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.
Showing posts with label a priori knowledge. Show all posts
Showing posts with label a priori knowledge. Show all posts
Monday, 6 November 2017
Thursday, 21 September 2017
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.
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.
Wednesday, 30 August 2017
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.
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.
Friday, 21 April 2017
Explaining the A Priori in Terms of Meaning and Essence
UPDATE (Nov 2019): A descendent of this post, 'Apriority and Essential Truth', has been published in Metaphysica.
It wasn't just the positivists who thought there was a tight connection between meaning and truth in the case of a priori propositions:
Noteworthy in this connection is that Kripke was not altogether gung ho about his severance of necessity from apriority:
Now, in Chalmers' epistemic two-dimensionalist framework, indicative necessity is itself explained in epistemic terms. But if we try for a more full-bloodedly semantic conception of it, we may get something more explanatory of the special epistemic status of a priori truths. The notion we are after is something like: a proposition is indicatively necessary iff, given its meaning, it cannot but be true. And the modality here is not supposed to be epistemic.
But what aspect of its meaning? Sometimes 'meaning' covers relationships to things out in the world, and even the things out there themselves. What we are interested in is internal meaning. Putnam's Twin Earth thought experiment - though this is not how he used it - lets us see the distinction we need here. We want to talk about meaning in the sense in which Earth/Twin Earth pairs of propositions mean the same. This can be articulated using the middle-Wittgenstein idea of the role an expression plays in the system it belongs to (see Wittgenstein (1974, Part I)).
So, what if we say that a proposition is indicatively necessary iff any proposition with its internal meaning must, in a non-epistemic sense, be true? Can indicative necessity in this sense be used to explain apriority?
Maybe not, since there are indicatively necessary truths which are indicatively necessary only because their instantiation requires their truth. Example: language exists. (Language is here understood as a spatiotemporal phenomenon.) This is indicatively necessary, because any proposition with its internal meaning must be true, if only because the very existence of that proposition requires it to be true. Its truth comes about from the preconditions for its utterance, but - you might think - not from the internal meaning itself. It is interesting to note that it is indicatively necessary, but it lacks the special character of a priori propositions whereby they, in some sense, don't place specific requirements on the world.
This situation pattern-matches with Fine's celebrated (1994) distinction between necessary and essential properties. Socrates is necessarily a member of the set {Socrates}, but that membership is not part of his essence, since it doesn't have enough to do with Socrates as he is in himself. Likewise, he is necessarily distinct from the Eiffel Tower, but this is no part of his essence. So let us throw away the ladder of indicative necessity and instead hone in on the notion of essential truth. A proposition is essentially true iff it is of its internal meaning's essence to be true (i.e. to be the internal meaning of a true proposition).
Thus, with encouragement from Gödel and Kripke, we can develop ideas from Chalmers, Putnam, Wittgenstein, and Fine, to yield:
To say that a proposition is a priori is to say that it can, in some sense, be known independent of experience. (You may need experience to get the concepts you need to understand the proposition, but you don't need any particular further experience to know that the proposition is true.) What is distinctive about these propositions which explains their being knowable in that peculiar way? It is that their internal meanings - their roles in language - are, of their very essence, the internal meanings of true propositions; any proposition with that internal meaning must be true, and not for transcendental reasons relating to the pre-conditions of the instantiation of the proposition, but as a result of that internal meaning in itself.
So we can have an account of apriority which explains it in terms of a tight connection between meaning and truth, freed of its accidental associations with conventionalist and deflationary views about meaning, modality and essence.
This is not to say that a priori propositions' truth is to be explained in a case by case way by considerations about meaning and essence. That would be to crowd out the real mathematical justifications of non-trivial mathematical truths. But explaining apriority in general in this way wards off misunderstandings which come from treating a priori truths too much like empirical truths. And that is what makes it an explanation.
References
Chalmers, David J. (1998). The tyranny of the subjunctive. (unpublished)
Fine, Kit (1994). Essence and modality. Philosophical Perspectives 8:1-16.
Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.
Putnam, Hilary (1973). Meaning and reference. Journal of Philosophy 70 (19):699-711.
Wittgenstein, Ludwig (1974). Philosophical Grammar. University of California Press.
It wasn't just the positivists who thought there was a tight connection between meaning and truth in the case of a priori propositions:
However, it seems to me that nevertheless one ingredient of this wrong theory of mathematical truth [i.e. conventionalism] is perfectly correct and really discloses the true nature of mathematics. Namely, it is correct that a mathematical proposition says nothing about the physical or psychical reality existing in space and time, because it is true already owing to the meaning of the terms occurring in it, irrespectively of the world of real things. What is wrong, however, is that the meaning of the terms (that is, the concepts they denote) is asserted to be something man-made and consisting merely in semantical conventions. (Gödel (1951/1995), p. 320.)Perhaps we should try to recover some insight from the idea, nowadays highly unfashionable within philosophy (but alive and well in the broader intellectual culture, I think), that a priori truths like those of mathematics are in some sense true owing to their meanings. Philosophers often used to express this by calling such propositions 'necessarily true', but since Kripke that sort of usage has been crowded out by another.
Noteworthy in this connection is that Kripke was not altogether gung ho about his severance of necessity from apriority:
The case of fixing the reference of ‘one meter’ is a very clear example in which someone, just because he fixed the reference in this way, can in some sense know a priori that the length of this stick is a meter without regarding it as a necessary truth. Maybe the thesis about a prioricity implying necessity can be modified. It does appear to state some insight which might be important, and true, about epistemology. In a way an example like this may seem like a trivial counterexample which is not really the point of what some people think when they think that only necessary truths can be known a priori. Well, if the thesis that all a priori truth is necessary is to be immune from this sort of counterexample, it needs to be modified in some way. [...] And I myself have no idea it should be modified or restated, or if such a modification or restatement is possible. (Kripke (1980), p. 63.)This may make it sound like the required modification would consist in somehow ruling out the problematic contingent a priori truths from the class of truths whose epistemic status is to be explained. But Chalmers' idea of the tyranny of the subjunctive suggests another route: try instead to find a different notion of necessity - indicative, as opposed to subjunctive, necessity; truth in all worlds considered as actual, rather than truth in all worlds considered as counterfactual - better suited to the explanation of apriority.
Now, in Chalmers' epistemic two-dimensionalist framework, indicative necessity is itself explained in epistemic terms. But if we try for a more full-bloodedly semantic conception of it, we may get something more explanatory of the special epistemic status of a priori truths. The notion we are after is something like: a proposition is indicatively necessary iff, given its meaning, it cannot but be true. And the modality here is not supposed to be epistemic.
But what aspect of its meaning? Sometimes 'meaning' covers relationships to things out in the world, and even the things out there themselves. What we are interested in is internal meaning. Putnam's Twin Earth thought experiment - though this is not how he used it - lets us see the distinction we need here. We want to talk about meaning in the sense in which Earth/Twin Earth pairs of propositions mean the same. This can be articulated using the middle-Wittgenstein idea of the role an expression plays in the system it belongs to (see Wittgenstein (1974, Part I)).
So, what if we say that a proposition is indicatively necessary iff any proposition with its internal meaning must, in a non-epistemic sense, be true? Can indicative necessity in this sense be used to explain apriority?
Maybe not, since there are indicatively necessary truths which are indicatively necessary only because their instantiation requires their truth. Example: language exists. (Language is here understood as a spatiotemporal phenomenon.) This is indicatively necessary, because any proposition with its internal meaning must be true, if only because the very existence of that proposition requires it to be true. Its truth comes about from the preconditions for its utterance, but - you might think - not from the internal meaning itself. It is interesting to note that it is indicatively necessary, but it lacks the special character of a priori propositions whereby they, in some sense, don't place specific requirements on the world.
This situation pattern-matches with Fine's celebrated (1994) distinction between necessary and essential properties. Socrates is necessarily a member of the set {Socrates}, but that membership is not part of his essence, since it doesn't have enough to do with Socrates as he is in himself. Likewise, he is necessarily distinct from the Eiffel Tower, but this is no part of his essence. So let us throw away the ladder of indicative necessity and instead hone in on the notion of essential truth. A proposition is essentially true iff it is of its internal meaning's essence to be true (i.e. to be the internal meaning of a true proposition).
Thus, with encouragement from Gödel and Kripke, we can develop ideas from Chalmers, Putnam, Wittgenstein, and Fine, to yield:
To say that a proposition is a priori is to say that it can, in some sense, be known independent of experience. (You may need experience to get the concepts you need to understand the proposition, but you don't need any particular further experience to know that the proposition is true.) What is distinctive about these propositions which explains their being knowable in that peculiar way? It is that their internal meanings - their roles in language - are, of their very essence, the internal meanings of true propositions; any proposition with that internal meaning must be true, and not for transcendental reasons relating to the pre-conditions of the instantiation of the proposition, but as a result of that internal meaning in itself.
So we can have an account of apriority which explains it in terms of a tight connection between meaning and truth, freed of its accidental associations with conventionalist and deflationary views about meaning, modality and essence.
This is not to say that a priori propositions' truth is to be explained in a case by case way by considerations about meaning and essence. That would be to crowd out the real mathematical justifications of non-trivial mathematical truths. But explaining apriority in general in this way wards off misunderstandings which come from treating a priori truths too much like empirical truths. And that is what makes it an explanation.
References
Chalmers, David J. (1998). The tyranny of the subjunctive. (unpublished)
Fine, Kit (1994). Essence and modality. Philosophical Perspectives 8:1-16.
Gödel, Kurt (1951/1995). Some basic theorems on the foundations of mathematics and their implications. In Solomon Feferman (ed.), Kurt Gödel, Collected Works. Oxford University Press 290-304. (Originally delivered on 26 December 1951 as the 25th annual Josiah Willard Gibbs Lecture at Brown University.)
Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.
Putnam, Hilary (1973). Meaning and reference. Journal of Philosophy 70 (19):699-711.
Wittgenstein, Ludwig (1974). Philosophical Grammar. University of California Press.
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