Wednesday, March 10, 2010

Omniscience and Dialetheism

(Thanks to M.C. for showing me the article.)

So in a 2007 article in Analysis*, Peter Milne suggests that omniscient beings would have to be dialetheists, given sentence S, below:

Sentence S: No omniscient being knows that which sentence S expresses.

His reasoning is that, if no omniscient beings exist, sentence S is "quite straightforwardly true." If an omniscient being exists, however, that being would have to both know and not know that which sentence S expresses. Thus, Milne seems to be suggesting, we have to choose between atheism and dialetheism.

Now, I am, to a slightly obnoxious degree, convinced of the deep irrationality of theism, but this particular argument doesn't strike me as a good one. Here's why: if this should force theists to accept true contradictions, then very mundanely familiar parallel arguments should force atheists to accept true contradictions as well. Whatever one thinks about the rationality of believing inconsistencies, the theist doesn't seem to be in any worse a boat than the rest of us.

Why, after all, should we think that an omniscient being would both know and not know the content of S? Because, presumably, if an omniscient being existed and didn't know the content of S, then S would be true, which would mean that omniscient beings would know it.

Why, however, does that last step--that "which would mean"--go through? Presumably because "some being is omniscient" simply means "some being knows the content of all true sentences."

Given this, on the assumption that an omniscient being exists, the claim that no omniscient being knows the content of S reduces to the claim that S isn't true. Put differently, it looks like S is the negation of the conjunction of the claim that there is an omniscient being and the claim that S is true...and is, thus, a pretty straightforward Contingent Liar. Presumably, if Liar sentences in general are problematic in some way--ungrounded, non-truth-evaluable, meaningless, whatever one's story may be--then S is problematic in exactly the same way. The fact that this particular Liar is rigged so as to have its paradoxicality not "kick in" unless an omniscient being exists changes nothing, just as, presumably, any plausible story about what makes Liars problematic should also apply to resolutely non-paradoxical cousins of the Liar like the Truth-Teller sentence ("This sentence is true.")


*"Omniscient Beings Are Dialetheists," Analysis 67.3, July 2007, pp. 250–51.


Emil O. W. Kirkegaard said...

I never heard of sentences expressing sentences. Don't you mean proposition, that is, which proposition the sentence expresses?

Ben said...

I don't see any reference in any of this to sentences expressing sentences. Milne talks about "that which the sentence expresses" w/o really specifying whether "that" should be understood in terms of propositions or in some other way, and I think also about the "content" of sentences, so, since nothing much seems to hang on this for the purposes of this argument, I follow his usage in the discussion of his article.

johnny_boy said...

You said:

"on the assumption that an omniscient being exists, the claim that no omniscient being knows the content of S reduces to the claim that S isn't true... and is, thus, a pretty straightforward Contingent Liar."

The Russell sentence R ('There is a set all of whose members are precisely the non-self-membered sets') is, modulo naive set theory, equivalent to a blatant contradiction P & ~P. But there is an interesting sense in which R does not reduce to P & ~P. For one, nobody ever thought P & ~P was obviously derivable in naive set theory. R, on the other hand, is. So from R, but not P & ~P, it was easy to see that naive set theory is inconsistent. That makes R interesting, and it also makes for paradox since naive comprehension seems plausible and so does R. But there's nothing plausible about P & ~P and, together with naive comprehension, nothing paradoxical about the pair's being inconsistent.

The same thing is going on with Milne's paper and it's not obvious (in fact it seems obviously false to me) that S is a "straightforward contingent liar".

Ben said...


Given naive set theory, a contradiction is entailed by R, but it's not by any means equivalent to the contradiction "there is a set that is a member of itself and not a member of itself," since the entailment only works in one direction. (This contradictory sentence no more entails R than it entails the claim that there is a set of all and only the ordinal numbers, which also entails it.) By contrast, given the assumption that there's a being who knows the content of all true sentences "this sentences is not true" and "the content of this sentence is not known by a being who knows the content of all true sentences" do seem to be equivalent--given the assumption just mentioned, the entailment works in both directions....if it's not known by the omniscient being, it's not true, and if it's not true, it's not known by the omniscient being.

More importantly, for there to be an interesting analogy here, Milne's paper would have to show that theism entailed inconsistency. This certainly seems to be his goal. But, while I'm quite friendly to the suggestion that this is true for other reasons, it looks to me like *this* doesn't show it. By analogy, a sentence that says that "the sentence written on the chalkboard is true" is paradoxical iff a paradoxical sentence is written on the relevant chalkboard, but this doesn't give us any special reason to suppose that a paradoxical sentence isn't written on the chalkboard, or that the belief that such a sentence is written on the chalkboard commits us to dialetheism, or anything like that. After all, we all know that *other* paradoxical sentences exist, so whatever problem the person who believes one to be written on the chalkboard has about the sentence "the sentence written on the chalkboard is true" isn't a bigger problem, or even a different problem, than the rest of us have about all the paradoxical sentences we *do* think exist. Similarly, given theism, Milne's sentence S is a paradoxical sentence, but it's not clear why the theist couldn't apply any solution that worked for standard paradoxical sentences to S. If there is no such good solution, that doesn't strike me as a bigger problem, or a different problem, than the rest of us have about regular paradoxical sentences. At the end of the day, I don't see that this has much of anything to do with theism.

Mickey said...

I agree with your overall analysis, Ben.

If I had to make one objection, it would be this. You said, "Thus, Milne seems to be suggesting, we have to choose between atheism and dialetheism.", but that seems incorrect. Consider that theists, such as myself, as "irrational" as our belief are, could just modify the understanding of 'omniscience' to something else which dodges the dilemma. Or, we could just abandon the idea of an omniscient God. Either strategy is a far cry from atheism. Further: theists would not even have to give up the idea that God is a perfect being. For nothing about being a perfect being entails omniscience unless omniscience is both possible in and of itself and compossible with God's other properties and the set of all external facts.

Ben said...


Fair enough. I guess "force us to choose between accepting dialetheism and rejecting classical theism" might have been a better formulation.

(In general, there's a case to be made that analytic phil of religion tends to smuggle in the assumption that the only conceptual options are no Gods or the omni-omni-omni God, and I'm probably guilty of aping that here.)

That said, I'm a bit curious about the suggestion that, if the paradox went through, theists could tweak their conception of omniscience to dodge the problem. I'm not sure I see how the move would work there.

(FWIW, my general stance on theism is for the usual boring reasons. I find classical theism irrational because of the Problem of Evil, and to a lesser extent because standard solutions to omnipotence paradoxes always seem a bit ad hoc to me, and I find theism in general a bit irrational for standard evidentialist sorts of reasons. That said, like I said in the post, I'm thoroughly unconvinced that Milne's paradox poses a special problem for those who believe in omniscient beings, even where "omniscient" is understood in the classical unrestricted sense, above and beyond the general problem posed to everyone by the Liar Paradox.)