A being X is omniscient iff:So for fun….
(i) for every true proposition p, X knows p, and
(ii) there exists no proposition p such that X believes p and p is false.
1. Let p be the proposition: “God does not know p.”That is, p is a self-referential proposition.
To put this into words, for those who are less inclined toward reading logic, the original definition of omniscience is, in language form: “An omniscient being knows all true statements, and does not know any false statements.” Now, orthodox Christians believe God is omniscient, so we can examine a specific statement, the statement: “God does not know this statement.”2. Suppose p is true.
3. Therefore, p is known by God, per (i).
4. If p is known by God, then p is false.
5. Therefore, if p is known by God, (ii) is false.6. Suppose p is false.
7. Therefore, p is not known by God, per (ii).
8. If p is not known by God, then p is true.
9. Therefore, if p is not known by God, (i) is false.
Typically, we would say the statement, “God does not know this statement” is either a true or false statement. If it is true, then God does not know the statement “God does not know this statement.” The result is that there is a true statement that God does not know. But the definition of omniscience includes in it the fact that “An omniscient being knows all true statements.”
On the other hand, if the statement is false, then we are saying God does know the statement “God does not know this statement.” But that means that God knows a false statement, which violates the second part of the definition: “An omniscient being…does not know any false statements.”
As I said, typically we would say the statement, “God does not know this statement” is either a true or false statement, but now we realize that the statement is neither true nor false; or rather, the truth value of the statement is in a constant state of flux. If it is true, then it is false; but if it is false, then it is true. So does God know these types of statements? The above definition of omniscience cannot tell us.
The result is that, at best, the above definition of omniscience is incomplete—it does not take into account statements that have variable truth-value. In other words, this definition of omniscience didn’t take Gödel into account.
Yep, this is a fine example of Gödel's incompleteness theory in action (used WRT omniscience).
ReplyDelete1. Our 'human' definition of omniscience is clearly too small.
There are properties of omniscience that haven't been captured in the definition presented above. It is doubtful any human definition of omniscience will pass the test as omniscience is likely beyond mere formulations.
To say "God IS omniscient" means that the only way to fully describe omnisciences is to fully describe God (because God is the only one who exhibits this property) - so it makes sense that our definition is incomplete.
2. Even though Gödel's theory is being used here to show our definition of omniscience is weak (incomplete), his theory is actually more of a threat to humanistic secularism than faith given that he fundamentally proves science cannot explain everything. In this example above, its actually showing logically that defining God precisely is impossible.
ἐκκλησία said:
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It is doubtful any human definition of omniscience will pass the test as omniscience is likely beyond mere formulations.
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I agree. Ultimately you have to choose either precision with incompleteness, or imprecision with completeness. For instance, I use the definition "Omniscience is knowing everything that is possible to be known regardless of truth-value." That is a more complete definition now, but it lacks precision. It doesn't tell us what thinks are possible to be known.
Heh heh. By "thinks" I mean "things" of course. It's always fun trying to post something right when you get a phone call :-D
ReplyDeleteWell, Peter, your argument is fun, but unconvincing.
ReplyDeleteYou see, you seem to be relying on the assumption that "proposition" is equal to "statement," which I do not believe was intended by the author.
You said that the claim was synonymous with “An omniscient being…does not know any false statements.”
But that can't be what the author means, since obviously if I say "I am having Pizza right now," then the author would believe that God knows that I am not eating pizza right now, but still knows that I just said I am. (How else would God keep track of lies?) Therefore, God *knows* a statement, "I am eating pizza right now," that is not true.
So say that God knows a true proposition means that He knows something that is true, and to say that God does not know a false proposition is to say that God never believes an untruth.
You also said "Typically, we would say the statement, 'God does not know this statement' is either a true or false statement," and then go on to have fun with the idea that God "only knows true statements." But, one wouldn't normally call a paradoxial statement either true or false. In context, that statement is the equivalent of "This statement is false" which is either nonsense or a paradox, and can neither be true or false.
Therefore, while it is true that that one statement, in most contexts, would be classified as either true or false, IN CONTEXT, it is more analogous to a statement which logical people can neither define as true or false.
This paradox comes up in many tales and riddles. For instance, in one story, the king told a wise man that he could make one statement, and if he decided that the wise man had spoken truth, the man would be hung. If we decided that the wise man had spoken something falsely, he would be beheaded. If the king could not decide, or didn't know, then the wise man would go free. The wise man thought and then said "You are going to behead me."
Without context, we might call that statement true or false, but in context, the King could not call it true and then make it true without breaking his word (which he wasn't willing to do) and couldn't call it false and make it false without also breaking his word. Therefore, he freed the wise man.
Similarly, the statement "the following statement is true" can usually be classified as true false, except in context when the following statement is "the previous statement is false."
In any case, it can still be maintained that God knows all truth, does not believe anything false, and also knows paradoxes.
On the other hand, Chuck Norris DOES know if the statement "This statement is false" is true or false.
This seems like it is just a fancier version of the ol' "This statement is false" game.
ReplyDeleteOr, in other words, "Let p be the proposition: p is false."
It's a meaningless "proposition."
Hi Skarlet,
ReplyDeleteActually, propositions are statements. In any case, if you don't like the word "statement" feel free to do a global replace with the word "proposition" and my argument still stands.
To clarify a few things, you said:
You said:
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You said that the claim was synonymous with “An omniscient being…does not know any false statements.”
But that can't be what the author means...
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What he said was:
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(ii) there exists no proposition p such that X believes p and p is false
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The only quibble you could introduce is that he says "believes" instead of "knows" there, but since knowledge is a form of belief anyway I don't see how that distinction helps. Indeed, in the document, he tells us: "Given the common assumption that the objects of knowledge are true propositions..." So it is clear he's using the word "know" as already having been established as "true", and I believe that's the only reason he didn't use the word "know" in the second half.
Be that as it may, it doesn't dodge the problem, for the proposition can simply be changed to "p is the proposition: 'God does not believe p'" and the problem returns.
You said:
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But that can't be what the author means, since obviously if I say "I am having Pizza right now," then the author would believe that God knows that I am not eating pizza right now, but still knows that I just said I am. (How else would God keep track of lies?) Therefore, God *knows* a statement, "I am eating pizza right now," that is not true.
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That's not quite accurate under what is being proposed. It would instead be something along the lines of:
1. Given, a false statement, s.
2. p is the proposition: s is a false statement.
Notice that p is true. In other words, God doesn't know "Skarlet is eating a pizza." Rather, He knows: "The statement/proposition 'Skarlet is eating a pizza' is false." Since that statement is a true statement about a false statement, it would satisfy the definition of omniscience provided.
Finally, to conclude, you said:
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In any case, it can still be maintained that God knows all truth, does not believe anything false, and also knows paradoxes.
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But adding "and also knows paradoxes" goes beyond the defintion given, proven that the definition given is, indeed, incomplete. After all, the definition provided says nothing of paradoxes.
Try to replace the symbol p with text that it represents, and you'll see the problem with self-referential propositions.
ReplyDeleteMesa Mike said:
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It's a meaningless "proposition."
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Ah, but it's not meaningless. It establishes the limits of certain structures of thought, demonstrating that there is no "complete" theorum that can account for all true statements. In other words, it proves that you will never be able to form enough axioms to be able to prove every single truth that can be known.
To say it's a paradox or to say it's meaningless is to miss the forest for the trees, then. See, there is nothing intrinsic to the proposition--especially if it's restructured the way that Skarlet did, which breaks it into two propositions--to make the statement "meaningless." It is the structure itself that determines that.
In a way, it's similar to the Cantor diagonal in math. Cantor proved that an infinitely-long list of numbers will be missing an infinite number of numbers from that list of numbers. Yet which numbers are missing is entirely a function of the structure of the original list! That is, it is the list itself that excludes those numbers, not anything inherent in math.
In a similar manner, Gödel Incompleteness Theorum shows that it is the axioms of a system itself that makes it impossible to prove certain true statements within the system itself. This is very profound and deep, not at all meaningless. :-)
Peter, I'd love for you to do some more posts involving GIT, since it has boggled my mind for several months now. Quick question, though: can GIT be applied to itself? There appears to me that GIT could almost be self-refuting, but it could be that I just don't get it.
ReplyDeleteRyan,
ReplyDeleteI might be able to do that. I'm not as good at his mathematical proofs, since he uses math that's beyond me; but people (such as Hofstadter) have used his theorums in many different ways that I can grasp and understand easily. In fact, I would recommend you read Hofstadter's "Gödel, Escher, Bach" if you're really interested in it. Of course, it goes well beyond Gödel's Incompleteness Theorum, but I find the book fun.
Let me modify that, then, and say that, rather than being meaningless, the meaning of such self-referential propositions is undefined.
ReplyDeleteSo, rather than say that the truth value of the statement is in a "state of flux," I'd say it is "undefined" -- much the same way that the square root of infinity is undefined.
But you're right. The given definition of omniscience only takes into account statements that are either true or false, but not ones that have undefined truth value.
On the other hand, so what? Does a definition of omniscience (or anything else for that matter) have to take into account that which is undefinable?
Mesa Mike asked:
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Does a definition of omniscience (or anything else for that matter) have to take into account that which is undefinable?
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Well, the "omni" part of omniscience would seem to suggest that :-D But to answer your question more seriously, it would depend on what you were using your definition for. If you're using it just to discuss something informally amongst friends, you can keep the terms loose. But if you're trying to make rigid arguments where the definition is critical, then it's important to keep the implications in mind.
BTW, I wouldn't use the word "undefined" though. If we're using math terms, I would say probably we're dealing with an "imaginary" aspect here (i.e, such as the square root of -1). Or possibly even "transcendent" in the sense of "transcendent numbers." Both are critical for an understanding of mathematical concepts in ways that something being "undefined" isn't. But I won't quibble that point with you :-)
"I would recommend you read Hofstadter's "Gödel, Escher, Bach" if you're really interested in it."
ReplyDeleteFlipped through it at the library a couple times, but it's a bit above what I can afford right now. I liked the inclusion of several of Zeno's Paradoxes.
Ryan said: Quick question, though: can GIT be applied to itself? There appears to me that GIT could almost be self-refuting, but it could be that I just don't get it.
ReplyDeleteRyan, Gödel's theorem says basically that any axiomatic system (such as logic or mathematics) is either:
1. Complete (but inconsistent); or
2. Consistent (but incomplete)
What does that mean?
1. Complete (but inconsistent) means that all true propositions that can be constructed from some given particular axiomatic system can be known, but that system MUST BE contain self refuting propositions (therefore it is inconsistent).
2. Consistent (but incomplete) means that some particular axiomatic system has no internally self-contradicting propositions, but that it MUST THEREFORE BE unable to establish all true propositions; which means there are some true propositions left out (hence incomplete).
The inability to establish all truth is a huge limitation to knowledge as it means some truths must be taken strictly as self-evident (which suggests some truth is only obtained by faith).
Gödel made no claim that his theorems were complete (and they are consistent). They have been proven to be consistent both with themselves and with our entire body of propositional logic.
This means that his theorem together with all propositional logic will only ever be incomplete in the sense that there will be some propositions that cannot be proven true.
But he himself, made no claim to be able to establish all truth, and he recognised them as being incomplete WRT to that larger body of logic.
So establishing as true, the inability to arrive at some truth by way of axiomatic methods does not mean that we cannot establish as true his particular theorems .
There are many such proofs that prove his theorem true, but as Mark Twain once wrote “Few things are harder to put up with than the annoyance of a good example “.
Gödel needed only to provide a single example of a true but unprovable statement to establish the truth of his theorems. which he did.
Let proposition P be the statement "This statement P is unprovable"
If the statement P is true, it cannot be proven (or it would be false, not true)
If the statement P can be proven true, it makes no sense, because the statement itself says that it cannot be proven true. It is self-refuting.
Thus, the proposition P can only be true (but incomplete, because it cannot be proven true); or
it can be self contradicting.
ἐκκλησία, thanks for that explanation. It was helpful.
ReplyDeleteI propose considering a new Reflective Paradox fallacy. The definition of omniscience being addressed seems a little askew anyway, but I think it stems from a failure to understand the difference between the logical univalence of the eternal Creator, the only one who could be omnicient, and the bivalence of the created order by which we attempt to apprehend eternal things, of which are the existential objects of omnicience.
ReplyDeleteInstead of GIT, how about MIT?
ReplyDeleteMIT = Magisterium's Incompleteness Theorem.