Grim's Cantorian Argument Against Omniscience

Philosopher Patrick Grim since the early 1980s has advanced an argument against the existence of God which turns on an incoherence he sees in the concept of omniscience, an essential property of God as understood by most orthodox incarnations of the Abrahamic religions.^[1] It runs as follows: There does not exist a set T of all distinct truths. For suppose towards a contradiction that T exists. Let f be a mapping from T into the set P(T) of all subsets of T, and consider the subset S of T of every truth q which is not contained in the set f(q). By definition of S, no truth q is mapped by f to S, and we conclude that f is not surjective. Since f is an arbitrary mapping from T into P(T), it follows that no such map is surjective. Now define a map g from P(T) into T. For each subset A of T (where A is a member of the set P(T)), define g(A) by the truth expressed by the sentence "A is a subset of T." Since every member of P(T) is distinct, then g maps them to distinct truths, which is to say g is injective. So there is an inverse map which induces a map g' from T into P(T), where g' is surjective. This contradicts our conclusion that no such map is surjective. It follows that our assumption must be false---that there can be no collection T of all distinct truths. However, in order to conceive the omniscience of God we must conceive that he knows all distinct truths. Since we can always conceive of known truths as sets, then conceiving the omniscience of God requires that we be able to conceive all truths as a set. Yet we have already shown that there is no such set, and so we cannot conceive all truths in this way. Therefore we cannot conceive the omniscience of God. Since omniscience is an essential property of God, this means we cannot conceive of God at all, and this prevents us from ever affirming that God exists.

I first encountered this argument on the internet, championed by an atheist mathematician who frequents a message board for religious debates. That it should find its greatest popularity among mathematicians seems natural, given that it is based on Cantor's power set theorem. At that time I reacted with amusement, struck by its obvious silliness. I asked Hans---that was the name of this mathematician---if he was serious, and although I don't recall his exact reply, it left me with the impression that probably he was not. Serious or not, though, Hans continues to this day to enthusiastically tout Grim's argument, and this led me to investigate its origins. For until recently I was not aware that this argument had ever been advanced by a professional philosopher, and when I discovered Grim had done so, I was simultaneously surprised and disappointed that such empty sophistry had found a home in the academic literature. I became even more dismayed when I discovered further that several more philosophers had published extensive responses to Grim. As if that weren't bad enough, it turns out that all but one of these responses which I took the time to read at any length have even deeper problems than Grim's original argument. This has contributed to my growing disillusionment at the desperate state of philosophy as an academic discipline.

John Abbruzzes published the worst of these objections to Grim in a 1997 paper,^[2] the first of which turns on a fundamental misunderstanding of Grim's argument. He writes:

Grim also makes a second assumption: that the multiplicity of truths, T, is a completed totality. This, however, need not be so, for the multiplicity of all truths may be, and in fact seems to be, infinite.^[3]

As Grim himself has pointed out, though, this criticism mistakenly limits Cantorian sets to finite collections, and this is not at all what we have in mind in the original argument.^[4] This of course calls into question Abbruzzes' remaining criticisms. Perhaps the least unreasonable of the bunch is that, according to Abbruzzes, Grim's argument can be adapted to deny the existence of the set of all propositions, where Abbruzzes insists that Grim quantifies over this set by asserting that there is no collection of all truths. In his own words:

His conclusion that 'There is no set of all truths' is equivalent to the universal proposition 'for all x, if x is a set, then x is not a set of all truths'. ...but [this] involve[s] universal quantification...over sets and truths...^[5]

Yet this analysis again seems obviously false. Even if we assume that Grim is asserting something logically equivalent to the statement "for all x, if x is a set, then x is not a set of all truths," this is not a case of quantification over truths or propositions, but over sets. We might instead try to follow in the spirit of Abbruzzes' criticism by pointing out that we are not justified in using T at all in our conclusion if we agree with Grim that T is incoherent. However, this is not at all required to feel the force of the argument. It is quite enough to conclude that the concept of omniscience shares the incoherence of T, and that this prevents us from ever coherently supposing that some entity is omniscient. In this way, Abbruzzes offers us nothing in the way of a valid or cogent critique of Grim.

Keith Simmons in his 1993 rebuttal to Grim discusses a more promising, but ultimately still unfulfilling consideration against his argument.^[6] I shall refer to this henceforth as the Liar objection, and it challenges Grim on the grounds that his argument depends on the allegedly question-begging assumption that

(1) There is a language in which the sentences of that language are too numerous to form a set.

To show that Grim in fact requires this key assumption, Simmons identifies another, seemingly innocuous assumption which Grim appears also to require, and generates from this a version of the liar paradox which has no apparent solution but to invoke (1). The reasoning goes something like this: Grim requires for his argument that for a given set, we can construct a distinct truth to correspond to the cardinality of that set. In particular, he must assume that for every cardinal number there is at least one distinct truth. To make this assumption, however, he must quantify over all cardinal numbers. Since cardinal numbers are too numerous to form a set, then whatever language we use to express his assumption must have the ability to quantify over such set-defying collections. It is therefore possible to make the following paradoxical statement:

(2) This sentence is not true in any language of any cardinality.

It's easy to see how (2) leads to a contradiction, whether it is true or false. Apparently, the only way Simmons sees to escape the paradox is to say that Grim's argument "stands above" the cardinal numbers. In other words, cautions Simmons, we need to assume on Grim's view that there is a language in which the sentences of that language are at least as numerous as the cardinal numbers. So (1) must be true, if Grim's argument is to succeed.

Though clever, this Liar objection has serious problems. First, I question the key assumption that we require a distinct sentence in order for each distinct truth to exist. While it may be the case that we need sentences in order to express truths, Grim's argument only requires that set-defying collections of truths exist, and not that they be expressible in some language. On the other hand, this position against truths depending on sentences is not at all obvious, and so we might be interested in the Liar objection as binding the success of Grim's argument to it, that is, to the assumption that truths can exist apart from sentences. Given uncertainty as to whether or not this assumption is accurate, the Liar objection, if otherwise successful, would limit the force of Grim's argument. Unfortunately, though, the Liar objection has further problems, and so it fails in this regard, as well. For if we are to interpret it as an objection to the truth of Grim's conclusion that omniscience is an incoherent concept, then we must further posit that (1) is false, since if it were true, then while Grim's argument might beg the question, on Simmons' view its conclusion would nevertheless be likewise true. Yet if (1) is false, and if Simmons' reasoning holds up, then it must also be the case that we cannot, as Grim does, use our language to talk about arbitrary cardinal numbers, which according to Simmons is tantamount to quantifying over them. Of course it seems clear that in fact our language is quite capable, one way or another, of expressing truths about set-defying collections like the cardinals. Simmons even does this himself when expressing his Liar objection, for example when he mentions entities such as "all cardinals." In defense of Simmons, we might say that we ought for this reason to take his argument as incoherent, but acknowledge that its incoherence somehow reveals a similar incoherence in Grim's argument. However, this recourse seems implausible since it entails a denial of the coherence of discussing arbitrary cardinals, which is required for modern mathematics. In short, Simmons' reasoning, though he aims it squarely against Grim, should it succeed, succeeds in undercutting the coherence of set theory. Yet it seems much more natural to affirm the coherence of set theory, and suppose instead that something must have gone wrong in the Liar.

Third, whatever solution we have to the similar paradox

(3) This sentence is not true in any language which exists.

must be sufficiently specific as to render it inapplicable to (2). Yet judging from his approach to (2), we might expect Simmons to prefer to object to (3) on the grounds that we cannot quantify over extant languages due to their numerousness. On this view, (3) is incoherent, and thus not paradoxical. This leads quickly to problems. For example, it commits Simmons to the implausible position that there are infinitely many extant languages; for since we can always quantify over finite quantities, then if there are finitely many extant languages then we can quantify over them. Next, if we consider the case where extant languages form a set, then Simmons must assume that there are certain sets too numerous over which any extant language may quantify, and this entails an even more direct challenge to the coherence of set theory than we have already mentioned. Otherwise extant languages are too numerous to form a set, which again seems implausible, and which Simmons seems to himself implicitly deny when he agrees with Bertrand Russell that the statement "to each set there corresponds a proposition" is untrue.^[7] So given these three counter-objections, it seems that the Liar is insufficient to overturn Grim's argument.

Most recently, Laureano Luna published in 2011 another creative yet inadequate defense against Grim's argument.^[8] Luna's objection to Grim had been foreshadowed by Abbruzzes' earlier remark to the effect that, whatever logic we human beings use, God needn't himself think in terms of set-theoretical objects in order to quantify the propositions he knows.^[9] Luna develops this idea further, writing that

if the traits of abstract objects depend on the peculiarities of the human mind, then the nonexistence of the set of all truths can be just a phenomenon relative to the human intellect (as a finite intellect, for instance) so that for other intellects (perhaps infinite ones) truths could form a completed totality. ...the nonexistence of certain totalities, such as the totality of all sets or the totality of all truths, is the consequence of a peculiar trait of the human mind, a trait related to its finiteness.^[10]

To put in in terms of the present discussion, suppose we can agree that Platonism is false or incoherent. Then, according to Luna, the properties of abstract objects like T depend in some way on the intellects which conceive them. In particular, the previous proof only shows that T fails to exist relative to our limited human minds. So, it could well be that relative to a superior mind such as God's, T may exist after all.

I'm a little suspicious of the leap from rejecting Platonism to supposing that all the properties of abstract object depend on the minds which conceive them. However, I don't want to criticize him on that point since, for reasons of my own, I do agree that the properties of abstract objects are indeed determined by our minds. Indeed, we can see this by noticing that abstract objects are completely characterized by definite conceptual procedures. So unless we have another way to make sense of abstract objects, it seems unavoidable that we should attribute to their properties a dependence on our minds. The real trouble with Luna's argument is that his conclusion that God might be able to make sense, in a manner of speaking, of the existence of T, does nothing to diminish the fact that we ourselves remain wholly unable to do so. There may well be a truth, call it u, which a super-intelligent being G knows, but which we are incapable as human beings of even comprehending. Unfortunately for us, if belief in the existence of G depends on a comprehension of u, then since u defies comprehension then we can never justifiably believe---and hence never know---that G exists. With this in mind, we can interpret Grim's argument along the following lines: since justified belief in the existence of an omniscient being depends on the our being able to make sense of T as a coherent concept, and since we find T to be incoherent, then we cannot justifiably believe, and hence we cannot know, that an omniscient being exists. Given this interpretation, it won't matter if some other being is able to make coherent sense of something like T. The fact remains, if Grim is correct, that we are not able; yet we must be the ones to make sense of T if we are ever going to justifiably believe in the existence of an omniscient being.

Even just to take the objection as far as we have, though, we must, as I alluded in the previous paragraph, speak loosely about T. After all, T does not really exist, even on Luna's view, with respect to human minds. In other words, neither the symbol 'T' nor the description which it abbreviates denotes anything of which we can make coherent sense. If T is incoherent to us, though, then any attempt on our part to appeal to T we shall find likewise incoherent. Yet Luna must appeal to T if he is to suggest that God can comprehend T. If Luna justifiably believes that God comprehends the existence of T, then he must find the sentence "God comprehends the existence of T" to be coherent. However, the incoherence (with respect to us) of T as a constituent part of the sentence infects the sentence as a whole, with the unhappy result that Luna believes something incoherent to him. If we take incoherence as undermining or otherwise preventing justification, then it follows immediately that Luna is unjustified in believing that God comprehends the existence of T. In short, when Luna asks us to suppose that God might comprehend T, he's asking us to suppose something which is at least incoherent to us.

Philosopher J.C. Beall also thinks he has discovered a way to avoid the force of Grim's argument. He contends that we ought to regard propositions as maps from possible worlds into truth values, and that on this view we cannot construct a collection of distinct truths corresponding to each member of P(T).^[11] I think it's fairly obvious, though, that the possible worlds interpretation of propositions is false. The reasons to reject this interpretation are numerous, but to give just one example we may turn to again to Beall, who acknowledges that on his view there is precisely one necessary truth; to disprove the possible worlds interpretation, then, we need only find two distinct necessary truths, for instance that Alvin is not identical to Patrick, and that 17 is prime. No doubt Beall would argue those sentences express the same proposition, but I think it's fairly clear that no such argument can ever be successful. A single super-proposition is simply not what we mean when we talk about necessary truths.

Despite this failing of Beall's approach, however, it's not too terribly far from what I think is the best response to Grim. The idea is not to claim that there is only one truth---for this is manifestly false---but rather that there is only one way reality actually is. In other words, if God has a perfect mental representation of the whole of the universe, past, present and future, then this is all He needs to know about in order for Him to satisfy orthodox doctrines relating to omniscience. So Grim is mistaken to suppose that omniscience involves the incoherent notion of T. Instead of thinking about knowledge as turning on individual truths, we can liken it to having an accurate picture of the world. To be omniscient, then, is something akin to having a perfect picture. Grim might object to this response by alleging that in order to know the way the reality is---in order to have that perfect picture---God must know every truth, i.e. he must know every element of T. Yet although this view might be intuitive, I see no good reason to think it is appropriate. Consider as an analogy the sentence "a baseball is smaller than a school bus." It expresses a true statement, and provided we have sufficient experience with American suburbia, we all know it to be true. However, it may never have occurred to us until just a moment ago to relate baseballs and school buses in just the way that sentence does, and pick out an arbitrary baseball as being smaller than an arbitrary school bus. Does that mean we didn't know prior to considering this sentence that a baseball is smaller than a school bus? Surely not. In this way, knowledge does not depend on logical constructions or expressions of particular statements. We don't need an omniscient being to hold in his mind a list of all possible truths. Instead, we merely require Him to have a complete understanding of the way reality is.

A defender of Grim might insist that on this view there are nevertheless truths which God does not know in the fullest sense. If we take truths to necessarily be about the world, then this of course contradicts the assumption that God has a complete understanding of the world. However, the only way I can see this objection to succeed is if there is a conscious creature which has a deeper apprehension of some truth than does God. For the only unique content to individual truths, as opposed to a general picture of the world, is the collective properties which mark them out for individuality. To see how this works, we can look at a sort of special case:

(4) The 76th digit of π is 6.

Consider a hypothetical reality R where no conscious creature thinks about (4) at any time in the whole existence of R. From our perspective outside R, we can easily verify that (4) is in fact true. Is then (4) also true from the perspective of a conscious creature within R? I would suggest that (4) is not a truth about R, but rather it is a truth about our perspective outside of R. Now, it may well be the case that the conscious creatures of R know an effective procedure with which, from our perspective outside R, we can say in some sense that they could discover the truth of (4). Even if they don't recognize this procedure for what it is, we can still say that there exists, at least beyond the boundaries of R, a procedure for them to find which would allow them to learn that (4) is true. However, all these things which we say about the truth of (4), and the possibility of conscious creatures inside R to discover (4), must be said outside of R, from our external perspective. For the creatures of R themselves, there is no truth about (4) as long as (4) never occurs to them. The "knowledge" that they are missing out on by failing to think about (4) is the knowledge of what could possibly be the case if only they took the time to investigate matters relating to (4). Yet if it really is the case, as we have stipulated, that nobody in R ever thinks about (4), then the only sense in which it could "possibly" be otherwise is from a larger perspective, i.e. from our external perspective. The God of R, if He exists, is not missing out on information about R by failing to know (4), because R is such that (4) is never conceived therein; from the perspective of the conscious creatures (including God) of R, (4) does not really exist to be conceived.

This view may seem radical, but I find nothing else tenable. To ease its strain on our intuition, we may consider the following sentence:

(5) God decrees an asteroid to destroy the earth.

It is in God's power to know (5), and yet God does not know (5) so long as he declines to decree an asteroid to destroy the earth. Is God missing out on knowledge because he does not know (5)? Clearly this is not the case. It's not that (5) is impossible for God to know, though. He could come to know (5) simply by making it the case that (5) is true. Yet we don't say for this reason that God is not omniscient. Similarly, the God of R could come to know (4) by conceiving the requisite procedures relating to (4). The moment the God of R conceives these procedures, (4) becomes true from within R. If God declines to conceive the procedures, though, which is to say that if God's mind does not operate in such a way as to calculate the 76th digit of π, then (4) is not true from within R. For there is nothing we can say about (4) which has any bearing on the conscious creatures of R except to relate R to our own perspective where (4) is true.

Now, I should note that I think this view is well-founded; for I can't make sense of its denial, nor do I think can anyone else. Even if we reject it, though, it should be clear that Grim's argument will still fail. For if we reject the view I just outlined in favor of one where we take truths to be so regardless of their conception by a real conscious creature, then there still seems to be no good reason to accept Grim's conception of omniscience. In fact, to the extent that Grim's conception is provably incoherent, we have good reason to reject it. This doesn't mean we can't make sense of omniscience in some other way, however. Philosopher and Christian apologist Alvin Plantinga, addressing Grim on this matter in a published dialog, explains succinctly:

My main puzzle is this: why do you think the notion of omniscience, or of knowledge having an intrinsic maximum, demands that there be a set of all truths? As you point out, it's plausible to think there is no such set. Still, there are truths of the sort: every proposition is true or false (or if you don't think that's a truth, every proposition is either true or not-true). This doesn't require that there be a set of all truths: why buy the dogma that quantification essentially involves sets? Perhaps it requires that there be a property had by all and only those propositions that are true; but so far as I can see there's no difficulty there. Similarly, then, we may suppose that an omniscient being like God (one that has the maximal degree of knowledge) knows every true proposition and believes no false ones. We must then concede that there is no set of all the propositions God knows. I can't see that there is a problem here for God's knowledge; in the same way, the fact that there is no set of all true propositions constitutes no problem, so far as I can see, for truth. So I'm inclined to agree that there is no set of all truths, and no recursively enumerable system of all truths. But how does that show that there is a problem for the notion of a being that knows all truths?^[12]

Grim's response to this objection is to point out that we have no semantics for quantifying over anything but sets. Yet Plantinga shows with his examples that indeed we do have just such semantics available to us---the only catch is that they are not formalized into a mathematical framework. So it seems plain to me that Plantinga's objection is perfectly successful, and that Grim has nothing to offer us in terms of a reason to accept his deliberately incoherent account of omniscience.

In sum, and at the risk of sounding overly harsh, I must once more express my profound disappointment at the audacity of Grim to advance such an obviously flawed argument, the willingness of numerous, conventionally "reputable" philosophical journals to publish it again and again, the eagerness with which his fellow philosophers took that argument seriously, and the miserable inadequacy, Plantinga excepted, of their various responses. I don't mean to suggest that these professionals are incompetent or unskilled, but it pains me to see them permit each other to publish arguments of such poor quality. It, along with similarly bad arguments found in the literature, has me worried for the discipline as a whole.


Notes:

[1] Grim, Patrick, "The being that knew too much," International Journal for Philosophy of Religion 47, 2000.

[2] Abbruzzes, John, "The Coherence of Omniscience: A Defense," International Journal for Philosophy of Religion, Vol. 41, No. 1, Feb., 1997.

[3] Abbruzzes, p32.

[4] Grim, pp147-8.

[5] Abbruzzes, p32.

[6] Simmons, Keith, "On An Argument Against Omniscience," Noûs, Vol. 27, No. 1 (Mar., 1993), pp. 22-33.

[7] Simmons, p26.

[8] Laureano, Luna, "Grim’s arguments against omniscience and indefinite extensibility," International Journal for Philosophy of Religion "Online First" 2011 Apr 23.

[9] Abbruzzes, p32.

[10] Luna, 2011.

[11] Beall, J.C., "A Neglected Response to the Grim Result," Analysis, Vol. 60, No. 1, Jan., 2000, pp. 38-41.

[12] Plantinga, Alvin, and Patrick Grim, "Truth, Omniscience, and Cantorian Arguments: An Exchange," Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 71, No. 3, Sep., 1993, pp267-8.