Defending Propositional Omniscience: A Way Back to Full Omniscience

Abstract: In this paper I will set out to defend a version of propositional omniscience. In doing so my task will be to establish the conditional that either the difficulties faced by propositional omniscience are not insuperable, or else that, if my general approach to dealing with them fails, no attempt to salvage propositional omniscience will ever succeed. The paper will deal first with challenges to the instantiability of any kind of omniscience, and then move on to dealing with challenges posed specifically to the propositional account of omniscience.

The question concerning the nature and extent of God’s knowledge is one with which analytic theologians have had to grapple, and one on which nothing approaching a general consensus has yet to emerge among them. It has led some to adopt open theism (i.e., to deny that God knows what the future will bring),[1] others to adopt the view that God is omnisubjective,[2] others to adopt the view that God both knows everything (including about the future) and yet is learning new facts every moment of every day,[3] and still others that God is factually omniscient even though there are truths that He can’t possibly know.[4] This diversity is less surprising when one appreciates how riddled the question is with philosophical puzzles about semantics, set theory, the nature of time, the nature of knowledge and the nature of propositional content. What is a little more surprising, perhaps, is that so many philosophical theologians have shrunk back from defending propositional omniscience in light of the proposed difficulties. I will argue that these difficulties are not insuperable and that, therefore, we ought to hold our ground and defend propositional omniscience. This paper can thus be read as an attempt to kick against the goads of the current sensus intellectorum.

Before diving into my defense, a word about motivations for defending the possibility[5] of propositional omniscience may be appropriate. First, propositional omniscience given theism has some strong intuitive appeal, for “being incapable of knowing all there is to know or being capable of knowing all there is to know and knowing less than that are conditions evidently incompatible with absolute perfection.”[6] For the theist, and perhaps especially for the perfect being theologian, this provides strong impetus for preferring propositional omniscience to any ‘weaker’ or more gerrymandered versions of omniscience unless led by necessity to do so. Second, it seems obvious to me that if propositional omniscience can be defended then it not only provides the most elegant solution to Fitch’s paradox,[7] but also (thereby) enriches the funds of natural theology by adding yet another argument for God’s existence to an already impressive deposit. Finally, among the advantages of propositional omniscience we could include that the most plausible alternative version of omniscience, namely ‘factual omniscience,’ follows from it, for “it is not possible to be propositionally but not factually omniscient.”[8] These reasons conjointly provide us with ample motivation for at least exploring how we might go about defending the coherence of propositional omniscience.

Having, I hope, justified my philosophical project to the reader’s satisfaction I will now proceed to offer a defense of propositional omniscience. I will defend a version of propositional omniscience, shortly to be defined, against two general kinds of attack; first, I will defend it against the charge of logical incoherence, and, second, I will defend it against challenges typically raised in the literature from what we might call ‘the problem of indispensable indexicals.’ I will take omniscience to be exemplified by a being S if and only if for any true proposition P, S knows P, and for any untrue proposition Q, S does not believe Q.[9]

Propositional Omniscience (PO)=def. For any proposition P, if P is true then it is known, and if P is not true then P is not believed.

In other words, if a proposition has the property of being true, then a PO-being[10] believes it, and if it either has the property of being false, or in any case does not have the property of being true, then a PO-being does not believe it.[11]

Many arguments against Omniscience have recently been registered in the academic literature, and although I do not have the space or the time to deal with all of them, I will raise at least two such arguments which I feel are particularly troubling. I take them to be the very best arguments against omniscience available, so that if they can be dispelled we have good reason to think that the others probably can be as well. The first such argument comes from Patrick Grim and is often referred to as a ‘cantorian’ argument against omniscience. God’s omniscience, he opines, must consist in his knowing all truths, but since “by Cantor’s power set theorem we know that the power set of any set is larger than the set itself,”[12] we can prove quite easily that there is no such thing as the set of all truths. For any such supposed set of all truths we can take its power set and generate new truths which belong to the set of all truths, and since we can do this indefinitely the set of all truths is not merely infinite, but indefinite, and therefore nonexistent.

Although this objection has a tremendous amount of prima facie force, responses to it abound in the literature, primary among which is the response of Alvin Plantinga who expresses his puzzlement at Grim’s argument by asking why we should think that “the notion of omniscience, or of knowledge having an intrinsic maximum, demands that there be a set of all truths.”[13] Although this response seems to me to be perfectly satisfactory, not everyone has been so easily convinced by it. In particular, Patrick Grim has replied in return that “the only semantics we have for quantification is in terms of sets,”[14] so that giving up the use of them altogether makes ‘omniscience-statements’ inexpressible “within any logic we have.”[15]

Supposing that one feels an inordinate attachment to set-theoretic language, there still remain ways in which omniscience can be safeguarded. Alexander Pruss, for instance, attempts to evade set-theoretic paradoxes by shaving down what he calls the “Big Conjunctive Contingent Fact”[16] or BCCF[17] to a BCCFOF: “let the Big Conjunctive Contingent First-Order Fact (BCCFOF) of a world be the conjunction of all contingent first-order propositions in that world, with any logical redundancies omitted in order to root out set-theoretic paradoxes.”[18] Thus, we might just say that God is first-order propositionally omniscient by knowing the set of true propositions in the BCCFOF (along with the set of all necessary truths), and be satisfied with that. This kind of response steps in the right direction, but on its own it obscures the fact that Grim’s paradox leads to much more severe problems than the (alleged) problem for omniscience, such as how we are to construe logically possible worlds. If logically possible worlds are maximally consistent sets of propositions then, given this cantorian argument, it follows that the actual world isn’t a logically possible world. This constitutes a definitive reductio ad absurdam, and so even in the absence of a solution to the problem, this cantorian argument cannot be accepted.

The typical solution adopted in the literature has become to construe worlds not as sets of propositions, but as “possibly true maximal proposition[s]… [which entail] every proposition with which [they are] consistent.”[19] If this construal of worlds, in response to set-theoretic paradoxes, dissolves the problem of the cantorian challenge, then it does so with the additional advantage of restoring the coherence of omniscience as well. God, to be omniscient, merely has to believe the world-sized-proposition which is true, and not believe anything with which it is inconsistent. In fact, this proposal sits well with the classical line taken by philosophical theologians that God’s knowledge is not discursive (i.e., divided into different regular-proposition-sized beliefs) but is an intuitive grasp of the truth as a simple[20] seamless whole. Immanuel Kant, for instance, writes:

“Now, however, we can also conceive of an understanding which, since it is not discursive like ours but is intuitive, goes from the synthetically universal (of the intuition
of a whole as such) to the particular.”[21]

At this point we have indicated enough philosophical avenues by which to evade the problem that we can rest reasonably assured that Grim’s argument poses no defeater for belief in (the possibility of) omniscience.[22]

A second argument which has invited the attention of philosophers and theologians more recently is the so-called ‘grounding’ argument against omniscience presented by Dennis Whitcomb. He illustrates the problem as follows:

“Suppose for reductio that someone is omniscient. Then his being omniscient is partly grounded by his knowing that he is omniscient (which is one of the knowings that helps make him all-knowing). And his knowing that he is omniscient is partly grounded by his being omniscient (for knowledge is partly grounded by the truth of what is known). Since partial grounding is transitive, it follows that his being omniscient is partly grounded by his being omniscient. But this result is absurd, for nothing can partly ground itself.”[23]

The notion of grounding here, Whitcomb suggests, is not merely one of bearing a supervenience relation, which he points out may hold symmetrically between two facts, just as “the facts about the surface area and the volume of a sphere each supervene on the other,”[24] but one of bearing a relation of dependence. However, it is absurd to think that any fact is (even partly) grounded by itself in the sense of depending upon itself, and therefore no being can be omniscient.

To lay out the argument more precisely, Whitcomb argues that five claims (including omniscience) are incompatible, and then defends the “truth of each of them except the claim that there is an omniscient being.”[25] These claims include transitivity (i.e., that if A grounds B and B grounds C, then A grounds C), irreflexivity (i.e., that if A grounds B then B does not ground A), that truth grounds knowledge, and that every fact of the form ‘∃x∀y’ is grounded by its instances. It turns out that God’s knowing that He is omniscient is an instance of His omniscience, but that His omniscience (at least partly) grounds His knowing that He is omniscient, which implies that His omniscience (at least partly) grounds itself (which is absurd).[26]

This argument has been addressed in at least two ways in the literature. First, Joshua Rasmussen, Andrew Cullison and Daniel Howard-Snyder have co-authored a paper presenting a powerful reductio of Whitcomb’s argument by way of parody. They put forward a “formally identical argument that concludes that one of the present co-authors does not exist”[27] but insist that, since this is absurd, “Whitcomb’s argument is unsound.”[28] They begin by defining a predicate ‘daniscient’ as knowing “all and only whatever propositions Dan Howard-Snyder happens to know.”[29] From here the parody proceeds with perfect parity:

“Suppose for reductio that Dan Howard-Snyder is daniscient. Then his being daniscient is partly grounded by his knowing that he is daniscient (which is one of the knowings that helps make him daniscient). And his knowing that he is daniscient is partly grounded by his being daniscient (for knowledge is partly grounded by the truth of what is known). Since partial grounding is transitive, it follows that his being daniscient is partly grounded by his being daniscient. But this result is absurd, for nothing can partly ground itself. Hence our reductio assumption is false.”[30]

Rik Peels has also contributed similar reductios,[31] though he has done better by being able, in addition, to “provide a diagnosis of where precisely the argument goes wrong.”[32] In his submission, Whitcomb’s argument fumbles because his “notion of grounding actually covers two distinct kinds of [grounding] relations”[33] which he fails to disambiguate.[34]

At this point we can rest assured that omniscience isn’t in as much trouble as one may have imagined if they merely skimmed the literature. A variety of difficulties present themselves, however, for the possibility of PO. Recall that PO is not satisfied by just any old kind of omniscience, but only by omniscience of a peculiar sort; namely, omniscience in the sense that there is no true proposition which an omniscient being fails to know. However, there are very many propositions which it seems are both possibly true, and not possibly (all) known by God (or any other potentially PO-being). For instance, consider propositions like “I am Tyler,” or “I am John.” These pose serious difficulties for PO, for they suggest that there are propositions the meanings of which are bound up with indexicals in such a way that no being could know all such true propositions.

Turning once again to Patrick Grim, we find the problem put succinctly as follows: “only I can use… ‘I’ [in a propositional expression] to index me – no being distinct from me can do so,”[35] and yet since neither he nor any of us are omniscient, it follows that no being is (propositionally) omniscient just in case ‘I’ is essential to the meaning of the proposition in which it figures. The contention here is that propositions like “I am Tyler” are both true and unknowable to anyone other than me. Since there is no way to translate an indexical like ‘I’ in a proposition without thereby changing the very meaning of the proposition, it seems that nobody other than me can know any propositions in which ‘I’ indexes me. Since I am not omniscient, it would follow that no being is propositionally omniscient. This whole argument depends, however, on a crucial assumption which I mean to challenge; namely, that propositions just are meanings.

It is not atypical among analytic philosophers to simply regard propositions as meanings. Pruss, for instance, writes that “propositions have their meanings essentially – indeed, propositions could even be thought of as identical with meanings.”[36] I want, in what follows, to challenge this assumption. I will make a start of doing so by drawing off of Darren Bradley’s defense of two-dimensionalism with respect to objects of belief.[37] On Bradley’s view, objects of belief have two dimensions; first they have content, and second they have a mode, (i.e., “the way in which [what is believed] is believed”[38]) so that on this view beliefs have “a content that is grasped by a role.”[39] Bradley’s concern is to account for belief-change over time, especially in light of “standard confirmation theory,”[40] according to which the only rational rule governing belief-change is conditionalization.[41] This does not account, however, for belief changes such as when the belief that “today is Sunday” becomes the belief that “yesterday was Sunday.” Such changes of belief over time involve no new evidence on which the conditional probability of a belief B changes, but surely that kind of change of beliefs is rational nevertheless.

What we need, Bradley stipulates, is “two rules of belief update – conditionalization and mutation,”[42] where mutation corresponds to the mode of a belief as it changes over time, and conditionalization corresponds to the content of a belief. So, if a present-tensed proposition is uttered at one time, and a past-tense proposition with the same very same truth-conditions is uttered at a later time, “then both sentences express the same belief,”[43] even though they “are apprehended with different roles”[44] by involving different modes. This gracefully explains why the belief that ‘the meeting is now’ can catalyze action in me which ‘the meeting is at noon’ cannot, even if one is true if and only if the other is true. Bradley argues that “the neat bifurcation I defend requires that content only changes by conditionalization… [and] this requires that mutation doesn’t affect content.”[45]

Although, as I have already indicated, Bradley’s purpose is to account for the turnover of tensed beliefs, I see no reason why his response would not work equally as well for the indexical ‘I’ as for any tensed indexical. Moreover, so long as we add that content is propositional content, and admit that some meaning (namely the meaning apprehended by a mode) is extra-propositional, there is no reason why we cannot concede that there will always be some semantic loss in translating “I am tired” to “Tyler is tired,” while maintaining that both of these sentences express the very same proposition.[46] The propositional content of “I am Tyler” as uttered by me, and “yes you are” as uttered by you, is identical; “on the traditional view, the same proposition is expressed in each case.”[47] All this view requires is a commitment to the (very unsurprising) thesis that meaning is, to some extent (and in at least some cases), psychologically determined.

I said at the outset, however, that I would defend a conditional claim; namely that if my approach to defending propositional omniscience fails then no approach will succeed. In order to defend propositional omniscience in light of the problem of semantically essential indexicals it seems that we must either dislocate proposotional content from semantics, or else argue that nothing essential to the meaning of a proposition I might assent to is lost if I fix the context of utterance by getting rid of personal indexicals. I see no hope of successfully doing the latter, so if the former approach does not work it looks like propositional omniscience will turn out to be indefensible after all. Supposing, for the sake of argument, that this were the situation in which we found ourselves, it seems to me that we ought to opt for factual omniscience, according to which “for every truth, God knows the fact which that truth expresses – a claim which does not entail that God knows every truth about every fact.”[48] After all, the difference between PO, as I have defended it, and factual omniscience, is really just a matter of semantics.

Some concluding remarks should figure in at this point. We have seen that the arguments which I suggested were the most powerful against omniscience have failed to pose insuperable difficulties for omniscience, and this should raise our confidence that all arguments against omniscience currently on offer fail to pose a genuine defeater for the belief that at least one being is omniscient. We have also seen that Darren Bradley’s two-dimensionalism concerning objects of belief carves out a dialectical space for preserving propositional omniscience in light of the problem of indispensable indexicals precisely by differentiating the content of a belief and the mode by which the belief is apprehended. If this can be done then we can defend propositional omniscience, and if it cannot then there is no way left for us to defend propositional omniscience (in which case, again, we should be content to adopt factual omniscience).

[1] See William Hasker, God, Time, and Knowledge, (New York: Cornell University Press, 1998).

[2] See Linda Zagzebski, “Omnisubjectivity,” in Oxford Studies in Philosophy of Religion 1 (2008): 231-248.

[3] See William Lane Craig, “Doctrine of God (Part 5),” Lecture, Defender’s Class, March 29, 2010. He states that: “But if God knows these tensed truths, then that means that his knowledge is constantly changing, as future-tense truths become false and the present-tense [version] becomes true.”

[4] See Brian Leftow, Time and Eternity, (New York: Cornell University Press, 2009), 326.

[5] Here, as elsewhere throughout the paper, by ‘possibility’ I mean possibility in any world relevantly similar to our own.

[6] Norman Kretzmann, “Omniscience and Immutability,” in The Journal of Philosophy (1966): 409.

[7] Berit Brogaard and Joe Salerno, “Fitch’s Paradox of Knowability,” in The Stanford Encyclopedia of Philosophy ed. Edward N. Zalta (Winter 2013 Edition), [].

[8] Brian Leftow, Time and Eternity, 318.

[9] This definition has a familiar ring to it, and I wonder if I’ve heard/read something similar to it in the work of either Plantinga or Craig. I could find no such reference, but I note that I have a curious itch here – I want to avoid any semblance of plagiarism, and so I should note that I have an uncomfortable suspicion that I may be, here, unconsciously regurgitating something very similar in prose to what one might find in Craig or Plantinga (or, perhaps, elsewhere?). As I say, I can find no such reference, and in any case the definition as stated really does proceed from my mind.

[10] (i.e., a being satisfying propositional omniscience, or a ‘propositionally omniscient being.’)

[11] If one suggests that within para-consistent logic there may be propositions which are both true and false, and therefore that omniscience is impossible if para-consistent logics possibly describe a world (accurately), my response would be that no logically possible world can be accurately described by a para-consistent logic.

[12] Patrick Grim, “Logic and Limits of Knowledge and Truth,” in Nous (1988): 349.

[13] Alvin Plantinga and Patrick Grim, “Truth, Omniscience, and Cantorian Arguments: An exchange,” in Philosophical Studies 71, no. 3 (1993): 267. I note that it is precisely in anticipation of this problem that PO as I have defined it is articulated in a way which makes no explicit or implicit set-theoretic commitments at all.

[14] Ibid., 269.

[15] Ibid.

[16] Alexander Pruss, The Principle of Sufficient Reason: A Reassessment, (New York: Cambridge University Press, 2006): 284.

[17] Ibid.

[18] Ibid., 238.

[19] William F. Vallicella, A Paradigm Theory of Existence: Onto-Theology Vindicated, Vol. 89. (Dordrecht: Kluwer Academic Publishers, 2002), 23.

[20] ‘Simple’ is here used in the sense of being non-composite.

[21] Immanuel Kant, “The Critique of the Power of Judgment,” translated by Paul Guyer and Eric Matthews, (New York: Cambridge University Press, 2002): 276.

[22] Also see: Keith Simmons, “On an Argument Against Omniscience,” in Noûs (1993): 22-33.

[23] Dennis Whitcomb, “Grounding and Omniscience,” in Oxford Studies in Philosophy of Religion Vol. 4, ed. John Kvanvig OUP (2012): 5.

[24] Ibid., 3.

[25] Ibid., 1.

[26] Ibid., 7.

[27] Joshua Rasmussen, Andrew Cullison and Daniel Howard-Snyder, “On Whitcomb’s Grounding Argument for Atheism,” in Faith and Philosophy 30, no. 2 (2013): 198.

[28] Ibid.

[29] Ibid., 199.

[30] Ibid.

[31] For example, he has argued by beginning with the assumption that “some person S knows that K,” where “‘K’ stands for the fact that < Someone has knowledge >,” but since this assumption is an instance of K, K appears to be grounding itself, from which it follows (given irreflexivity) that no person knows that K.

[32] Rik Peels, “Is Omniscience Impossible?,” in Religious Studies 49, no. 04 (2013): 481.

[33] Ibid., 487.

[34] For want of space I refer readers interested in the details to the paper itself, and in particular to pages 487-489.

[35] Patrick Grim, “Against Omniscience: The Case from Essential Indexicals,” in Nous (1985): 154.

[36] Alexander Pruss, The Principle of Sufficient Reason, 45.

[37] I do not necessarily endorse the particulars of his view, such as the insinuation that all beliefs have both a content and a role, or that

[38] Darren Bradley, “Dynamic Beliefs and the Passage of Time,” in Attitudes De Se, ed. A. Capone & N. Feit  (University of Chicago, 2013): 294.

[39] Ibid., 301.

[40] Ibid., 302.

[41] Ibid.

[42] Ibid., 303.

[43] Ibid., 301.

[44] Ibid., 295.

[45] Ibid., 303.

[46] Just as the B-theorist might concede to the A-theorist that some meaning will inevitably be lost when translating tensed expressions to tenseless expressions without thereby conceding that there are propositions whose truth-makers include a fact about what time it objectively is.

[47] Patrick Grim, “Against Omniscience,” 153.

[48] Brian Leftow, “Time, Actuality and Omniscience,” in Religious studies 26, no. 03 (1990): 309.


One thought on “Defending Propositional Omniscience: A Way Back to Full Omniscience

  1. Thanks for the posts; your apologetics are both helpful and insightful. Your work will be an important tool for those after truth and will provide resources for depeening their intellectual understanding of the Catholic faith.

    Your skepticism (and worry) of hearing propositional omniscience defined elsewhere is warranted; W.L.Craig mentions it throughout his podcasts (that’s why the literature you went through didn’t entail it). But, I found it too on a Question of The Week:

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