Skip to main content.
May 19th, 2008

Analyticity and Intuitionism

Here’s a little argument that was inspired by some things Williamson says in chapter 3 of “The Philosophy of Philosophy”. It’s not at all the way Williamson intended his arguments to be used I guess.

  1. Any logical truth is true in virtue of meaning facts alone.
  2. Timothy Williamson is a philosopher is not true in virtue of meaning facts alone.
  3. Any disjunction with exactly one true disjunct is true in virtue of whatever the true disjunct is true in virtue of.
  4. So, Timothy Williamson is a philosopher or Timothy Williamson is not a philosopher is not a logical truth.

The premises could use being tidied up a little bit, but I think there’s something close to this in Williamson. Of course, he rejects (4), so he’s more interested in the argument from (2), (3) and the negation of (4) to the negation of (1). (Not that he would be quite as cavalier in the formulation of the argument as I’ve been.) Still, I think it’s a pretty interesting argument this way.

When I first saw this in Williamson, I thought, wow there’s a nice argument against the law of excluded middle. But now I’m worried that a structurally similar argument could, in principle, be run against the law of non-contradiction. I’ll leave it as an exercise for the reader to figure out the best way such an argument would go. I’m leaving it as an exercise in part because I’m not quite happy with any of my attempts, and in part because I’m too lazy. But unless I’m confident that no such argument could be used to reject LNC, I’m not going to be using this argument against LEM. And as of now, I’m certainly not confident of that.

Posted by Brian Weatherson in Uncategorized

12 Comments »

This entry was posted on Monday, May 19th, 2008 at 12:00 pm and is filed under Uncategorized. You can follow any responses to this entry through the comments RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.

12 Responses to “Analyticity and Intuitionism”

  1. P.D. Magnus says:

    Perhaps I am missing something, but I don’t see why I should accept 3. I would accept this variant, call it 3a: “Any disjunction with two independent disjuncts, exactly one of which is true, is true in virtue of whatever the true disjunct is true in virtue of.” A sentence and its negation are not independent, so 3a won’t help the argument.

    The only thing that worries me about 3a is that it appeals to logical independence, which I am not sure I can entirely explicate.

  2. Jonathan Ichikawa says:

    I think that “in virtue of X alone” may need clarification. On a weaker reading, maybe it’s sufficient for “p is true in virtue of X alone” that X is a sufficient truthmaker for p; that any time X obtains, p is true in virtue of X.

    On a stronger reading, “p is true in virtue of X alone” demands, not only that X is a sufficient truthmaker, but that there is no other fact that is also a truthmaker for p.

    Only on the stronger reading is the argument valid; a promising move for one wanting to resist the argument, I think, would be to claim that premise (1) is only true on the weaker reading.

  3. Brian Weatherson says:

    PD,

    Two related reasons. First, we might think of the truth tables as giving not just an algorithm for figuring out truth values of compounds, but of really giving the constitutive facts in virtue of which the compound is true. If so, the truth table for A or B tells us that it is true, if it is, in virtue of A being true. And the truth tables don’t come with a caveat – only good in case of logical independence.

    Relatedly, standard truthmaker theory says that the that the truthmaker for a disjunction with one true disjunct is the truthmaker for that disjunct.

    I’ll have to think more to have something to say about Jonathan’s comment…

  4. Gillian Russell says:

    The argument equivocates on ‘alone.’ ‘True in virtue of meaning facts ALONE’ in the first premise suggests something like ‘the meaning facts are sufficient to guarantee the truth of the sentence, no matter what the other facts are’. If that’s what you mean in P1 and P2, those premises are both true but the conclusion doesn’t follow: just because one empirical fact is also involved in determining the truth of the disjunctive sentence, it doesn’t follow that meaning facts wouldn’t have been sufficient to determine its truth whether that empirical fact held or not.

  5. Gillian Russell says:

    Just read Jonathan’s comment – I think he’s basically right.

  6. Alan says:

    Does “meaning facts alone” in (1) range over all possible worlds for logical truths, whereas that is impossible for a world-bound statement about TW? (This seems to me to be equivalent to JI’s statement put in a worlds-related way.)

  7. Charles Stewart says:

    If I understand the argument correctly, it would also show that if A is a logical tautology and B is a contingent fact, that “A or B” is not a non-logical truth, since among the grounds for the truth of “A or B” are the grounds for the truth of B, and so the grounds for the truth or “A or B” don’t contain only meaning-fact-based truthmakers.

    This example seems to support Jonathan Ichikawa’s objection to (1). Instead (1) maybe should say that a proposition is logically true if it contains any meaning-fact-based truthmakers. Other truthmakers are welcome to the party, wherever they come from.

  8. Aidan McGlynn says:

    If A is a logical tautology and B is a contingent fact, then ‘A or B’ isn’t a disjunction with exactly one true disjunct, and so 3 is just silent. So the argument doesn’t show that ‘A or B’ is not a logical truth.

    (Am I right in thinking ‘not a non-logical truth’ was a typo?)

  9. fitelson says:

    Brian — was the argument involving LNC you had in mind something like the following (this would appear to be the “dual” argument to your LEM argument, anyway)?

    1. Any logical falsehood is false in virtue of meaning facts alone.
    2. Timothy Williamson is a non-philosopher is not false in virtue of meaning facts alone.
    3. Any conjunction with exactly one false conjunct is false in virtue of whatever the false conjunct is false in virtue of.
    4. So, Timothy Williamson is a philosopher and Timothy Williamson is a non-philosopher is not a logical falsehood.

    If so, then I think this brings out (perhaps more starkly) some of the problems raised by commentators above.

  10. Brian Weatherson says:

    Branden’s suggestion is what I had in mind for the equivalent argument against LNC. Though I think we need some extra step to get to the really bad conclusion, namely that ~A is true in virtue of whatever A is false. And that I’m not so sure about. Maybe ~A is true in virtue of what makes A refutable. And if A is p & ~p, then what makes A refutable might not be what makes it false. At least that’s what I think I’d say if I was defending intuitionism.

    Aidan is right that premise 3 as stated isn’t vulnerable to criticisms concerning disjunctions with two true disjuncts. But I think it might be worth thinking harder (than I did) about those cases in order to get at what’s most puzzling about the case.

    If we have a disjunction with two true disjuncts, arguably there are two things in virtue of which the disjunction is true: the truth of the first disjunct and the truth of the second disjunct. In that case, it is potentially misleading to talk about what the sentence is true in virtue of. Perhaps one of the things it is true in virtue of is a meaning fact, and one isn’t. But this is very different from the way in which “Williamson is a philosopher” is true in part in virtue of a meaning fact (that the sentence means that Williamson is a philosopher) and in part in virtue of a non-meaning fact (that Williamson is not a philosopher).

    Now the question is, should we think that p v ~p is in the same boat? That is, there are two somewhat distinct things in virtue of which it is true. The first of these is that it has a true disjunct, which is not a meaning fact. The second is that its meaning is a logical truth (if it is). These don’t combine to form some larger story in virtue of which p v ~p is true. Rather, they are distinct grounds for the sentence’s truth, just like a disjunction with two true disjuncts has two distinct grounds for its truth.

    That could well be correct; I certainly don’t have a defence of that argument against such a criticism off the top of my head. It is interesting to work through whether a similar complaint can be made to some of the arguments against metaphysical conceptions of analyticity Williamson makes in Chapter 3 of “The Philosophy of Philosophy”.

  11. miguelhoeltje says:

    Consider the following argument:

    1. Any true disjunction is true in virtue of (at least) one of its disjuncts being true.
    2. ‘Timothy Williamson is a philosopher’ is not true in virtue of one of its disjuncts being true.
    3. Any disjunction with exactly one true disjunct is true in virtue of whatever the true disjunct is true in virtue of.
    4. So, ‘Timothy Williamson is a philosopher or Timothy Williamson is not a philosopher’ is not a true disjunction.

    I take 1, 2, and 3 to be clearly true, while 4 is obviously incorrect. Hence the argument is invalid. Yet the argument seems to be structurally analogous to Brian’s. Hence, Brian’s argument seems also to be invalid. What did I miss?

  12. miguelhoeltje says:

    Ok, that was way too quick – I just noticed what I missed. Nevermind.

Leave a Reply

You must be logged in to post a comment.