Rigidity and General Terms: Two Different Analogues of the Singular Case

This post wrestles with and begins to settle on a view about the confusing issue of how Kripke's notion of rigidity may apply to general terms. 

One analogue of rigidity for general terms: how about we think of it as a rigid connection between properties (or alternatively, an additional connection between the "rigid" term and a further property).

So for example, 'water' in the first instance picks out the property of being water, which is tied rigidly to the property of being (mainly composed of) H20.

Or 'cat' is tied in the first instance to the property of being a cat, and that is tied rigidly to the property of being an animal.

But now there is another kind of thing which seems different, and comes up with sentences like 'John has the property we talked about yesterday'.

Suppose the property we talked about yesterday was the property of having the property that is discussed in Book A. And suppose Book A contains a discussion of the property of redness. Now 'has the property we talked about yesterday' rigidly designates the property of being a property we talked about yesterday, but it also non-rigidly designates the property of having the property that is discussed in Book A, as well as the property of redness.

This motivates the picture of, behind a predicate, a stack of properties, where the top one is designated rigidly, and the ones below not.

Problem: we might want to say that 'cat' rigidly designates a certain kind of animal. And I may then want to rephrase that as: 'cat' rigidly ascribes animality. 

But doesn't the 'rigid' bit here fall away? Take the phenomenological, underlying-nature-neutral counterpart of 'cat' - 'catty thing'. Now even if in our world all the catty things are cats, it doesn't sound right to say that 'catty thing' ascribes the property of being a cat at all - it's not that it ascribes that property, only non-rigidly. 

So now it is beginning to look like the distinction we are after here is between a term merely covering things with property P, vs. ascribing to them property P.

But then that seems wrong when we go back to 'John has the property we discussed yesterday', since if what we talked about yesterday was the property of redness, there is a sense in which that sentence ascribes redness to John. 

This is hell!

But this whole problem, occupying the last few paragraphs, perhaps only arises from mixing together two different analogues of rigidity that we get when we look at predicates.

It may be protested that  'John has the property ...' is not a property ascription syntactically at all, but rather a 2-place relational statement with a non rigid second term.

Be that as it may, we can still classify 'has the ...' as a predicate and can still talk about a rigid/non-rigid distinction. And so I think we need to recognise that there are at least two quite different things going on here - two different things which are a bit similar to rigidity/non-rigidity as applied to names.

One is the difference between 'has the property discussed earlier' and 'is red'. Another is the difference between 'is water' and 'is watery' (one brings being composed of H20 along with it in counterfactual scenario descriptions, and the other doesn't), or 'is a cat' and 'is a catty thing'. 

One reason the second analogue may be counterintuitive if presented as a kind of rigidity is that in the case of singular terms, rigidity is associated with simplicity (both syntactic and semantic), but in the case of predicates (let's look at 'is gold', 'is water', 'is a cat' and put aside 'has the property...') it's the opposite. The "non-rigid" predicates just don't take any further property along with them, but the rigid ones do. I.e. 'is a catty thing' or 'is catlike' just picks out one property, but 'is a cat' is tied to the further property of being an animal.

Actually, the 'any' in 'any further property' is probably wrong! Maybe all predicates rigidly take some further properties along with them. So this second sort of "rigidity" we can talk about in connection with general terms should be thought of as relative to whatever further property is in question. (For instance, 'is a pencil' is arguably counterfactually locked to 'is a physical object'. So 'is a pencil' rigidly picks out physical objects - we might want to say something like that.)

One thing that is emerging here is that the 'has the property discussed earlier' vs. 'is red' thing is one distinction which pattern-matches with Kripke's discussion of rigidity as applied to names, but there is also another thing going on - predicates dragging further properties along with them in counterfactual scenario descriptions - which actually corresponds better with Kripke's informal applications of the notion of rigidity to general terms.

Now it looks like the general-term-"rigidity" considerations in Naming and Necessity are actually closer to the "necessity of constitution" and similar considerations than they are to the "necessity of identity" considerations.


Background Reading:

Online:

- Section 4.2. of the SEP entry for Rigidity: https://plato.stanford.edu/entries/rigid-designators/#ObjAppRigTerForKinPro

Also:

- Kripke, Saul (1980). Naming and Necessity. Harvard University Press.

- Soames, Scott (2002). Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford University Press

- Salmon, Nathan (2004). Are general terms rigid? Linguistics and Philosophy 28 (1):117 - 134.

- Linsky, Bernard (2006). General Terms as Rigid Designators. Philosophical Studies 128 (3):655-667.

- Martí, Genoveva & Martínez-Fernández, José (2011). General terms, rigidity and the trivialization problem. Synthese 181 (2):277 - 293.

- Schwartz, Stephen P. (2002). Kinds, general terms, and rigidity: A reply to LaPorte. Philosophical Studies 109 (3):265 - 277.

- de Sa, Dan López (2007). Rigidity, General Terms, and Trivialization. Proceedings of the Aristotelian Society 107 (1pt1):117 - 123.

- Marti, Genoveva (2004). Rigidity and General Terms. Proceedings of the Aristotelian Society 104:131-148.

- Zouhar, Marián (2009). On the Notion of Rigidity for General Terms. Grazer Philosophische Studien 78 (1):207-229.

- Orlando, Eleonora (2014). General terms and rigidity: another solution to the trivialization problem. Manuscrito 37 (1):49-80.

- Gómez-Torrente, Mario (2004). Beyond Rigidity? Essentialist Predication and the Rigidity of General Terms. Critica 36 (108):37-54.

- Kosterec, Miloš (2018). Criteria for Nontrivial General Term Rigidity. Acta Analytica 33 (2):255-270.

Update on my Necessity and Propositions account (and my haste to declare it false)

In some recent posts here I have discussed propositions like 'Air is airy' (due to Jens Kipper) which we know to be necessarily true, but only because we know empirically that air is not a natural kind, and hence that all there is to being air is being airy, and 'Eminem is not taller than Marshall Mathers' (due to Strohminger and Yli-Vakkuri), which we know to be necessarily true, but only because we know empirically that Eminem is Marshall Mathers, in relation to the account of necessity defended in my thesis. That account says that a proposition is necessarily true iff it is in the deductive closure of the set of true inherently counterfactually invariant propositions. (Roughly, a proposition is ICI if it does not vary across counterfactual scenarios when held true. For more detail see Chapter 5 of my thesis.)

At first, I reacted by thinking that such propositions show that account to be false. I then came up with another account, based on the idea of a counterfactual invariance decider. I still find this new account more elegant, but I soon came to have doubts about just how threatening they are to the ICI-based account in my thesis.

I have recently realised that the ICI-account fares even better in the face of these examples than suggested in the post mentioned above. There, I suggested in effect that 'All there is to being air is being airy' could be argued to imply 'Air is airy' on a suitably rich notion of implication, thus saving the ICI-account, and similarly that 'Eminem is Marshall Mathers' could be argued to imply 'Eminem is not taller than Marshall Mathers' on a suitably rich notion of implication.

But, I have realised, no such rich notion of implication is required! We just need to conjoin the empirical proposition which decides the modal matter with the proposition whose modal status is in question. 'Air is not a natural kind and air is airy', or 'All there is to being air is being airy and air is airy', are both true and ICI, and they both - very straightforwardly, by conjunction elimination - imply the desired proposition. For the Eminem case we have 'Eminem is Marshall Mathers and Eminem is not taller than Marshall Mathers'. So there was never a serious problem for the ICI-account after all! 

Admittedly, these impliers do perhaps seem a bit "clever", a bit artificial in some way, and this - together with not requiring any appeal to implication at all - is why I still think the CI decider account is more elegant. 

One thing that I think went wrong in my thought process around this is that I got a kind of kick out of concluding that my original account was false. Doing so made me feel like a virtuous philosopher, open to changing their views. But I am glad that I now have a more elegant account, and the notion of a CI decider. (I wonder: Would the CI decider account still have come to me if I had not overreacted and thought my original account falsified? Or did my foolishness here cause me to come up with the CI decider account?)

Sticking Up for 'Essence = Necessity + Intrinsicality' in the Face of Zylstra's Argument

Followup: Contingent Examples of Term-Relative Intrinsicality?

UPDATE 11/12/2017: The more I think about Zylstra's argument, the more I think I've been overly critical, and not sufficiently open to changing my views. I have moderated some of the worst excesses by editing the below a little bit. I continue to think about the lessons which we should draw from Zylstra's argument, and may come back to the matter in a future post. One thing which has just begun to bother me is that, if we try to take the lesson to show that we'd better make intrinsicality term-relative when it comes to relations, is that the stuff which comes to mind when trying to explicate the resulting notion of "intrinsicality" - I found myself thinking things like 'x bears R to y intrinsically if part of what it is to be x is to be R-related to y' - just ends up sounding like a characterisation of essence; the necessity-ish bit seems to come of its own accord. So maybe there are grounds here for serious doubt about the overall E = N + I approach to essence.

An interesting new paper by Justin Zylstra attempts to cast doubt on the project of analyzing essence in terms of necessity plus something else. As Fine famously pointed out, it is plausible that the set {Soctrates} essentially contains Socrates but that Socrates does not essentially belong to {Socrates}. Being a member of that set does not have enough to do with Socrates as he is in himself, we might say, to count as an essential property of Socrates. Nevertheless, Socrates necessarily belongs to {Socrates}; in no possible world do we find Socrates but not the set containing him.

So essential properties aren't just the necessarily-possessed properties, or so it seems. Fine makes the further proposal that we give up trying to analyze essence in terms of necessity and instead go the other way around. But others have accepted that the essential properties aren't just the necessarily-possessed ones, but sought to supplement the analysis of essence in terms of necessity. I am sympathetic to this approach, and particularly to the idea - prominently defended by Denby - that essence = necessity + intrinsicality. Let's call this the E = N + I approach.

(Denby, it is important to note, favours an account of intrinsicality on which the property of containing Soctrates is not intrinsic, but extrinsic, to {Socrates}. This leads him to push back against the prima facie plausible Finean thesis that containing Socrates is essential to {Socrates}. In my view, this was a mistake on Denby's part, and we should instead try to understand 'intrinsic' in such a way that it does come out true that the property of containing Socrates is intrinsic to {Socrates}.)

You can imagine my interest in Zylstra's paper, which is supposed to cast serious doubt on this approach. Here I want to explain why I think it does no such thing. I won't reconstruct Zylstra's detailed and technically sophisticated argument in full. To fully assess what I'm saying, in particular to verify that I speak the truth about what Zylstra does in his paper, you'd have to look at the paper.

To understand why Zylstra's argument goes as wrong as I think it does, it helps to note that he aims his criticisms more generally at any attempt to supplement a necessity-based analysis of essence so that it delivers the goods on Fine's celebrated examples, provided it is of a certain general form. He intends this form to cover the E = N + I approach. The trouble is, it is very easy to formulate a version of that approach which does not take general form in question.

The central problem with Zylstra's handling of the E = N + I approach is that he considers only Denby's version, which proceeds as if the relevant notion of intrinsicality can be treated as a sentential operator. It is intrinsic that p. But no friend of the E = N + I approach should want to do that.

The whole point of bringing in intrinsicality, I would have thought, is that it is plausibly intrinsic to {Socrates} that it contains Socrates, but not intrinsic to Socrates that he is contained by {Socrates}. But if we represent our idea of intrinsicality as a sentential operator, all we can say is:

It is intrinsic that Socrates is a member of {Socrates}.

or

It is intrinsic that {Socrates} contains Socrates.

or whatever.

Now, this doesn't really even make sense without explanation, but putting that aside, and assuming that such claims will either be true or be false, Zylstra is able to show that an analysis of essence in terms of necessity and this weird intrinsicality sentential operator can't deliver the goods.

But so what? This just shows that the relevant notion of intrinsicality can't be captured as a sentential operator! Indeed, in his last section, entitled 'A glimmer of hope', Zylstra suggests that instead of supplementing a necessity-based analysis of essence with a notion that can be expressed as a sentential operator, we might be able to use an operator that takes a sentence and a noun phrase and produces a sentence:

Recall that the Supplemented Necessity Analysis involved an existentially bound variable O that functions syntactically as a monadic sentential operator. But nothing prohibits us from introducing a further type of variable Xt that functions syntactically as a binary term-sentence operator. (Zylstra (forthcoming), Section 5.)

Considering as he is all analyses of the relevant, sentential-operator form, rather than just the weird instrinsicality-as-a-sentential-operator instance, he never comes back to consider that maybe the E = N + I approach should be pursued with a binary term-sentence operator. (Another reason for Zylstra's neglecting to do this, perhaps, is that it is Denby's version of the approach that Zylstra considers, and that version - ill-advisedly, as I suggested in a parenthesis near the beginning of this post - fails to deliver the intuitive Finean verdict that containing Socrates is essential to {Socrates}.) But really, that's just the natural view when you think about this. The weird sentential-operator form is just an especially bad version of the E = N + I approach which no one sympathetic to that approach should allow.

I conclude that Zylstra's new paper poses no real threat at all to the E = N + I approach to understanding essence. Rather, the lesson that the friend of the E = N + I approach should draw is that intrinsicality is not to be expressed using a monadic sentential operator. Nor will it do to think of it, in general, as something which relations possess or fail to possess tout court. A relation like the set-membership relation is, so to speak, extrinsic on Socrates’s end but intrinsic on {Socrates}’s end.

In a way, this is really just a criticism about emphasis. Rather than presenting his argument as if it were a serious threat to the E = N + I approach, and then offering a 'glimmer of hope', Zylstra should, in my view, have just presented his argument as showing something instructive about how a friend of the E = N + I should, and should not, try to formulate it.

References 

Denby, David A. (2014). Essence and Intrinsicality. In Robert Francescotti (ed.), Companion to Intrinsic Properties. De Gruyter. pp. 87-109.Author-archived version currently available open-access at http://philpapers.org/rec/DENIAE-3.

Fine, Kit (1994). Essence and modality. Philosophical Perspectives 8:1-16.

Zylstra, Justin (forthcoming). Essence, necessity, and definition. Philosophical Studies:1-12. Currently available open-access at the author's Academia.edu page, the URL of which is currently http://vermont.academia.edu/JustinZylstra.

Two-Dimensional Semantics and Counterfactual Invariance Deciders

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.

Old Account May Not Be False After All, But New One Still Better (and New Frontier: Relation to Two-Dimensionalism)

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.

A New Account of the Conditions Under Which a Proposition is Necessary

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.

Strohminger & Yli-Vakkuri Improve (in Some Respects) Upon Kipper's Bombshell (and My Account of Necessity May Be False)

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.

Notes on Counterfactual Scenario Descriptions, Sources of Modal Error (or Uncertainty), and Deep Puzzles of Modality

One interesting thing about my account of subjunctive necessity is the way it separates two kinds of things that we could be wrong or confused about in our judgements of subjunctive modality: the truth of what we're holding true for the purposes of a counterfactual scenario description (CSD) and the genuineness of that CSD.

For example, I might think 'Hesperus is not Phosphorus' is subjunctively possible because I falsely believe that Hesperus is Phosphorus. Or I might be acquainted with some strange animals I call Toves and then not feel sure whether 'Toves are animals' is necessary simply because I am not sure whether the things I am acquainted with are animals.

By contrast, there are modal questions which do not centre - at least not in any definite way - on the truth or otherwise of what is being held true. For example, granting that I am human, could I have been an animal? Or how about a Neanderthal? Or a bank account?

And cosmic questions about whether there could have been less matter or energy, or perhaps just one atom in a void? And here the question arises: how do you know what you have to know in order to know whether something is necessarily the way it is or not? (In some cases, that seems clear. E.g. you have to know whether Hesperus is in fact Phosphorus in order to know whether it necessarily is. But in others it really doesn't.)

Another puzzling thing stems from the way, in my account, you can have CSDs which aren't possible, since the things held true for them aren't true. For instance, if I think wrongly that Hesperus isn't Phosphorus (or even just grant that for the sake of argument), I will be prepared to produce CSDs involving Hesperus not being Phosphorus, and these may be perfectly genuine. This strikes me as an important virtue of my account - i.e., that it is some sort of advance, giving us a fruitful way of talking and thinking philosophically about modality. It is hard to say exactly why. One thing is that it enables us to bracket off distracting sources of modal uncertainty and error, perhaps allowing us to focus better on the stuff which really bothers us philosophically about modality. 

In any case, this thing - about there being genuine CSDs which, despite being genuine, aren't possible because false things are held true for them - gives rise to some puzzlement in its own right. When we can go different ways on the question of whether Hesperus is Phosphorus or Clark Kent is Superman - questions of the identity or distinctness of things - it seems like our underlying way of thinking about things, our conceptual apparatus, is basically the same. And to a fair, but perhaps lesser extent, going different ways on the question of the underlying nature of cats, or Toves, or water also seems to leave our conceptual apparatus largely the same. (There may be something wrong or lacking in this description.) We get the sense that we can flip the switch either way on these things quite readily, and continue in much the same way in either case when it comes to grasping a range of genuine CSDs which arise on the assumption that things are the one way or the other. On the other hand, what of things which are - in a conceptual sense, I want to say - far from true? 

Things get puzzling very quickly once such questions come into view. If I somehow hold it true that humans are cats, can I then produce genuine CSDs according to which that is true? It is hard to know what to say. One thing is that there might be an issue about whether we can really hold such things true, or perhaps better, whether it makes sense to talk of holding such things true. But to that it may always be replied - OK, but we can sometimes do something here, in these cases where you might worry about whether we can really hold the things in question true or not.

Are there two ways, then, of getting out of the sphere of genuine CSD-hood? One by holding things true which are either true, or not a big deal or problematic to hold true, and then going further and further away from actuality, so to speak, until you say things we might hesitate to call CSDs (e.g. holding it true that I am human, but then talking about a scenario where I'm a cat), and another by holding far-out things true?

Another worry concerns what might be called modal encroachment. The idea that, if we learn more about some things, we might realise that some things aren't possible that we thought were. And there is a question here about whether that could affect what we think about genuineness of a CSD, or whether what we formerly thought were not only genuine CSDs but possibilities (i.e. that the things being held true for those CSDs are the case) can always be retained as genuine CSDs by holding the right false things true.

I feel that with these issues my account, which could seem merely logic-choppy and perhaps trivial in a way, begins to make contact with some of the deeper puzzles surrounding modality. 

In a future post I want to try to explore how our ideas might be prone to shifting and slipping without our realising it when we philosophize about modality. For instance, the obscure way the stakes can seemingly be raised in some way by the question of 'But could that really have happened?'. I also hope to make some progress on puzzles concerning 'whether the ground of modality is in us or the world', by trying to better uncover the thought processes underlying that unsatisfactory question. Perhaps then we will see better what the real issues are in this thicket of philosophy.

Postscript (or seedling for next time):

Dim hypothesis re. Kripkean showing of necessity of identity: it shows that things couldn't have been otherwise in a deep way by showing that they couldn't have been otherwise in a shallow way. 

I.e. in a certain frame of mind, we might think 'What do we know about how, and the extent to which, things really might have been different?'. A frame of mind with a sense of cosmic mystery, open to underlying system we have little or no inkling of. Then the Kripkean arguments come along and say 'Well, whatever the truth is about that, things certainly couldn't have been such that Hesperus isn't Phosphorus'.

It is notable that Kripke's results are necessities, or denials of possibility. This leaves it open that we have a way of thinking, or a concept of modality, on which all the Kripkean necessities are necessary as required, but where there is leeway which then disappears on some deeper view.

An Adventure in Linking Necessity to Apriority

[UPDATE: These ideas have led to a paper, 'Linking Necessity to Apriority', in Acta Analytica.]

There is an important link between necessity and apriority which can shed light on our knowledge of the former, but initially plausible attempts to spell out what it is fall victim to counterexamples. Casullo (2003) discusses one such proposal, argues that it fails, and suggests an alternative. In this post, I argue that Casullo’s alternative also fails, suggest another, argue that that fails too, and then suggest another which I hope is correct.

First proposal

Kripke (1980) showed that it is not always knowable a priori whether a proposition is necessarily true. But, you might think, perhaps it is always knowable a priori whether a proposition has whatever truth value it has necessarily or contingently. To use Casullo’s (2003) terminology, while Kripke showed that knowledge of specific modal status (necessarily true, contingently false, etc.) is not always possible a priori, this leaves open the possibility of apriori knowledge of general modal status (necessary or contingent - and on this usage of ‘necessary’ and ‘contingent’, truth value is left open). Perhaps that is the link we are after between necessity and apriority.

The claim that general modal status is always knowable a priori entails the following:

(1) If p is a necessary proposition and S knows that p is a necessary proposition, then S can know a priori that p is a necessary proposition.

(The second conjunct of (1)’s antecedent sidesteps the worry that some necessary propositions may be such that it is unknowable that they are necessary.)

Casullo, following Anderson (1993), argues convincingly that this is false. Consider:

(1X) Hesperus is Phosphorus or my hat is on the table.

This is a necessary proposition, but for all any S could know a priori, it could be necessarily true (if the first disjunct is true), contingently true (if the first disjunct is false but the second true), or contingently false (if both disjuncts are false). So (1) can’t be right.

Second proposal

In an interesting effort to avoid the problem affecting (1), Casullo introduces the notions of conditional modal propositions and conditional modal status:

Associated with each truth functionally simple proposition is a pair of conditional propositions: one provides the specific modal status of the proposition given that it is true; the other provides its specific modal status given that it is false. Associated with each truth functionally compound proposition is a series of conditional propositions, one for each assignment of truth values to its simple components. Each conditional proposition provides the specific modal status of the proposition given that assignment of truth values. Let us call these propositions conditional modal propositions and say that S knows the conditional modal status of p just in case S knows all the conditional modal propositions associated with p. (Casullo (2003), p. 197.)

His proposed link between necessity and apriority is as follows:

(2) If p is a necessary proposition and S knows the conditional modal status of p, then S can know a priori the conditional modal status of p.

Casullo dubs this ‘a version of the traditional account of the relationship between the a priori and the necessary that is immune to Kripke’s examples of necessary a posteriori propositions’ (Casullo (2003), p. 199). It handles (1X) nicely. Calling (1X)’s disjuncts ‘Hesp’ and ‘Hat’, its associated conditional modal propositions will run as follows:

If Hesp is true and Hat is true, (1X) is necessary.
If Hesp is true and Hat is false, (1X) is necessary.
If Hesp is false and Hat is true, (1X) is contingent.
If Hesp is false and Hat is false, (1X) is contingent.

These are plausibly knowable a priori, as required by (2).

But consider:

(2X) Everything is either such that it is either not Hesperus or is Phosphorus, or such that it is either on the table or not my hat.

While it contains connectives, this is not a truth functional compound in the relevant sense, since it does not embed any whole propositions. So on Casullo’s proposal, (2X) will be associated with just a pair of conditional modal propositions. Which ones? A problem here is that there is no very clear positive case for any pair (the account, after all, was probably not formulated with (2X) in mind), but I think it is clear that the only candidate pair which could stand a chance is:

If (2X) is true, it is necessary.
If (2X) is false, it is contingent.

(After all, (2X) is true and necessary, so the other available choice for first member couldn’t be right, and the second member of the pair seems true and knowable a priori.)

Instantiating Casullo’s proposal (2) on (2X), we get:

If (2X) is a necessary proposition and S knows the conditional modal status of (2X), then S can know a priori the conditional modal status of (2X).

But it seems clear that the first conditional modal proposition for (2X), i.e. that if (2X) is true, it is necessary, could not be known a priori. So (2) can’t be right either.

Third proposal

What strikes one initially about the disjunctive counterexample to the first proposal is that it has a component whose general modal status is knowable a priori. But this isn’t true of the counterexample to the second proposal; it has no component propositions at all. What is true about both counterexamples is, not that they have cromponent propositions whose general modal status is knowable a priori, but that they are implied by such propositions.

Let us say that a proposition p possesses a priori necessary character iff it can be known a priori that p is a necessary proposition, i.e. that p has whatever truth value it has necessarily.

Now, I submit that if a proposition whose general modal status is knowable at all is necessarily true, then it is in the deductive closure of a set of true propositions possessing a priori necessary character.

How, though, to generalize this so that it covers all necessary propositions (i.e. necessarily false propositions as well as true ones)? For a few weeks, I thought this would work:

If a proposition whose general modal status is knowable at all is necessary, then it is either in the deductive closure of a set of true propositions possessing a priori necessary character, or it is in the deductive closure of a consistent set of false propositions possessing a priori necessary character.

To cast the point in a form similar to (1) and (2) above:

(3) If p is a necessary proposition and S knows that p is a necessary proposition, then p is either in the deductive closure of a set of true propositions which S can know a priori to be necessary, or it is in the deductive closure of a consistent set of false propositions which S can know a priori to be necessary.

But I have just recently realised that this is false as well.

The problem lies with necessarily false propositions. Requiring consistency of the set of false propositions that implies a putative necessary proposition rules out necessarily false propositions that contradict themselves. E.g. 'It is both raining and not raining' is, and can be known to be, a necessary proposition, but it is not implied by any consistent set of false propositions of apriori necessary character. On the other hand, removing the consistency requirement causes the account to overgenerate, at least on a classical conception of implication; 'I had toast for breakfast' is implied by the set of false propositions of a priori necessary character {'2 + 2 = 4', 'not-(2 + 2 = 4)'}, since that set implies any proposition whatsoever.

Fourth proposal

Now, without wanting to rule out that we could specify a special implication-like relation which behaves as desired, I have nevertheless tentatively given up on bringing in consistency to get a general result which covers not only necessary true propositions but necessarily false ones as well. Instead, I think the thing to do is to exploit the idea that a necessarily false proposition's negation is necessarily true, giving us:

(4) If p is a necessary proposition and S knows that p is a necessary proposition, then either p or its negation is in the deductive closure of a set of true propositions which S can know a priori to be necessary.

Maybe this one is true! Please let me know, by comment or email, if you see a problem.

[UPDATE 31/08/2017: Trouble has arisen.] [UPDATE 2020: The trouble led to new ideas but on reflection does not threaten the core idea here. The paper that grew from this material discusses and deals with the examples that initially seemed to me to vitiate the core idea.]

Thanks to Albert Casullo for helpful and encouraging correspondence on this topic.

References

Anderson, C. Anthony (1993). Toward a Logic of A Priori Knowledge. Philosophical Topics 21(2):1-20.

Casullo, Albert (2003). A Priori Justification. Oxford University Press USA.

Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.

The Pre-Kripkean Puzzles are Back

Yes, but does Nature have no say at all here?! Yes.

It is just that she makes herself heard in a different way.

Wittgenstein (MS 137).

Modality was already puzzling before Kripke - there’s a tendency for the potted history of the thing to make it seem like just before Kripke, philosophers by and large thought they had a good understanding of modality. But there were deep problems and puzzles all along, and I think many were alive to them.

There is a funny thing about the effect of Kripke’s work which I have been starting to grasp lately. It seems like it jolted people out of certain dogmas, but that the problems with those dogmas were actually already there. The idea of the necessary a posteriori sort of stunned those ways of thinking. But once the dust settles and we learn to factor out the blatantly empirical aspect from subjunctive modality - two main ways have been worked out, more on which in a moment - the issue comes back, and those ways of thinking and the problems with them are just all still there.

(When I was working on my account of subjunctive necessity de dicto, I thought of most pre-Kripkan discussions of modality as irrelevant and boring. Now that I have worked that account out, they are seeming more relevant.)

What are the two ways of factoring out the aposterioricity of subjunctive modality? There is the two-dimensional way: construct “worlds” using the sort of language that doesn’t lead to necessary a posteriori propositions, and then make the truth-value of subjunctive modal claims involving the sort of language that does lead to them depend on which one of the worlds is actual.

This is currently the most prominent and best-known approach. However, it involves heady idealizations, many perplexing details, and various questionable assumptions. I think the difficulty of the two-dimensional approach has kept us in a kind of post-Kripkean limbo for a surprisingly long time now. Except perhaps in a few minds, it has not yet become very clear how the old pre-Kripkean problems are still lying in wait for us. I have hopes that the second way of factoring out will move things forward more powerfully (while I simultaneously hope for a clearer understanding of two-dimensionalism).

What is the second way? It is to observe that the subjunctively necessary propositions are those which are members of the deductive closure of the propositions which are both true and C, where C is some a priori tractable property. (On my account of C-hood, the closure version of the analysis is equivalent to the somewhat easier to understand claim that a proposition is necessary iff it is, or is implied by, a proposition which is both C and true. On Sider’s account of C-hood this equivalence fails.)

My account of subjunctive necessity explains condition C as inherent counterfactual invariance, which in turn is defined using the notion of a genuine counterfactual scenario description. And it is with these notions that the old-style puzzles come back up. Sider’s account has it that C-hood is just a conventional matter - something like an arbitrary, disjunctive list of kinds of propositions. (Here we get a revival of the old disagreements between conventionalists and those who were happy to explain modality semantically, but suspicious of conventionalism.)

What are these returning puzzles all about? They are about whether, and in what way, meaning and concepts are arbitrary. And about whether, and in what way, the world speaks through meaning and concepts. Hence the quote at the beginning, and the quote at the end of this companion post.

Reflections on My Claim that Inherent Counterfactual Invariance is Broadly Semantic

This post presupposes knowledge of my account of subjunctive necessity de dicto as expressed in my thesis and in a paper derived from it which I have been working on. (I hope my self-criticism here doesn't cause any should-have-been-blind referees to reject the paper. A revise-and-resubmit verdict I could live with.) Here I try to take a next step in getting clear about the status and significance of the account.

In my thesis and derived paper, I propose that a proposition is necessary iff it is, or is implied by, a proposition which is both inherently counterfactually invariant (ICI) and true, and explicate this notion of ICI.

I claim that ICI is broadly semantic, and put this forward as a key motivation and virtue of the account. I don’t provide much argument for this claim - the intention, I suppose, was that this would just seem self-evident. But I have become increasingly aware of the importance of the fact that this could be challenged, and the importance of getting clearer about the underlying primitive notion of a genuine counterfactual scenario description (CSD).

I do provide one reason, near the end of my presentation of my account, for thinking that ICI is broadly semantic given my preferred approach to propositions and meaning. But there are two reasons for wanting more. One is that it may be hoped that my claim that ICI is broadly semantic could be justified independently of my particular approach to propositions and meaning, where I advocate understanding what I distinguish as the ‘internal’ component of meaning as role in language system. A second, perhaps more suggestive, reason for wanting more is that, even given my preferred approach, the argument I give is basically this: ICI is explained in terms of how a proposition - its negation, really - behaves in certain contexts - namely CSDs. But here of course I have to single out genuine CSDs.

And here’s the thing. (At least, the following seems to be right.) For my claim that ICI is broadly semantic to hold water, the notion of genuineness of a CSD had better be broadly semantic. For it is not enough for a notion to be broadly semantic that it can be characterized in terms of appearance in certain sorts of linguistic context C, where C-hood is blatantly extra-semantic. For instance, we may say a proposition has G iff it (or its negation, to make this more like the ICI case) doesn’t appear in any description which has the property of being written in some notebook I have in my room. In that case, it is plain that whether or not a proposition has G is not a matter of its meaning or nature.

So, I now think that the little argument I give at the end of my presentation of my account, about how my particular approach to propositions and meaning ‘fits well’ with the notion of ICI as broadly semantic only goes so far, and that as an argument that given my particular approach the notion of ICI is or should be seen as broadly semantic, it is weak, since it gives no reason to think that the all-important notion of genuineness of CSD is broadly semantic.

Further, I think it is clear that I want to put forward my account, and I think the account has theoretical value, independent of whether a case can be made that genuineness of CSD is broadly semantic. And so my whole presentation of why my account is interesting and of its motivation is somewhat crude. As a story about what caused it, and the specific things I was thinking, it may have some interest. But as a way of situating the theory and giving a sense of what its value (within philosophy) consists in, it is crude and not really to the point. I do of course hint at other sources of interest (e.g. that the account clarifies the relationship between the notion of necessity and those of truth and implication), and don’t rest everything on the ‘semantic hunch’, but I do perhaps give it too prominent a place - or at least, an incompletely justified place.

So, is the notion of a ‘genuine counterfactual scenario description’ broadly semantic? And what does it mean to be broadly semantic? I may follow up with a post addressing these questions more thoroughly, but for now a couple of remarks. Whatever it is to be broadly semantic, it is not to be conventional in any sense. The idea is perhaps better gotten at, in some ways, by saying that genuine CSD-hood is a conceptual matter. But really I need to roll up my sleeves and investigate this more closely - it is not merely a question of hitting on some formulation. Finally, I propose that the following passage from §520 of Wittgenstein’s Investigations seems very to-the-point when it comes to the questions and difficulties I find myself coming up against here, and may help me plumb the depths of the matter:

So does it depend wholly on our grammar what will be called (logically) possible and what not,—i.e. what that grammar permits?”—But surely that is arbitrary!—Is it arbitrary?—It is not every sentence-like formation that we know how to do something with, not every technique has an application in our life [...].