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.
‘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.
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