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.