bd-from-kg

This is an attempt to prove that the concept of omniscience, and hence the Christian concept of an omniscient God, is logically incoherent. I’m not sure whether it has been presented before as an argument against the existence of God, but the paradox itself is hardly original with me – it was discovered by Russell around 1900. Anyway, it goes like this:

God must know, for every proposition p, either that it’s true or that it’s false. Not only that, but He knows all of these things simultaneously, and thus knows, of the class TP of all true propositions, that each proposition in it (and no other) is true, and of the class FP of false propositions FP that each element of it is false. So God’s mind must contain the concepts of TP and FP themselves. It follows that God’s mind must include the concept of P, the class of all propositions, since this is just the union of TP and FP. [Note: the point of this is just to establish that P exists as a conceptual entity – a “set” if you will – since it exists as a concept in God’s mind.]

Now for any set of propositions m, let T(m) be “every proposition in m is true”. [Note: it doesn’t seem to matter just what T(m) asserts; the only thing that matters is that T(m) is a proposition, and that if m and n are different sets of propositions, T(m) and T(n) are also different.]

Now God’s mind must include, for each such m, the knowledge of whether T(m) is itself one of the propositions in m. And thus God’s mind must include the knowledge of exactly which T(m)’s are members of their corresponding m’s. That is, He knows the contents of w* == {m: T(m) is not a member of m} and hence of w, whose elements are T(m) for every m which is an element of w*. That is, w consists of all of the propositions T(m) which are not elements of the set m to which they refer. [Note: Once again the statements about “God’s mind” are used only to establish the existence of w as a conceptual entity – i.e., a “set”.]

Since w is a set of propositions, we also have the proposition T(w). Now consider the question whether T(w) is a member of w.

If T(w) is an element of w, then it isn’t, because w consists of precisely those T(m) which are not elements of the corresponding m.

But if T(w) is not an element of w, then it is, again because w consists precisely of those T(m) which are not elements of the corresponding m.

This is an unavoidable consequence of the fact that w is a set – i.e., a conceptual entity - which is in turn an unavoidable consequence of the fact that God knows which propositions of the form T(m) are elements of their corresponding m. And this in turn is an unavoidable consequence of the fact that God knows, for any set of propositions m, whether T(m) is an element of that set.

In short, the logical paradox is an unavoidable consequence of the assumption that God is omniscient.

So God cannot be omniscient. In fact, the concept of omniscience is logically incoherent.

Therefore the concept of an omniscient God is logically incoherent.

Comments?

[Edited to fix definition of w, in which crucial "not" was omitted.]

[ February 07, 2002: Message edited by: bd-from-kg ]</p>

GunnerJ

::Sweeps hand, palm down, over his head, saying, "Whoosh!"::

Vorkosigan

My head hurts. I lost you around t(w). English, please. I assume that this proof would not apply to an entity that is not omniscient.

Also, the union of the classes TP and FP would not include propositions that were meaningless. Or would those be false under your conditions?

Let's see if I can rewrite this in English....

There is a set of propositions, called Miami. There is another proposition, called T-Miami. T-Miami lies outside of Miami, but it says that everything in Miami is true. T-Miami could be in Miami, or it could be elsewhere. Only God knows. God knows this about every such proposition. For example, he knows T-Atlanta is in Atlanta, and T-Detroit is in Detroit. Not all T's are actually in the set, it might be that T-Dallas is not in the set Dallas and T-Miami is not in Miami, but T-Pittsburgh is in Pittsburgh.

Let's assume that T-Dallas is not in Dallas, but T-Miami is in Miami.

Now we form another set out there called ALTUSA. ALTUSA includes all the T's about the cities, but only those T's that are not located in the cities. So it does not include T-Atlanta and T-Miami and TR-Detroit, because those are in the cities. It DOES contain T-Dallas, because it isn't in Dallas. It DOES contain T-Seattle, because T-Seattle cannot be found in Seattle.

Now, ALTUSA contains all those propositions, the T's. Now there's another T, a meta-T. It describes ALTUSA. T-ALTUSA says that all the propositions in ALTUSA are true.

Now, if we looked in ALTUSA, would we find T-ALTUSA there? Hmmmm.....

T-ALTUSA cannot be in the set ALTUSA, because ALTUSA contains only those T's that are NOT in their sets. Like T-Dallas. So, we kick it out. It belongs OUTSIDE ALTUSA because it is a T that is not in its set.

Problem: ALTUSA MUST contain T-ALTUSA, because it is a set of all T's that are NOT in their sets. So the universe kicks it back. It belongs in T-ALTUSA.

We look at it again. Doesn't belong, since ALTUSA contains all T's that are NOT repeat NOT part of their sets. So we boot it out....and the universe kicks it back....and we boot it out....

Wow!

Nope....I couldn't rewrite it in English. But it's at least clear to me.

Michael

[ February 07, 2002: Message edited by: turtonm ]</p>

Kenny

This appears to be along the same lines as the paradox described by Patirck Grim in the following exchange with Alvin Plantinga.

<a href="http://www.sunysb.edu/philosophy/faculty/pgrim/exchange.html" target="_blank">Truth, Omniscience, and Cantorian Arguments.</a>

God Bless,
Kenny

xoc

w* is where I am confused. I'm either not familiar with the notation or w*==, or something else is going on. Does * imply multiply and == mean equivelent? Or are you saying that w is NOT equivelent to T(m)?

Vorkosigan

Thanks for the site cite, Kenny. An excellent bit of reading, even if my head aches even more now.

Kenny

Well, if you have any extra Tylonal, by all means share

bd-from-kg

xoc:

w* is just a name I picked for a set closely associated with w. It could just as well have been w'. I should probably just have said w == {T(m): T(m) is not a member of m} and avoided defining w* entirely. As for ‘==’, this just means “is defined as”. I should have used ‘≡’, but didn’t realize until now that this symbol is available in the standard symbol set.

turtonm:

Sorry, this argument really can’t be presented in any reasonable way without using symbols. But you seem to have gotten the gist of it. However, I’m not sure that you really grasped the import, which is that there can be no such thing as the totality of all truths, or of all propositions (or of all true propositions) or anything of the sort. But the existence of an omniscient God seems to intrinsically involve the existence of such a totality (i.e., God’s mind contains all truths, so the totality of all truths, being a definable part of the contents of God’s mind, must form a “completed whole&#8221 ) . Actually it doesn’t matter whether one refers to such a totality as a “set” or something else; any “completed whole” of this kind gives rise to various logical paradoxes, such as the one I presented.

To all:

At the risk of increasing the sale of Tylenol even further, let me attempt to help clarify what’s going on here.

A couple of simpler examples of this kind of argument (both of which are closely related to this one in different ways) might be helpful. Anyone who is unfamiliar with these is going to have a very hard time understanding the argument I presented.

The first is Russell’s (original) paradox, which shows that there cannot be a set which contains all sets as elements (and which therefore showed that one cannot throw around terms like “set” casually, or assume that any collection that can be “defined” can actually be conceived).

Before starting, we should note that the standard notion of a “set” allows a set to be an element of itself. For example, let N be the set of all sets that I’m thinking about right now. Then N is an element of itself. Now let S be the set of all sets. Clearly S, being a set, is an element of S. So far, so good. But now let V be the set of all sets that do not contain themselves as elements. This set obviously exists (if S does) since it is a clearly definable subset of S. But does V contain itself as an element? If it does, it doesn’t, because V contains only sets that do not contain themselves as elements. But if it doesn’t, it does, because V contains all sets that do not contain themselves as elements. This contradiction shows that V cannot exist (or more precisely, that the concept of V is logically incoherent, like the concept of a square circle). But since the existence of S unavoidably implies the existence of V, S cannot exist either (i.e., the concept of a set of all sets is logically incoherent).

The second classic argument of this kind is Cantor’s “diagonalization” proof that the power set of any set is strictly larger than the set itself. Thus, let S be any set and P be the set of all subsets of S. Now if S is finite, P is obviously larger than S: if S has n elements, P has 2^n elements. But if S is infinite, the meaning of “larger” is not immediately clear. In fact, come to think of it, the meaning of “the same size” isn’t all that clear either. Cantor gave a clear meaning to these terms which is used by all mathematicians nowadays.

Cantor's solution is as follows: we say that a set Q is at least as large as a set R if there is function that “maps” Q onto R. That is, Q ≥ R (in size) if there is a function f such that (i) for any element q of Q, f(q) is an element of R, (ii) for any element r of R, there is a q in Q such that f(q) = r. And of course Q = R (in size) if Q ≥ R and R ≥ Q. (Note that in this formulation the mapping does not have to be one-to-one.)

So, how do we know that for any set S (even if it’s infinite), the set P of all subsets of S is larger than S itself? Why, because S is not “at least as large” as P! In other words, there is no function f that maps S onto P. And how do we know that? Well, suppose we have a function that supposedly does this. We’re going to define a subset z of S (i.e., an element of P) that f doesn’t map any element of S to. We do this as follows: any element s of S is an element of z if and only if s is not an element of f(s).

Now let s be any element of S. Does f(s) = z? No, it does not, because s is an element of f(s) if and only if it is not an element of z. So f(s) cannot be identical to z. Since this is true for all s, f does not map anything to z, and so does not map S onto P. And since this is true for all f, there is no mapping of S onto P. So P is strictly greater than S in size.

It should be clear that the argument I presented originally uses ideas from both of these proofs; all of them have the “diagonalization” idea in common. An argument involving the “set of all propositions” is inevitably a little more complicated because one has to “invent” propositions corresponding to subsets of the supposed “set of all propositions”.

It may be worth noting that Cantor’s argument also has a more direct bearing on the question of whether the concept of God is logically coherent. Thus, God is often defined (e.g., for purposes of the Ontological Argument) as something like “that than which nothing greater can be imagined”. It seem “obvious” at first sight that this is a coherent concept, just as it seems “obvious” that the concept of a “set than which no larger set can be imagined” is logically coherent. But the latter concept is not logically coherent: for any set, a larger can be imagined, namely its power set. This seems to create serious doubt as to whether the concept of a being “than which none greater can be imagined” is really logically coherent, even aside from the question of whether omniscience is a logically coherent concept. In fact, I suspect that this can be turned into a rigorous argument. But since God can be defined in other ways, and since the Ontological Argument can be shown to be fallacious on other grounds, it may not be worth the bother.

It might be worth noting that in all of these cases there is an element of self-reference, and this is both what makes the arguments confusing at first sight and what makes some of the concepts involved logically incoherent. Thus the “set of all sets” must contain itself as an element, and among the things that an omniscient being would have know would be the contents of its own mind – in other words, what it knows. But while the concept of the “set of all sets” can be dispensed with without throwing out the concept of a “set”, the idea that an omniscient being would have to know the contents of its own mind cannot be dispensed with without twisting the concept of an “omniscient being” beyond all recognition. More about this later.

[ February 08, 2002: Message edited by: bd-from-kg ]</p>

Kenny

I think the shaky premise in the argument is that God’s omniscience demands that there actually be a set of all truths. As Plantinga points out on the link I provided, God’s omniscience could simply be conceived of the property of “knowing all true statements and believing no false ones” without having to say that God retains the set of all truths in His mind (since there is no such set). It is not immediately obvious that the coherence of the property of “knowing all true statements and believing no false ones” implies the existence of a set of all true propositions.

God Bless,
Kenny

bd-from-kg

Kenny:

It’s nice to see you actually posting something substantive for a change instead of just posting yet another link to an incredibly long, complicated discussion and then disappearing into the night.

That said, you’re quite right that Plantinga denies that the existence of an omniscient being implies the existence of a set of all true propositions, or of all truths, or whatever. And his objection does not merely apply to the term “set”, but to any term that refers to such a totality, whether it’s called a class, or a collection, or whatever. But I don’t see how this objection can be taken seriously. If God exists as an actual, concrete entity, then the contents of His mind must exist as an actual, completed thing – i.e., a totality. And this means that the collection of propositions that He knows to be true (for example) must also exist as a completed whole, as a part of God’s mind. It seems to me that to deny this is to deny that God’s mind has actual, concrete existence.

It might be objected that, while an omniscient being would have to know, of each proposition, whether it is true or false, it wouldn’t necessarily have to know, for example, of each collection of propositions, whether every element of it was true, or exactly what propositions are in each such collection. But this leads to serious problems.

For example, in college I often sat in the card section at a football game. Everyone got a card that was white on one side and black on the other, along with instructions about which side to hold up when any of various patterns was “called”. Now it was quite possible to “know”, for each card and each pattern, which side was to be held up for that pattern. But that is not the same thing as knowing what the pattern was. For this one had to “understand” the significance of each of these sets of instructions, and this involved knowing all of the “blacks and whites” for that pattern simultaneously and picturing the result of holding up the cards in just that way.

Thus knowing all facts of the form “for pattern K, card M should be held white side up” did not constitute knowing “everything there was to know” about the cards and the patterns: in particular, it did not constitute knowing what the patterns were – i.e., what picture or phrase would present itself when each pattern was “realized” or displayed. In other words, knowing everything about the card patterns involved knowing global facts about the cards, and the instructions included with them.

Here’s another example (this time involving infinite sets). It appears that Goldbach’s conjecture is probably true, but it is possible that there is no finitary proof of it. (At any rate, we know from Godel that there are such propositions about the integers, so we might as well take GC as an example.) Now imagine a being who knows, for every even integer, a pair of primes whose sum is that integer, but does not know that there is such a pair for every even integer. Could one seriously claim that such a being knows “everything” about integers? Of course not! To know everything about integers it would have to know, for every such global proposition about the existence or nonexistence of a particular pattern, whether that proposition is true or not regardless of whether there is a finitary proof or disproof. The same sort of thing is true of collections of any cardinality, or for that matter, of the class of all cardinalities. Thus, for example, God must know whether there are cardinal numbers that have certain sets of properties. (There are some interesting questions of this kind for which the answer is unknown to us humans, but God must presumably know the answer.) And He would have to know this even though the class of all cardinal numbers does not constitute a set.

Thus, if God is truly omniscient, He has to know, for every global proposition about a collection of objects (concrete or abstract) whether that proposition is true. In particular, He would have to know, for every proposition about a collection of propositions (such as that it contains at least one true proposition), whether it is true. It seems to me that this is more than enough to allow my proof to go through.

This is the best I can do at this time, having only had time to skim the article you cited. (I guess I’m not as fast a study as turtonm, who managed to go from barely being able to understand the OP to understanding the extremely complicated argument between Plantinga and Grim in just a few hours.) In a week or so I may have a better grasp of what P&G are trying to say.