Tuesday, August 05, 2008

Liars, Divine Liars, and Semantics

Let ‘L’ name the sentence, “This sentence is not expressing a truth.” L seems to be saying that L is not being used to say anything true. If so then, if L is expressing a truth then L is not expressing a truth, whence L is not expressing a truth. But then, L seems to have been expressing a truth—that L is not expressing a truth—after all. In short, L is a paradoxical Liar sentence.
......Letting ‘N’ denote (for brevity) the property of not expressing a truth, L seems to say that L is N, but hardly straightforwardly. If we let T be “L is N” then I can say that maybe L only seems to say that L is N because of its resemblance to T. Note that if L is expressing a truth—if it is not N—then L is N (paradoxically), but if T is not N then it only follows that L is N (relatively straightforwardly).
......On this (fairly popular and) Traditional Resolution, L is a special sort of nonsense (and hence not true), its apparent sense being due to its resemblance to T (which is simply true). A related paradox is therefore Moore’s. Let G be the sentence “George believes that S is certainly P, but S is not P.” G may be true, but if George utters G there is something odd about it (even with “George believes” replaced by “I believe), and were George omniscient he could only say G by lying.
......There are many other ways to resolve the Liar (e.g. Kripke’s, e.g. see subsection 3.2 of the recent SEP entry on Self-Reference), but this is certainly one possibility. And why should we expect there to be a unique resolution? Natural linguistic entities are usually only partially (and fuzzily) defined—maybe such incompleteness facilitates their flexibility?—and Liar sentences are relatively artificial, so it may well be that different resolutions yield different (but equally legitimate) extensions of our languages.
......Atheistic uses of Divine Liars (as follows) have therefore begged the question (so far as I can see). Let ‘DL’ name the sentence “God knows that this sentence is N.” If DL is not N—if it is expressing a truth—then God (an omniscient being, here) exists, and DL is N. So DL is N. But if DL is N then God should know that DL is N, which is what DL seems to be saying; so the atheist is tempted to conclude that God does not exist.
......A theist, on the other hand, could regard DL as some evidence that such appearances are deceptive, at least when it comes to sentences like the Liar (and to a lesser extent with Moore’s paradox), and an intelligent agnostic would be unable to make much of DL in the absence of compelling reasons to regard a different way of resolving such sentences (e.g. Kripke's) as the only correct one (not just the conventional one, as that would certainly beg the philosophical question).
......Incidentally it is usual to use non-well-founded names to define Liars (as Grim defined his Divine Liar, see second comment below), e.g. F = “F is N,” which creates additional problems for this approach (e.g. saying “F is N” is then to speak nonsense), so note that the atheist would additionally need to justify such an extension of the natural process of naming, to reap any benefit from such problems. (Prima facie it would be wise to analyse Liars and such naming seperately.)


Andrew Cullison said...

Isn't it also open to the theist to hold that omniscience is knowledge of all true propositions (that are knowable).

Consider (D)

(D) is true iff (D) is not known...

Some have used this to argue that God does not exist...but it seems like (D) is a proposition that is metaphysically impossible to know - a similar response can be invoked by the theist and maintain that omniscience is restricted to metaphysical possibility.

It seems like a natural move given that this is already done for omnipotence.

Enigman said...

I don't know; if (D) is that set-off biconditional, why can't (D) be false? If the biconditional is false then (D) needn't be known... Anyway, Grim's sentence was:

(4) God doesn't believe that (4) is true.

If (4) is true then (4) is a truth that God doesn't know. But (4) is knowable, e.g. I might know that (4) is such that God couldn't know it. One problem is that it doesn't seem to be part of the meaning of 'omniscience,' that an omniscient being could fail to know something that I know.

We should rather call God something else, in such a case ('superscient'?). Now, I might know some truth via epistemic luck, so that since God's epistemic standards are so much higher than mine, so he wouldn't know that truth, but maybe that's why (4) uses 'believe' and not 'know.' Maybe...

Anyway, if (4) isn't true, for whatever reason - or if it doesn't express, or isn't expressing a truth, in some strengthened version of (4) - then God clearly doesn't have to know it, and so a similar problem is then that God wouldn't believe that (4) was true.

That's a problem because if it makes sense for me to say that - that God doesn't believe that (4) is true - then prima facie (4) makes sense; and indeed, it seems to be true and knowable (by me, at least). It could then seem that God should also know (4).