Wednesday, April 02, 2008

Subtracting Self-Reference

Brandon recently linked his old post on the Propositional Depth Response to the Liar Paradox:
Every meaningful statement must be assumed to have a determinate propositional depth...

L: [L] is false.

This has no determinate propositional depth. If we assume that L has a propositional depth of n, we find that, since L embeds itself, it must have a propositional depth of n+1.

I like this sort of account. (Alex recently suggested that one might arbitrarily choose whether or not to believe the proposition P: I believe [P]. My immediate response was to doubt that there really is any proposition here. The 'depth' account can explain why.)

A standard objection to this sort of view is that there would seem to be some true self-referential propositions. In an old guest post, Rad Geek suggested:
EM: [EM] is true or [EM] is false.

Now, it's not entirely clear to me that the above constitutes a wholly meaningful claim. (What is this '[EM]' it speaks of? I get stuck in an infinite loop if I try to fill it out.) But perhaps we can apply a lesson from Yablo and say that it is partly true. (I doubt his truthmaker account can actually accommodate this, but never mind that for now.)

Subtract out the self-reference, and what remains ("__ is true or __ is false") is true about logical forms, i.e. insofar as it claims that the law of excluded middle holds. The particular application is meaningless, but we can abstract away from that part of what's said.

Another puzzle case:
M: [M] is true and grass is green.

We certainly don't want to say that this is wholly meaningless. It's partly true: grass is green. Again, it seems that the thing to do is simply to subtract away the meaningless self-referential component.

P.S. Towards the end of his post, Brandon worries that the propositional depth solution commits us to the view that "whether the sentence has the same meaning, or any meaning at all, depends on purely contingent facts about the world that we may not be aware of." We should embrace this result, though, as Michael Sprague once pointed out to me:
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.


  1. What about pairings of sentences, like "The next sentence is true", "The last sentence is true"? If you think that these also fail the propositional depth test, then aren't you committed to rejecting all forms of holism about meaning?

  2. There are no doubt some true self-referential propositions.

    H: H contains four words.

    We must ensure that the self-referential sentence H (and infinitely many others) don't come out as meaningless on your account.

    The fourth sentence of the second comment on the "Subtracting Self-Reference" post at Philosophy, et cetera is false.

    It is also quite unappealing to hold that the prior sentence is meaningless, I think.

  3. Brian - H is a self-referential sentence. (Propositions don't contain words.) The propositional depth is still zero. See Brandon's post for more detail on this.

    Alex - I think those pairings fail the test, since they lead us into an infinite loop. I'm not sure whether "all forms of holism" involve such infinite loops.

  4. Ok. I see that the sentence

    (0) There is a ghost in the parlor.

    expresses a proposition that makes no further reference to a proposition. So has a depth of 0. And the sentence

    (1) The proposition expressed by (0) is true.

    expresses a proposition that makes reference to a further proposition. So has a depth of 1. And so on.

    Now what about a proposition that makes reference to itself.

    (P) The proposition expressed by (P) is a proposition.

    Lets call the proposition expressed by (P), 'Pete'. Pete looks like this:

    \\Pete, propositionhood//

    Isn't Pete a proposition? It seem to me like a Pete is true proposition with infinite depth. But I am not sure how these things are calculated exactly.

    (P) expresses a proposition that makes reference to a proposition, that makes reference to a proposition, that makes reference...etc. So, (P) expresses a proposition!

    But anyway how does this have any impact on the LIAR SENTENCE which makes no reference at all to a proposition?

    This sentence is false.

    There is no reference to a proposition, so it has a depth of 0. Just like" This sentence contains five words".

  5. Richard I'd be careful saying that propositions and sentences are so separated. Certainly for those who follow Frege they are. But one could well argue that a proposition is a sentence translated into an other kind of sentence or that a proposition is a sentence asserted or who knows what else.

    Of course someone who says, "this proposition has four words" is themselves committed to a particular way of viewing propositions that many (most?) would not accept. Which was your ultimate point.

    I think the idea of depth can be dealt with by simply refusing to allow self-referential propositions. i.e. a self-referential appearing sentence might be best broken down into different parts.

    So "this sentence is false" is simply bad because a sentence isn't true or false. If one adopts the assertion view then you get A asserts that their assertion is false. So I'm not sure one can avoid the problem too much. Although there are ways of dealing with this going back at least to Peirce.

  6. On holism:

    Well, it seems that you are committed, for instance, to the view that definitions of truth and falsity can't both make reference to one-another.

    Take the two propositions: "A statement is true when it is not false", "A statement is false when it is not true". These cannot be even part of the meaning of these terms on your account. (I'm obviously not saying that these are the whole meaning of these terms, but they are part, and that is sufficient to make them have an infinite depth.)


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