Tuesday, March 01, 2005

Contextual Impossibility

I want to reconsider something from my Adolf desire paradox:
Adolf desires that more desires be thwarted than fulfilled. [A 'desire that P' is 'fulfilled' if P is in fact true, 'thwarted' if P is false.] Further suppose that - apart from this desire of Adolf's - there are in total an equal number of fulfilled and thwarted desires in the world. It follows that Adolf's desire is fulfilled if and only if it is not fulfilled.

The problem with Adolf is that he was put in a context where his desire became self-referentially paradoxical. This situation could potentially arise whenever you have a desire that refers to 'all desires'. But it's logically impossible that a desire could have its own thwarting as a fulfillment condition - that's equivalent to a statement having its own falsity as its truth condition! So I think we must hold all such desires to be impossible. (It would seem odd to say they are possible some times but not others, depending on the external context. Why should the existence of an internal desire depend on the external world?)

Patrick suggested to me that the analogous liar sentence is not always impossible. Imagine a sheet of paper headed with the sentence: "Most statements on this page are false." But suppose the rest of the page contains an equal number of true and false statements. The original statement would now be true iff it is false. In other words, the context converts it into the liar sentence. Contradictions are impossible. Since it's impossible to have a statement truly asserting a contradiction, we must conclude that the apparent sentence is in fact meaningless (it refers to no proposition) in this context.

But we do not want to say the original sentence is always meaningless - that would seem clearly mistaken. For example, we can imagine the rest of the page is instead filled with false statements, in which case the original statement will be plainly true. It is not plausible to claim the statement is meaningless in such a harmless context. It is not really a liar sentence. It is merely a potential liar sentence. Whether the paradox is realised or not depends on the context.

This suggests that Adolf's desire should similarly be possible in appropriate contexts. It is unproblematic to desire that more desires be thwarted than fulfilled, so long as the sum of other desires is not exactly balanced between the two.

As I noted before, it does seem odd to say a desire is possible in some contexts but not others. But given the quasi-indexical nature of this desire, its context-dependence is perhaps not so surprising after all. In a sense, the meaning of "most statements on this page are false" will vary depending on which page it refers to. Similarly, the content of a desire that "most desires be thwarted" will depend on the background set of all desires that it refers to. In most contexts, the desire will be unproblematic (contrary to my earlier suggestion). Only in a few specific contexts will a paradox arise. So whether such desires are possible will depend on their external context.

Hopefully that doesn't sound too ad hoc. I guess an alternative solution to the liar paradoxes would simply be to accept true contradictions and adopt a paraconsistent logic, but I'm a little reluctant to do that.

11 comments:

  1. "Most statements on this page are false" yields a contradiction only in conjunction with the supposition that the rest of the page contains an equal number of true and false statements. So what is really "meaningless" is the utterance "Most sentences on this page are false, and the supposition is true." This is the internally inconsistent claim; the original claim is self-consistent and (I think) clearly meaningful.

    Similarly, to desire that more desires be thwarted than fulfilled is perfectly self-consistent; it's only made inconsistent when conjoined with certain suppositions.

    Speaking generally: I think it is somewhat misleading to talk about "contexts" here. Almost any proposition can be made problematic by being placed in certain contexts. "My grandmother is unhappy" yields a contradiction when conjoined with the supposition that my grandmother actually is happy. So I suppose we should say that "My grandmother is unhappy" is "meaningless" in such contexts. But it clearly isn't meaningless, in that context or in any other. The only thing unique about its being uttered in that context is that it cannot possibly be true.

    Similarly, I think, there's nothing meaningless about "Most sentences on this page are false," even in contexts in which the rest of the page contains an equal number of true and false statements. What would be meaningless would be to claim that most sentences on this page are false *and* to claim, at the same time, that that context obtains.

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  2. This reminds me of the "This Page has been deliberately left blank" in bursary exams. It must be taken as though that statement was not actually part of the page. And the same applies to "Most sentences on this page are false". It must be taken as an objective statement that is external to the page. The statement should be taken as exclusive of itself.

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  3. Greg, that was more or less my earlier proposed solution (see my post on 'immoral value'). But although that's an option, it also seems possible to interpret these claims inclusively. For example, "Most sentences [inclusive of this one] on this page are false" seems meaningful, and even possible in the right contexts. So I've tried to propose a solution which can account for this.

    David, I disagree. Your grandmother scenario does not threaten to yield a true contradiction, but merely a false one. As you say, "The only thing unique about its being uttered in that context is that it cannot possibly be true." That is unproblematic. It's meaningful to say "P and ~P", because it is simply false. It is not both false and true. Liar sentences are quite different from your everyday contradictions. They are paradoxical in a way that normal contradictions are not.

    More generally, I think it is unhelpful to put 'suppositions' in place of contexts. In effect, you are trying to move the background contexts inside the claims. But this is misleading and, I think, quite deeply mistaken. To claim "P", in context C, is not the same as claiming "P in context C". To say (1) "this sentence is false" is NOT the same as saying (2) "S is false and S is this sentence". (2) is not paradoxical; we can deny the right conjunct and conclude that (2) is a simple falsehood. But (1) is genuinely paradoxical; if it is false then that makes it true! This is a serious difference, so we must conclude that (1) and (2) are quite different claims. You cannot add contextual suppositions as an explicit part of the claim. They must remain implicit in the background context. (Put another way, we do not need to claim that a context obtains. It simply must be the case that the context does, in fact, obtain!)

    "Similarly, I think, there's nothing meaningless about "Most sentences on this page are false," even in contexts in which the rest of the page contains an equal number of true and false statements."

    Let's say I give you such a page, and ask: "Is the first sentence of this page true or false?" What do you answer?

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  4. Wahoo, you've come up with another liar paradox. Okay, a few questions, starting with possible internal flaws and working towards external ones.

    Firstly, under your conception of desire-fulfillment, is it actually possible to desire that most of your desires get thwarted? If every desire has a motive force, then one will simply seek to maximise their fulfillment - is it sensible to say that it's even possible for a person to desire what you suggest? Is that going against the whole idea of desire fulfillment as an explanation for human motivation?

    Secondly, there can easily be something meaningless about "most sentences on this page are false". A), some sentences may be meaningless, thus neither true nor false. B), there may be an even number of sentences, leaving it possible that "most" in not a valid concept for all possible circumstances.

    Thirdly, it's not the same thing, necessarily to say
    "I desire that most of my desires are thwarted"
    versus
    "For most of my desires, I more greatly desire that they are thwarted"
    This may have some implications; I'm not sure.

    Fourthly, I still don't believe in the law of the excluded middle, and it baffles me why anyone does, except perhaps when talking about physics. Linguistically speaking, it's clearly silly.

    Cheers,
    -MP

    p.s. just ideas to throw around - I had little desire, so to speak, to try to defeat your logic. It seems sound enough. I'm just teasing out alternative meanings and implications etc...

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  5. Or, alternatively, "All of philosophy can be reduced to this false sentence"

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  6. Richard,

    I agree that to claim "P," in context C, is not the same as to claim "P in context C"; I think that was my point in my last comment, though perhaps not.

    I see your point now about how (1) is paradoxical while (2) isn't. I didn't see that before.

    Here's a thought:

    Propositions normally imply that whatever state of affairs makes them true, obtains. So "Most sentences on this page are false" implies that there are more false sentences on this page than true ones. Perhaps propositions also imply that whatever state of affairs makes them non-paradoxical, obtains. In that case "Most sentences on this page are false" implies that there is not an equal number of false sentences on this page as true ones on this page.

    If so, then in answer to your last question, I would say: The first sentence on such a page is neither true nor false. But this (I would say) does not make it meaningless. The sentence in question clearly has truth conditions. It is true or false if the rest of the page does not contain an equal number of true and false sentences. So I would say that the sentence in question *implies*, in a certain way, that the rest of the page does not contain an equal number of true and false sentences. In this way, I would say, the sentence contradicts the facts of the matter, but it remains meaningful.

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  7. In general when one makes a statement it is oriented towards what existed before that statement not at the same time or after.

    Hence it in general does not refer to itself Since it makes litle sense that someone would actually state a paradox in any real situation it would be like me saying Me Coffee want and you taking that to mean that I want coffee as oposed to it being a nonsense sentance or somthing about what my cup of coffee "wants" (which it may be technically)

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  8. First, I'm not sure why liar sentences are meaningless to begin with. It seems to me that something can be impossible to be true because its truth would make it false and impossible to be false without its falsity implying its truth and yet still have a meaning. I know what the liar sentences means. I just don't think describing it as meaningless is the right way to go about it. People have been abandoning talk of sentences being meaningless for other philosophical puzzles when it was simply positivism that made people say they were meaningless (e.g. sentences about God or even anything about metaphysics). Why do we still talk of liar sentences as meaningless?

    Second, the problem as you stated it won't go. This can easily be fixed, but you need to be more careful in how you say it. You wanted an invlusive sense of "most statements on this page are false", but the inclusive sense just makes it come out false, but so is its denial. Slightly fewer than half o of the statements are false and slightly fewer than half are false. It's only if you don't count the current sentence that you get half of each. That's easy enough to fix, but as it is it's not right.

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  9. Sorry for the typos. That's what I get for trying to comment while getting my son ready for school. The only one that makes a difference is "Slightly fewer than half o of the statements are false and slightly fewer than half are false". The second 'false' should be 'true'.

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  10. It's worth noting that all liar sentences are dependent on context. For example, an instance of the liar sentence next to an arrow pointing to another sentence (like, say, "All ravens are orange") may be true. Context determines the sentence to which "this sentence" refers; the truth or falsity of that sentence then determines the truth of the liar.

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  11. Ha, nice point Michael! (I'd never thought of that before.)

    Jeremy, I don't understand why the inclusive version doesn't work. Suppose the top sentence is true. So slightly fewer than half on the page are false, so the top sentence itself is false. Then suppose it is false. Then over half of the page's sentences are false, so it is true. So the top sentence is true iff it is false, just as I wanted. Where's the problem?

    Fair point about the 'meaninglessness' response though. I discussed this a bit in comments to previous posts. If you allow it to be a genuine proposition, it's hard to see how one can avoid contradiction in case of "this sentence is not true". Giving up bivalence doesn't help here. Besides, what do you do in case of liar-desires, which seem to have a concrete reality? (Is it fulfilled or not? Saying 'both' or 'neither' seems particularly illegitimate here.)

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