Philosophical objections to omniscience

There are some very recondite objections to divine omniscience based on indexical reference as well as the liar paradox and set theory. I'm going to quote some sections from Graham Oppy's Describing Gods: An Investigation of the Divine Attributes (Cambridge, 2014). Although Oppy is an aggressive atheist, he proceeds to debunk these particular objections to divine omniscience. The quoted material begins right after the break:

Indexicals
It is not clear that ‘the problem of the essential indexical’ does create a serious problem for the standard analysis of omniscience. The idea is that ‘the problem of the essential indexical’ provides us with cases in which all of the instances of the schema p only if x knows that p are true, and yet in which x is not omniscient. So, for example, while a putative omniscient being O can know that I am making a mess, ‘the problem of the essential indexical’ is supposed to establish that there is no way that O can know what I know, when I know that I am making a mess. But why should we suppose that the alleged fact that O cannot know what I know when I know that I am making a mess somehow counts against the claim that O is omniscient? Truly bizarre hypotheses aside, it is reasonable to hold that it is not even logically possible for another being to share my first- person perspective on the world; and that it is not even logically possible for a being that fails to share my temporal location to share my temporal perspective on the world; and that it is not even logically possible for a being that fails to share my spatial location to share my spatial perspective on the world; and so forth. So, unless we are prepared to accept that it is impossible for there to be an omniscient being in a world unless there is only one first-person perspective on that world, it seems that we do bet- ter to say that ‘the problem of the essential indexical’ does not provide us with cases in which all of the instances of the schema p only if x knows that p are true, and yet in which x is not omniscient.

What it is like

It seems plausible – to me, anyway – to suppose that it is no limit on omniscience that I lack abilities to make certain kinds of discriminations and judgements on the basis of my experiences, without further recourse to external standpoints of assessment, merely because I happen not to undergo the relevant kinds of experiences.

The divine liar

As everyone knows, discussion of liar paradoxes is fraught with difficulty. When Grim rehearses difficulties with the adaptation of a wide range of possible responses to the standard liar paradox – i.e. to the analysis of the claim:

(*) (*) is not true

– it is, I think, not too hard to agree with him that none of these responses seems intuitively satisfying. That is, it is not too hard to agree with Grim that appeal to truth-value gaps, or truth-value gluts, or many-valued logics, or failure to express a proposition, or hierarchy, or whatever, does not seem to provide a satisfactory account of the divine liar. But, of course, we are only going to agree with Grim about these matters if we also suppose that appeal to truth-value gaps, or truth-value gluts, or many-valued logics, or failure to express a proposition, or hierarchy, or whatever, does not seem to provide a satisfactory account of the standard liar paradox. And therein, I think, lies the rub.

Grim himself supposes that, in the end, an appeal to hierarchy emerges as the strongest candidate for an adequate treatment of the simple liar paradox. Moreover, he claims that, on any satisfactory hierarchical approach, it turns out that there can be no set of all truths. So, according to Grim, while it is plausible to suppose that there is a hierarchical approach that provides a satisfactory treatment of the standard liar, it is also plausible to suppose that there is no satisfactory treatment of the divine liar: for, as he goes on to argue in the later parts of his book, there are good reasons to suppose that, if there is no set of all truths, then there is no omniscient being.

On the one hand, if we disagree with Grim about the prospects for hierarchical cases, then it seems to me that the divine liar provides no reason at all to think that there cannot be an omniscient being. The liar paradox is a problem for everyone; until we have found a satisfactory resolution to it, it would be premature to suppose that a satisfactory solution to the standard liar paradox will not also provide a satisfactory solution to the divine liar that is consistent with the possible existence of an omniscient being.

On the other hand, if we agree with Grim about the prospects for hierarchical cases, then the divine liar provides no independent reason to think that there cannot be an omniscient being: for the whole weight of the argument is now thrown onto the prospects for developing a satisfactory account of omniscience if there is no set of all truths, or proposition about all propositions, or the like. Against Grim’s claim that consideration of the liar paradox and its ilk can teach us that there can be no coherent notion of omniscience, it seems to me that – as things stand – the liar paradox is simply mute on the question of the coherence of the notion of omniscience.

The paradox of the knower

The points that we made in our discussion of the divine liar apply with equal force to the case of the paradox of the knower. There are a number of responses that have been suggested to the paradox of the knower – for example, to claim that truth and knowledge are only properly predicated of propositions and not, as the paradox of the knower supposes, of sentences; or to claim that truth and knowledge are only properly construed in terms of sentential operators rather than, as the paradox of the knower supposes, in terms of predicates; or to claim that truth and knowledge are only properly construed in terms of a hierarchy of predicates rather than, as the paradox of the knower supposes, in terms of a single predicate – none of which is evidently satisfying. If we suppose that none of these approaches is satisfying, then we have no reason to suppose that there is any particular problem that is made here for the notion of omniscience, since we shall need to wait to see what future work on this problem delivers. If – contra Grim – we suppose that one of the first two approaches is satisfying, then it seems that we can defuse the argument to the conclusion that the notion of omniscience is incoherent. And if – along with Grim – we suppose that only hierarchical approaches hold out any prospect of a genuinely satisfying treatment of the paradox of the knower, then the whole weight of the argument will be thrown onto the prospects for developing a satisfactory account of omniscience if there is no set of all truths, or proposition about all propositions, or the like. So, for all of the elegance of the paradox of the knower, it provides no particularly pressing reason to suppose that there is something wrong with the standard analysis of omniscience. If there is a pressing reason to suppose that there is something wrong with the standard analysis of omniscience, then it must reside in the remaining ‘Cantorian’ and ‘Gödelian’ arguments.

Expressive incompleteness and internal incompleteness

While there are features of Grim’s argument from expressive incompleteness that will be addressed in our discussion of the ‘short and sweet’ Cantorian argument, there is one key point that seems to me to be worth making here. Even if we grant to Grim that we understand his language- independent accounts of ‘self-reflectivity’ and ‘expressive completeness’, it is quite unclear why we should suppose that it follows from these claims that, in a system that is both self-reflective and expressively complete, there must be more properties of objects than there are objects. Indeed, even if we grant that there is a property for each member of the ‘power set’ of the identified objects, it seems that it is still open to us to insist that there are ‘proper class many’ objects, and ‘proper class many’ properties of objects in any system that is both self-reflective and expressively complete. Prima facie, then, it seems that Grim’s argument from expressive incompleteness is not so much as valid.

As in the case of the argument from expressive incompleteness, there are features of Grim’s argument from internal incompleteness that will be addressed in our discussion of the ‘short and sweet’ Cantorian argument. However, one point that I will note here is that the argument from internal incompleteness depends upon the assumption that, for any self- reflective system, there is a set of expressible predicates, and it also depends upon the assumption that there is a corresponding set of predicate objects. If – following the lead of the objection that we have already raised against the argument from expressive incompleteness – we suppose that there are ‘proper class many’ expressible predicates in a self-reflective system and/ or that there are ‘proper class many’ corresponding property objects, then we shall reject one or both of these assumptions. Hence, as before, there is good prima facie reason to suppose that Grim’s argument from internal incompleteness is not so much as valid.

Of course, in the light of the above objections, one might be given to wonder whether one can make sense of talk about there being ‘proper class many’ things. This is one of the central questions that emerge when we turn to consider the ‘short and sweet’ Cantorian argument that is the true cornerstone of Grim’s attack on the standard analysis of omniscience.

Cantorian quandaries

There are two parts to Grim’s ‘short and sweet’ Cantorian argument, each of which is open to challenge. On the one hand, one might challenge Grim’s argument for the claim that there cannot be a set of all truths by challenging the use that Grim makes of Cantorian set theory; on the other hand, one might challenge the contention that, if there is an omniscient being, then what that being knows constitutes a set of all truths.

As Mar observes, there are alternatives to Cantorian set theory in which ‘Cantor’s power set theorem’ fails. So, for example, in Quine’s ‘New Foundations’, there is a universal set that has the same cardinality as its own power set. While one might concede to Mar that there are some who do not find Quine’s ‘New Foundations’ artificial and strongly counter-intuitive, it seems to me that there is a reasonably strong argument from the widespread acceptance of Cantorian set theory by working mathematicians to the conclusion that we ought not to give up on ‘Cantor’s power set theorem’. On the other hand, as I noted, there is much that we do not understand about the foundations of set theory; it is not inconceivable that we might come to have good mathematical reasons for modifying those parts of Cantorian set the- ory that are required to underwrite ‘Cantor’s power set theorem’. While – as Grim argues – there is no serious competitor to Cantorian set theory that is currently on the market and that is consistent with the claim that there is a set of all sets, it can hardly be said that there are compelling reasons to suppose that no such competitor could be developed.

There is much more to be said against the contention that, if there is an omniscient being, then what that being knows constitutes a set of all truths. On the standard Cantorian picture, there is a universe of sets, but there is no set of all sets. Consequently, it is not very hard to come up with the suggestion that, while there is a universe of truths, there is no set of all truths: why shouldn’t what goes for sets go for truths as well? Indeed, one might think that there is a Cantorian argument for the conclusion that there is a set of all sets just in case there is a set of all truths. For consider. Since each set can be mapped onto the truth that that set has the particular members that it has, there are at least as many truths as there are sets. Since each truth can be mapped onto the set that has just that truth as its sole member, there are at least as many sets as truths. So, there are just as many sets as there are truths, and we can establish a one–one mapping between them. But, then, the axiom of replacement guarantees that there is a set of all sets just in case there is a set of all truths.

Suppose, then, that there is a universe of truths. What is there to stop us from supposing, further, that there is omniscient being that knows every one of the truths in the universe of truths? True enough, if we are to talk about a ‘universe’ – or a ‘proper class’, or a ‘collection’, or an ‘absolute infinity’ – of entities, then we need to develop a theory of ‘universes’, or ‘proper classes’ or ‘collections’, or ‘absolute infinities’. Moreover, as Grim emphasises, it may not be easy to navigate a path between, on the one hand, overly restrictive principles of comprehension whose adoption would cripple standard mathematics, and unrestricted principles of comprehension that simply return us to the Cantorian arguments that we are hoping to escape. However, there is no obvious reason why we should not suppose that an appropriate theory of this kind could be developed. And, in any case, if there is no such theory to be given, then what are we to make of Cantorian claims about the universe of sets? In our formulation of Cantorian set theory...we took it for granted that our variable ranged over sets, i.e. we took it for granted that the ‘domain of quantification’ for Cantorian set theory is the universe of sets. If we are not entitled to suppose that there is a universe of sets, then – it seems to me – we cannot even begin to formulate Cantorian set theory.

Grim considers the possibility that one might suppose that one can have quantification without totalities, i.e. without sets, or classes, or universes, or domains, or absolute infinities, or the like. Against this suggestion, he makes several objections. First, he claims that the only formal semantics that we have commits us to the existence of set-theoretic domains of quantification. And, second, he claims that it is possible to reformulate his argument without recourse to anything more than the use of quantifiers.

I think that there is an evident problem with the appeal to considerations about formal semantics: if Grim is right, then it seems that we cannot arrive at a coherent understanding of the semantics of Cantorian set theory. After all, Cantorian set theory quantifies over sets, and yet it denies that there is a set of all sets. If formal semantics requires a set-theoretic domain of quantification, then the combination of Cantorian set theory with formal semantics for that theory leads to contradiction. Not good. Rather than suppose that formal semantics requires set-theoretic domains, it seems to me that we do better to suppose that some domains of quantification are not sets; in any case, it seems clear that anyone who wishes to make use of Cantorian set theory had better be entitled to an assumption of this sort.

There are various questions that one might raise about this argument. One might suspect that talk about mappings, one–one mappings, functions and the like is really just talk about sets under other names. But if that is right, then it seems doubtful that this argument really ‘has no recourse to anything other than quantification’. However, as Grim observes, we can trade in all of this allegedly suspicious talk for talk about properties and relations. So perhaps we should be prepared to concede that Grim’s argument need not trade in anything other than quantification.

A more serious problem with Grim’s argument arises when we ask about the conclusion that we are supposed to be able to draw from it. If the conclusion is that there cannot be a proposition that is genuinely about all propositions, then it seems that that conclusion must be self-defeating: for it is, itself, a proposition that purports to be genuinely about all propositions! In the face of this difficulty, Grim claims that the conclusions that he states ought properly to be interpreted as denials of the coherence of the basic notions involved therein: all attempts to reason about ‘a set of all truths’ or ‘a proposition about all propositions’ end in a tangle of contradictions. (Grim also considers the possibility that one might make judicious use of ‘scare-quotes’ in the rendering of his conclusions; but it is quite unclear how this would work, and Grim does not undertake to offer any illustrative examples.) But if Grim is right in claim- ing that we can make no sense of quantification over all propositions, or all sets, or the like, then – as we noted above – we lose much that we should not want to lose. In particular, we are driven to the conclusion that (the standard quantificational formulation of) Cantorian set theory is itself incoherent, a conclusion that Grim should surely not want to embrace.

While it is clear that there are further matters to be resolved here, it seems to me that Grim’s Cantorian arguments do not strongly support the claim that we ought to revise the standard definition of omniscience. Perhaps Grim’s discussion of ‘other ways out’ in the cases discussed earlier – truth-value gaps, truth-value gluts, many-valued logics, hierarchies, redundancy theories and the like – make it plausible that we currently have no idea how to construct a fully satisfying theory of quantification over all propositions, and the like. But it does not seem unreasonable to suppose that there is a satisfactory theory of this kind to be discovered; and it also does not seem unreasonable to suppose that, when we have discovered a theory of this kind, we shall then see that the apparent difficulties that arise for the standard analysis of omniscience fade away. At the very least, it seems to me that it would be premature to give up on the standard analysis of omniscience in the light of the arguments that Grim has presented thus far: if there is a devastating objection to the standard analysis of omniscience that derives from considerations about quantification over certain kinds of totalities, it remains to be clearly established that this is so.