A Dialogue on Mathematical Propositions

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.

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.

Kipper's Bombshell

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.

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.

Caught in the Act of Confusing Subjunctive Necessity and Apriority: The Importance of the Sprigge Quote in Naming and Necessity

One of the big questions surrounding Kripke's innovations in Naming and Necessity is the extent to which, with his doctrines about necessity and the necessary a posteriori, Kripke corrected false views about necessity, as opposed to just emphasizing a neglected notion of necessity on which 'necessary a posteriori' is a non-empty category. Also unclear is the extent to which pre-Kripkean thinkers were confusing the Kripkean notion of subjunctive necessity with other notions, or just not giving that notion much attention. It's not clear, for instance, that Putnam in 'It Ain't Necessarily So' ever invoked subjunctive necessity.

This makes the following quote from Sprigge, which Kripke uses in N&N (p. 111), particularly interesting. It seems to provide a clear case of a philosopher confusing subjunctive necessity, on the one hand, with either indicative necessity or apriority on the other hand:

The anti-essentialist says that there would be no contradiction in a news bulletin asserting that it had been established that the Queen was not in fact the child of her supposed parents, but had been secretly adopted by them, and therefore the proposition that she is of Royal Blood is synthetic. In this way the anti-internalist parries the argument of the internalist by suggesting with regard to each proposed internal property of the particular in question, that we can quite well imagine that very same particular without the property in question. For a time he is winning. Yet there comes a time when his claims appear a trifle too far fetched. The internalist suggests that we cannot imagine that particular we call the Queen having the property of at no stage in her existence being human. If the anti-internalist admits this, admits that it is logically inconceivable that the Queen should have had the property of, say, always being a swan, then he admits that she has at least one internal property. If on the other hand he says that it is only a contingent fact that the Queen has ever been human, he says what it is hard to accept. Can we really consider it as conceivable that she should never have been human? (Sprigge (1962), p. 203.)

It seems pretty clear that here Sprigge takes the possibility of the bulletin - the possibility of finding out that Elizabeth II is not actually born of royal blood - as tantamount to it being the case that things could have gone such that she was not born of royal blood.

So, this quote makes it seem almost certain to me that someone - namely Sprigge - was actually confusing subjunctive necessity with either apriority or indicative necessity. Further questions are how widespread the confusion was around the time Sprigge wrote, and whether this was a relatively new thing at the time. Is it the case that, by the time Sprigge wrote the above but not for long before that, the notion of subjunctive or counterfactual necessity was "in the air", was salient, and so this sort of confusion is a relatively short term phenomenon occurring only in the lead-up to N&N (and shortly after, while people had yet to digest Kripke)? Or is the confusion something we can find much earlier evidence of?

References

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

Putnam, Hilary (1962). It ain't necessarily so. Journal of Philosophy 59 (22):658-671.

Sprigge, Timothy (1962). Internal and external properties. Mind 71 (282):197-212.