold 'Hesperus is Phosphorus' true. If CI deciderhood had been defined by referring to the genuine CSDs in which P (the potential decider) *and* Q (the potentially decided) are held true, rather than just P. )

The notion comes into its own with non-ICI propositions:

'Hesperus is Phosphorus' a positive CI decider for 'Hesperus is Phosphorus or my hat is on the table'; if you hold the former true, you won't let the latter vary across CSDs.

'Hesperus is not Phosphorus' is a negative CI decider for 'Hesperus is Phosphorus or my hat is on the table'; if you hold the former true, you will let the latter vary across CSDs (depending on whether my hat is on the table or not in the scenarios being described).

'Hesperus is Phosphorus' is a negative CI decider for 'Hesperus is not Phosphorus or my hat is on the table', and 'Hesperus is not Phosphorus' is a positive CI decider for 'Hesperus is not Phosphorus or my hat is on the table'.

Furthermore, this apparatus gives us good things to say about Kipper's counterexample to my old account:

'Air is not a natural kind' is a positive CI decider for 'Air is airy'; if you hold the former true, then the latter won't vary across CSDs.

(UPDATE 26/10/17: This last claim may be faulty, because you could perhaps hold 'Air is not a natural kind' true and also hold 'Air is not airy' true. This depends on how 'airy' is defined - is it part of its meaning that to be airy is to have the actual properties that *air *has, whatever those are? Then maybe you couldn't really coherently hold it false. But if it's defined in terms of a list of properties that we think air has, then you could get all skeptical the way Kripke does with cats and hold it true that air actually doesn't have these properties and it's some elaborate ruse which makes us think it does. This could be gotten around by narrowing our attention in the definition of positive CI deciderhood to genuine CSDs where, not just P (the potential decider) is held true, but also Q (the potentially decided). However, I don't think that this is required to save the analysis as formulated in this post, since we can just give up on 'Air is not a natural kind' itself being a CI decider (at least all by itself) of 'Air is airy' and instead appeal to 'All there is to being air is to be airy' or even just 'Air is airy and is not a natural kind'.)

(Likewise for the Strohminger/Yli-Vakkuri example: 'Dylan is Zimmerman' is a positive CI decider for 'Dylan is at least as tall as Zimmerman'.)

I think a good account of necessity can now be given as follows:

A proposition is necessary (i.e. necessarily true or necessarily false) iff it has a true positive CI decider.

Note: it seems plausible that CI deciderhood is an *a priori* tractable matter; whether some P is a CI decider for some Q, and if so whether it is a positive or a negative decider, seem to be the sort of thing we can work out *a priori*. What we might not be able to know *a priori* is the truth-values of P and Q.

I will keep working on the best way to present this sort of approach, but I think the essentials are now in place.
*This post contains my efforts to fix my account of subjunctive necessity *de dicto* in the wake of troublesome examples from Kipper, Strohminger, and Yli-Vakkuri. Further work is needed before I can give a good, freestanding presentation of the new account that seems to be emerging. Still, I have hopes that here I have finally found an approach capable of yielding a true analysis of this notion. Note that the talk of the 'link' refers to this less ambitious project which shares some features with my project of giving an analysis of subjunctive necessity *de dicto*.*

The Strohminger/Yli-Vakkuri example seems more straightforward as a counterexample when it comes to Casullo’s proposal, but maybe less so than Kipper’s original ‘Air is airy’ when it comes to my link and (more importantly) my account, which appeal to implication.

The reason for that is that you can argue that there is a good sense of ‘implication’ on which:

‘Bob Dylan is Robert Zimmerman’

implies

‘Bob Dylan is at least as tall as Robert Zimmerman’

And then, if you wanted to defend my link or my account from this sort of counterexample alone, it would be enough to make a good case that there is such an implication relation, and do an adequate job of characterising this relation or conveying the idea of it.

And the idea is that this implication relation would be just as must-ish, just as free from non-a priori elements, as a more austere formal notion of implication where only subject-neutral terms are allowed to play a role in making the implication hold.

But, would that move work to defend my link and account from the Kipper example ‘Air is airy?’. If not, we shouldn’t bother, and should instead look for a fix which handles both sorts of examples as neatly as possible.

Another desideratum for a fix would be to retain an isolable a priori element, or elements (as ICI and implication are supposed to be in my original account).

OK, so that is a reasonable basis on which to proceed. We know roughly what we want and have a couple of ideas for how to perhaps modify the link and account. Now, to consider whether a parallel move - parallel to claiming that ‘Dylan is Zimmerman’ implies, in the relevant sense, ‘Dylan at least as tall as Zimmerman’ - could be made with ‘Air is airy’.

So what would the argued implier be?

‘Air is not a natural kind’?

‘Air is heterogenous’?

‘Air has no particular underlying nature’?

These suggestions all provide the missing piece of information, which you would need to conclude that ‘Air is airy’ is necessary. But it seems weird to say that they imply ‘Air is airy’ in any natural sense. On the way of thinking behind Kipper’s example (which I propose to just work with as much as possible, since I want a way of defending my account or a successor to it which does not rely on rejecting that way of thinking), ‘Air is airy’ is a priori in any case, and is a sort of allusion to an unpacking of ‘air’ (in 2D terms, its A-intension). All the extra information, that air isn’t a natural kind or isn’t homogenous or whatever, gives us is that this thing, which we could already know to be a priori, is necessary, since we aren’t fixing on some underlying nature in our use of ‘air’ in counterfactual scenario descriptions.

Now, there is a feeling that, when you don’t know whether air is a natural kind or not, you’re as it were in a superposition between two notions, not quite having either. Following this idea, you might wonder whether ‘Air is airy’ might not be ICI after all, in the way we understand it once we know that air isn’t a natural kind. It’s just that, before we knew that, we used ‘Air’ with an incomplete or a not-completely-determinate meaning, such that on one filling in - that which we would get if air turned out to be a natural kind after all - ‘Air is airy’ comes out contingent, and on the other - the (presumably) actual one - the same sentence comes out necessary.

I am inclined to think that there is some sort of internally consistent position to be had that way, but the worry is perhaps that then the position ends up being about things which aren’t normally the things we are most interested in, or that it ends up employing unusual concepts, e.g. an unusual conception of ‘proposition’ which individuates them in a way finer than comes naturally (e.g. so that ‘Air is airy’ is a different proposition depending on whether we believe air is a natural kind or not). And then we might also get weird results, such as that ‘Air is a natural kind’ is *a priori* (as we mean it once we know it!).

So, what I am more interested in is sticking with a way of thinking which allows that we already have the proposition ‘Air is airy’ on board, fully-fledged, before we know whether air is a natural kind, and that discovering that it isn’t gives us a partly a posteriori basis for our knowledge that it’s necessary. I want an account of necessity which allows for this, if possible.

Still, and especially when combined with my doctrine of flexible granularity, it is nice to know that - as a last resort - I can defend my account more or less as it stands by insisting on “rich” implication (to handle the Dylan/Zimmerman example) and by holding that the account works with the Kipper stuff provided a granularity fine enough to distinguish ‘Air’ before and after the discovery. Likewise, Strohminger & Yli-Vakkuri have problems with the Kipper case, problems which I could try to go along with. But none of this is satisfying.

One key fact which seems highly relevant is this: while it seems false that, however you hold ‘Air is airy’ true, you will hold it fixed across CSDs, it seems true that if you hold it true by, or as well as, holding true the truth that air isn’t a natural kind, then you’ll hold it fixed.

And it feels relevant that that auxiliary thing you need to hold true is true.

And this phenomenon of being able to hold these things true in different ways, and also in a more agnostic way, matches the disjunctive examples that motivated the implication clause (e.g. you can hold ‘Hesperus is Phosphorus or my hat is on the table’ true without holding any particular disjunct true (agnostic case), or by holding the first disjunct true (as well as the second if you like) (ICI case), or by holding the first disjunct false and the second true (non-ICI case).

So it is possible that by accommodating this stuff in a different way, the implication clause would become superfluous. Which would be good, as otherwise excessive complexity threatens.

As a rough first pass for that strategy, if we try to synthesize a concept covering the underlying things that you can hold true when you hold one of these tricky cases true in a non-agnostic way, we could use that. But note that it seems wrong to think of them in general as ‘supporting’ propositions, I think (which might otherwise have been a clue to how to characterize the notion we need). That sounds right for disjunctive cases and the Dylan/Zimmerman case, but not for the ‘Air is airy’ case. That doesn’t need support, I feel like saying. You can already fully know, off your own bat, that air is airy, while being agnostic about its natural kind status.

The concept we need could be expressed with the phrase ‘counterfactual-invariance decider’. Since your holding these things true decides whether the proposition whose necessity is in question is counterfactually invariant for you or not. E.g. ‘Air is not a natural kind’ is a (positive) CI decider for ‘Air is airy’. ‘Hesperus is Phosphorus’ is a (positive) CI decider for ‘Hesperus is Phosphorus or my hat is on the table’ (and ‘Hesperus is not Phosphorus is a negative CI decider for it).

So, when we’re in ‘agnostic’ mode with respect to P we’re not holding true any of its CI deciders.

This feels like good progress! So, maybe all the stuff about implication has been a dead end. What’s really going on is that we need to zero in on true CI deciders, and it just so happened in the disjunctive cases (and arguably in the Dylan/Zimmerman case given rich implication) that these implied the propositions in question. And there’s something nice about that, since we’ve seen Casullo make an analogous mistake: he thought component propositions were the key to the linking problem, and I thought I was very clever to “realise” that it’s really implication. Now I’m thinking that may be wrong too, and the real, deep thing is CI-decidership. And it feels like this time this won’t turn out wrong. The notion is tailor-made!

So now, I think I can have a true account of necessity running something like:

A proposition P is necessary iff it doesn’t vary across CSDs for which a true CI decider of P is held true.

Or perhaps we should simply say:

A proposition is necessary iff it has a true positive CI decider.

(Note that this is now, unlike my old account which purported to give necessary and sufficient conditions for necessary truth, an account of necessity (as in necessary truth or falsity). Note also that we can hold that propositions are often CI deciders for themselves - if positive, then they’re ICI. But ICI itself might no longer play a role in the analysis. There’s still a plausibly a priori element here too: given propositions P and Q (not necessarily distinct) it is a priori whether P is a CI decider of Q and, if it is, how it decides the matter. It’s also plausibly a priori whether a given proposition has a positive CI decider. What isn’t a priori in general is whether these propositions and their deciders are true.)

I will try to investigate this further, and work on ways of clearly conveying and explaining the idea of a CI decider, and packaging the account as a whole. But I have a good feeling about it.

*This post is quite compressed and relies on things explained in the previous posts on the task of linking necessity to apriority, as well as alluding to my account of necessity as expressed in my PhD thesis. In a future post, I intend to explain and explore what these developments mean for my account of necessity.*

There has recently appeared an unpublished manuscript on PhilPapers (PDF available here at time of writing) which contains even stronger counterexamples to both Casullo's and my proposed link between necessity and apriority. It is by Margot Strohminger and Yuhani Yli-Vakkuri.

Strohminger & Yli-Vakkuri argue that Kipper's examples are contentious, relying on dubitable assumptions about natural kind terms and perhaps even embracing what they call 'Chalmersian two-dimensionalist ideology'. They provide even simpler examples of propositions whose general modal status cannot be known *a priori* (and, relevantly for me, these examples also don't seem to be *implied* by propositions whose general modal status can be known *a priori*). For example:
Bob Dylan is at least as tall as Robert Zimmerman.

This is necessary, since Bob Dylan *is *Robert Zimmerman. But for all we can know *a priori*, Dylan and Zimmerman are distinct, in which case this proposition would not be necessary, but contingent.

But hold on a minute! My link appeals to implication, and I said that the example above isn't implied by a proposition whose general modal status is knowable *a priori*. But can't we say that it is implied by 'Bob Dylan is Robert Zimmerman', which we can know *a priori* to be necessary? Yes, we can - although here we need a notion of implication which takes into account the meaning of 'is at least as tall as' - or at least the fact that it's a certain kind of comparative expression - rather than just the meanings of subject-neutral particles like 'or', 'all' and 'some'. So, from the point of view of disproving Casullo's proposed link, this example may be the best available so, but from the point of view of disproving my proposed implication-involving link, Kipper's natural kind examples may still have an edge.

It seems to be a very exciting time to be thinking about these issues! So far, in this post and the last, I've been talking about how these examples affect my proposed link between necessity and apriority. But the situation is more serious than that for me. The centrepiece of my PhD thesis was an account of the conditions under which a proposition is necessarily true (I've blogged about this account quite a bit here). And these developments, as far as I can tell, may well show that account to be false. This is very momentous for me, as I worked on that account for several years and considered it to be maybe my best bit of work.

I can't believe I didn't think of the example above in connection with my account! I even considered a very similar example when making a side point about using my notion of a genuine counterfactual scenario description (used in my account of necessity) to arrive at a definition of rigid designation which is in some ways more fundamental than the Kripkean one.

Stay tuned for more on whether and how these developments affect my account of necessity, and what can be done about it if they do.
In a recent post (and an article I am working on), I arrived at the view that if a proposition can be known to be necessary (i.e. necessarily true or false) then either it or its negation is in the deductive closure of a set of true propositions with *a priori *necessary character - i.e. propositions which are such that it can be known *a priori* that they are necessary.

There is a new article by Jens Kipper forthcoming in *Analysis*, 'On what is apriori about necessities', which seems to make serious trouble for this view (as well as its ancestors). Here is the problem in my own words:

Kipper zeroes in on the fact that with some terms, such as - plausibly - 'air' and 'water', it is not *a priori* whether they pick out a natural kind or not. It turns out that 'air' doesn't pick out a natural kind, and that 'water' does. Now, let 'airy' be a predicate that applies to a stuff when it exhibits the superficial characteristics that air has in our world, and similarly for 'watery'.

Kipper's bombshell is to point out that 'Air is airy' is plausibly necessary (since air doesn't have some underlying nature which makes it air - rather its being air is basically just a matter of its being airy) but 'Water is watery' is plausibly not necessary (since something with water's underlying nature, i.e. being comprised mainly of H20, could have existed in quite different conditions where it isn't watery). And it definitely seems like these things could not have been known *a priori*.

I have mixed feelings realising this. I was really happy with my proposed link between necessity and apriority. I still feel inclined to suppose that there is something in the idea. But I cannot deny the simplicity and insightfulness of Kipper's bombshell.

Kipper considers and casts serious doubt on a view that tries to escape the bombshell by claiming that meaning change occurs when we discover whether a term like 'air' doesn't pick out a natural kind. I am pretty sympathetic to Kipper's rebuttal of this, and am inclined to look elsewhere for a way of saving, or repairing, my link. Could I perhaps figure out a way of getting propositions which clearly do have *a priori* necessary character and which imply propositions like 'Air is airy'?

I will post again on this matter once I get a better view of the situation.