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About

I'm Greg Restall, and this is my website. I work in Philosophy at the University of Melbourne. [Email: greg at consequently.org; Skype: greg_restall; Post: Department of Philosophy, University of Melbourne, Parkville 3010, Australia.]

Writing

These are the three last modified entries on my writing page.

“Truthmakers, Entailment and Necessity 2008,” an addendum to “Truthmakers, Entailment and Necessity,” to appear in Truth and Truth-making, edited by E. J. Lowe and A. Rami, Acumen, 2008. →
[with Rebecca Kukla and Mark Lance] Appendix to Rebecca Kukla and Mark Lance ‘Yo!’ and ‘Lo!’: the pragmatic topography of the space of reasons, Harvard University Press, to appear. →
“Curry’s Revenge: the costs of non-classical solutions to the paradoxes of self-reference,” in The Revenge of the Liar, ed. JC Beall, Oxford University Press, pages 262–271, 2008. →
“Anti-Realist Classical Logic and Realist Mathematics,” under revision. →
“Proof Theory and Meaning: on second order logic,” to appear in the Logica 2007 Yearbook, Filosofia. →

Classes

In Semester 2, which starts on July 31, I’ll be teaching an honours seminar 161-438 Logic and Philosophy, in which we cover proof theory and its applications to semantics.

Events

AAL2007: the annual conference of the Australasian Association for Logic, University of Melbourne November 9 to 11, 2007.

Recent Past

University of Melbourne Philosophy Undergraduate Workshop, University of Melbourne September 21 to 23, 2007.

Logic Colloquium 2007, Wrocław, Poland, July 14-19, 2007.

1st GPMR Workshop on Logic & Semantics on Medieval Logic and Modern Applied Logic, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany, on June 28-30, 2007.

Logica 2007, Hejnice Monastery, Czech Republic, 18-22 June 2007.

Heart of Philosophy Café talk and discussion on “What Marx, Freud and Nietzsche have taught me about belief in God”. Tuesday May 8, 7--9pm in the Merrick's General Store.

A Philosophical Poll: on a priori knowledge of possibilities

I’d like to guage some philosophical opinions. (I don’t have many of my own. Most of them have been worn down by years of logical abuse.)

First, some pre-requisites:

Let’s understand ‘possibly’ as a metaphysical sort of possibility. If it helps, think of it as truth in some possible world, no matter how outlandish.
Let’s understand a priori knowledge in the way that all of the people who talk about a priori knowledge understand the term. Plausibly, we can’t know that Hesperus is Phosphorus a priori, but, plausibly, we can know the Chinese Remainder Theorem a priori.

Here’s the question. Can we have a priori knowledge of any proposition of the form

possibly ~p

where p is true? (We don’t need to know a priori that p is true.) Speaking rather briskly, do we have a priori knowledge that fatalism is false? (Where ‘fatalism’ is the thesis that everything that is true is metaphysically necessary.) This is fudging a scope distinction, commuting a propositional quantifier under the knowledge operator, but that doesn’t seem like too much of a worry. If you think that there’s a proposition ‘@’ true in the actual world alone (of any of the possible worlds accessible from the actual world) then, the question reduces to this:

Can we know a priori that possibly ~@?

What do you think? What do the Standard Positions on these topics (a priori knowledge, necessity, semantics, etc.) say about this? Is there anything that we obviously have a priori knowledge of that clearly a priori entails possibly ~@? (That’d be interesting: I can’t think of any examples right now…)

Anyway, if you’re the kind of person to have opinions on these sorts of issues, let me know what you think in the comments.

Posted 04:56 PM on November 1, 2006

Comments

“Is there anything that we obviously have a priori knowledge of that clearly a priori entails ~@?”

You mean “entails possibly ~@”, right? (~@ itself is actually false, so won’t be entailed by anything we know!)

How about this: I know a priori that if things had turned out differently, then @ would have been false. But on the standard analysis of counterfactuals, this entails that there is a possible world where things are different and @ is false.

If you don’t trust counterfactuals, we can pull a similar move using any a priori claim of non-supervenience. (That’ll give us a new world where the one thing is different despite the other staying the same. E.g. Semantic externalism entails some kind of “Twin Earth” — though I guess it needn’t be the XYZ one.)

On some views (e.g. Chalmers’ 2-D modal rationalism; and I think Lewis held something similar?) we - or at least ideally rational agents - can have a priori knowledge of every true modal proposition (assuming they’re all necessary truths, as per S5). I’m pretty sympathetic to this view, though I don’t think it’s the “standard” view anywhere outside of Canberra!

Nomological Necessitarians would presumably take the other side, since it’s a posteriori whether the laws of nature are deterministic. (They could still avoid fatalism by allowing the starting conditions to possibly differ, I suppose. But I don’t imagine they’d say this is a priori.)

Richard , November 1, 2006 07:04 PM

Thanks, Richard — that’s helpful. (I’ve fixed my error in leaving ‘possibly’ out of the relevant clause in the final question.)

The issue for me arises out of different readings of 2D semantics. If 2D semantics is genuinely two dimensional, then it’s obvious: if fatalism is false, it’s fixedly actually false.

In a straightforward 2D model in which sentences are evaluated at world-world pairs, where possibility is an S5 operator across one axis, and @ is true at a point (x,y) only if x = y, then in any model with more than one world, ~possibly@ is true at every point whatsoever.

So, on the reading of a priori knowability as fixed actual truth in these kinds of models, it’s a priori knowable that ~possibly@.

(The crucial part in the argument, in general, is that the points are pairs, and that the accessibility relation for fixed actuality commutes with the relation for necessity/possibility.)

If all goes well, I’ll talk about this (and other modal things) at the RSSS in December…

Greg Restall , November 1, 2006 07:22 PM

Oh, and on the substantive claim, concerning counterfactuals: isn’t the “if things had turned out differently, then ~@” true in a fatalist interpretation with no other modally accessible worlds? It’s a counterfactual with a necessarily false antecedent, and those are — on the standard reading of these things — all true.

Greg Restall , November 1, 2006 07:26 PM

Oh, yeah, my mistake. Easy enough to fix though: we know that “if things had turned out differently, then @” is false! (Or we can clarify that the other counterfactual is known to be non-vacuously true.)

Sounds like an interesting talk. A pity I’m leaving this month…

Richard , November 1, 2006 08:47 PM

I don’t think the fix is easy. The accepted reading for the negation of a would counterfactual is with a might counterfactual. ~(were A to be the case, B would be the case) is equivalent to “were A to be the case, ~B might be the case.” And here, I’m not sure that we know a priori that “were things to have been different, then it might have been the case that ~@”. At least, it doesn’t seem like much of an advance on “it is possible that ~@.”

Greg Restall , November 1, 2006 10:19 PM

Oh, and on the more general point where you say that we

can have a priori knowledge of every true modal proposition (assuming they’re all necessary truths, as per S5). I’m pretty sympathetic to this view, though I don’t think it’s the “standard” view anywhere outside of Canberra!

I think you shouldn’t mean that. One of the virtues of the 2D picture is the way that not all necessary truths (e.g., necessities of the form “it is necessary that a = b) are true in every point in the model. Neither are they true on all diagonal points, so they’re not a priori knowable.

Greg Restall , November 1, 2006 10:23 PM

I’ve always had a bit of a problem with the idea of “logical possibility” - I’ve never entirely understood what we can take it to mean, or what its implications really are. Chalmers uses it in a (to me) rather suspect way in the logical-possibility-of-zombies bit of The Conscious Mind. I guess in a way it comes down to whether one believes that Bertrand Russell could have been a fried egg, in any possible world (I don’t think so).

Anyway, I don’t think we can have a priori knowledge about even the possibility of a physical event occurring, somehow. (We can have instincts about, say, causality (following Kant, I think), but given that some of our instincts can turn out to be incorrect, I’d say that’s not real knowledge.) I would also tend to presume that anything of which we can have a priori knowledge would sit in the camp of “not possible that not p”. But that’s not what you’re asking.

I don’t believe that determinism and fatalism are the same thing, and it is a matter of scope (of a different sort) isn’t it? (I’m not really up on philosophical terminology these days, so I may be using an incorrect layman’s interpretation of that word). I mean: even if any particular state of the entire universe implies only one “following” state (which presumably is nonsense anyway), that doesn’t mean that macro-events are fated, or are unavoidable. But I guess I’m thinking of “determinism” as the scientific-materialist mechanist thesis, which I take to be pretty much what you call “fatalism” above, whereas I take “fatalism” to mean that all macro events were foretold or “caused” by something prior, and that we have no “choice” in some sense.

To apply that here, what I mean is that all other things being equal, I don’t think we can know a priori that a particular event “could have gone the other way”. We can know that a swimming race could have been won by the other guy, if this & that circumstance were different, but is that kind of “possibility” only a product of our ignorance of the fine details? Is it only because we’re letting so many variables, well, vary?

This haziness in our ordinary reasoning is really what bequeaths us our freedom, but it doesn’t really apply to the esoteric world of second-order logic. So I don’t know if I’m just talking crap or not ;) but maybe my half-layman’s view is of some interest. Or maybe it’s just hopelessly confused!

(oops, HTML formatting lost, but it was only italics)

Peter Hollo , November 1, 2006 10:48 PM

I’d have said it was fairly standard, among friends of the a priori, to say that lots of modal facts, including plenty of the sort you’re interested in, are knowable a priori.

E.g., the proposition that I’m standing (=p) is false. But I know it’s possible that I am standing. How do I know this? Plausibly, by some a priori means - e.g. by noticing that it’s conceivable that I be standing.

Suppose we deny the a priori knowability of this kind of fact, and demand that I gather empirical evidence before I can know that possibly I am standing. What sort of evidence could I get, given that p is false? Hard to say. Testimony perhaps, but how did my testifier find out herself? If we don’t accept that it’s a priori, therefore, a wide-ranging error theory, probably with sceptical implications, looms.

(Incidentally, I’m not meaning to commit to these views myself!)

Carrie Jenkins , November 1, 2006 11:23 PM

Yeah, I’m assuming that any modal proposition can be expressed in semantically neutral vocabulary, so that the primary and secondary intensions coincide (in being true at all points). Details here, but the upshot is that we have a kind of a priori access to the entire space of possible worlds: for every possibility, there’s a semantically neutral statement of it that is a priori knowable.

(That’s not to deny that there are also non-neutral statements that won’t be so knowable; but they’re less interesting for our purposes.)

Re: counterfactuals, do we really need all that analysis? I was hoping to get away with the following simple argument: (1) If the antecedent is impossible, then the counterfactual is vacuously true. But, arguably, (2) we have pre-theoretic knowledge that the counterfactual is false. Hence, (3) this knowledge entails that the antecedent is not impossible, i.e. that there about possible worlds where it’s true that “things turn out differently”, contra fatalism.

I guess the analysis highlights how the second premise is, in a sense, “question begging”. It includes the conclusion within it. But it’s also overwhelmingly plausible on independent grounds, so I don’t think that’s such a problem.

(One problem with my argument is that it “proves too much”, since analogues can be constructed using counterfactuals with genuinely impossible antecedents, e.g. “if 7 were one greater, it would still be prime” — which is intuitively just plain false, perhaps causing trouble for the standard analysis…)

Richard , November 2, 2006 12:52 AM

I do think there are some problems with your analysis, Richard. For a start, while we might have pre-theoretic knowledge about logical possibility, that’s by no means synonymous with physical possibility. It could be that the possible worlds in which things turn out differently are those in which the physical laws are different.

So the question is, when you’re evaluating the possibility of ~p (where p is true in this world), how much are you allowed to vary? And does it matter?

I’m pretty sure that a priori truths such as the primality of 7 are ones for which we couldn’t a priori know that they could be false. We could argue about, say, the Axiom of Choice, but that would just be silly ;)

Peter Hollo , November 2, 2006 09:07 AM

A couple of observations:

(1) As long as it’s a priori that there’s more than one possible world, it’s a priori that possibly ~@ (where here @ is understood as in comment 2 above, i.e. as quasi-indexically expressing a maximal proposition about the world of the speaker). One doesn’t need two-dimensionalism to get that. One just needs the plausible claims that (i) it’s a priori that @ holds in the actual world and that (ii) it’s a priori that @ is false of any non-actual worlds. So the question seems to reduce to the question: can one a priori exclude the hypothesis that there is exactly one possible world?

(2) Many two-dimensionalists and even non-two-dimensionalists hold some version of the following claim: when a sentence S is semantically neutral, then S is a priori iff its necessary. (Here semantic neutrality can be understood various ways for a non2Dist, but for a 2Dist it requires that all rows of S’s matrix are the same.) Many further hold that this principle itself is a priori. It follows that it’s a priori that if (semantically neutral) S is not a priori, then ~S is possible. Given the further plausible premise that there’s some neutral (a posteriori) truth S [e.g. ‘there are many philosophers’] such that it’s a priori that S is not a priori, we can conclude from these principles that there’s some false S such that it’s a priori that ~S is possible.

Of course one could resist this argument either by denying the 2D principles or by denying the “further plausible premise”. To do the latter, one might hold that all truths about the world are a priori (so that there’s only one conceivable world for the 2D intensions to range over), or else that there are some a posteriori truths, but none is a priori a posteriori.

djc , November 2, 2006 09:10 AM

Thanks, everyone.

David: the discussion in (2) is helpful and a different way of thinking from how I was approaching it. I’d got so far as what you went through in (1). The neat thing is that some motivations of 2D-like considerations motivate different answers here. Shall think some more about that.

Richard: yes, I slipped in my discussions of identity. If you think that any sentence has a semantically neutral equivalent, then you’re OK. But of course, a priori knowability doesn’t transfer across this “equivalence”.

And on the counterfactual question: I suppose it seems to me that — when I’m questioning whether or not fatalism is a priori knowable — I’d be just as happy to question whether or not the relevant counterfactual “if things had turned out differently, then ~@” is indeed a priori known to be false. But I agree, counterfactuals with impossible antecedents are hard to evaluate…

Carrie: they’re nice examples, which force us to be more specific on what we take a priori knowledge to be. I suppose my targets are Those Folks who think that a 2D semantic story is some kind of explication or analysis or something of what’s a priori knowable. Your examples make me wonder whether there’s a whole lot of knowledge of possibilities bound up with practical agency and our knowledge of our agency. How much of that is a priori in the relevant sense is obscure to me…

Peter: I think I used to be highly skeptical about metaphysical possibility, but now that I’ve got some ideas of how it works, I’m happier. I think it makes sense, and is actually a very important concept, deeply implicated in how we use many of our concepts. And, I think, the rules governing its behaviour are relatively straightforward. My interest at the moment is in trying to figure out what I say to the other elements of the 2D enterprise.

Greg Restall , November 2, 2006 08:30 PM

I don’t have anything to add, except to point out a typo:

“Let’s understand ‘possibly’ as a metaphysical sort of possibility. If it helps, think of it as truth in every possible world whatosever.”

I assume you mean “some possible world” rather than “every possible world whatsoever”.

Kenny Easwaran , November 4, 2006 12:24 PM

Thanks, Kenny! I’ve fixed the mistake.

Greg Restall , November 4, 2006 01:31 PM

Thanks Greg! I’ll definitely read your papers when I get a chance soon.

Peter Hollo , November 4, 2006 07:10 PM

I am a beginner so I hope my comment wouldn’t be “painfully” off the topic!

Consider the following example @1. (@1 and @2 are both propositions which are “true in actual world alone” and are “knowable”.)

@1: I don’t know that I don’t know that @2. (= I am double-ignorant of/about @2.)

(Maybe we can substitute @2 with a general proposition (e.g., q), but it is not clear for me whether @1 remains as a @-kind proposition, i.e. remains true “in actual world alone”.)

Rephrasing the original question:

(1) Can we know a priori that possibly ~@1?

(2) Can we know a priori that possibly ~ {I don’t know that I don’t know that @2}?

(3P) Can we know a priori that possibly ~ {I am double-ignorant of @2}?

(3N) Can we know a priori that it is not necessary that {I am double-ignorant of @2}?

Now given the “pre-requisites” and the fact that @1 and @2 are knowable, it seems the answer to the question is an affirmative one: our double-ignorance of knowable matters, by definition, is known a priori that can be removed. One knows a priori that there are no metaphysical necessities to keep him/her in the state of double-ignorance of/about a knowable proposition such as @2.

Hoss Parwas , November 24, 2006 12:08 AM