Archive for October, 2021


Can even an omniscient, omnipotent God have certain knowledge?

Or even an omniscient, omnipotent, omnibenevolent one, if that isn’t a contradiction (it seems to me to be, but I’m not certain).

First, we should establish what we mean by “certainty” or “certain knowledge”. An initial (but wrong) attempt would be that a subject S knows a proposition P “with certainty”, or “certainly”, or “has certain knowledge of P”, if S knows that P, and P cannot be false, in the sense that it is necessarily true, or true in all possible worlds, or in general that P in some relatively ordinary modal logic.

This is wrong, because it implies that all knowledge of necessary truths is certain knowledge. But that’s not the case; for instance I know that Khinchin’s theorem (the one with Khinchin’s constant in it) K, “for almost all real numbers x, the coefficients of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x”, is true. But my knowledge is not at all certain; my only evidence for it being true is basically the Wikipedia page and a tweet, and that is far from sufficient for certainty.

One could describe all sorts of situations in which the theorem might be false and still be in Wikipedia and on Twitter: the known proofs of the theorem might have a subtle flaw, the whole thing could be an internet prank, etc. Now if the theorem is actually true, it’s necessarily true, so the worlds described by those situations aren’t possible worlds (in the rather strong sense of mathematical or logical possibility), but they are “for all that I know possible”, and that’s the relevant property here.

In my undergraduate thesis I called this superset of possible worlds “conceivable worlds”, and that seems like a good-enough term for this little essay. The basic idea is that S knows P (for reasons R) iff S believes P, and P is true in all reasonably nearby conceivable worlds in which S believes P for R. (“Reasonably nearby” isn’t directly relevant here yet, but you get the idea.)

Note that here I’m saying only that I can’t be certain that K is true. I can definitely know that it’s true, and I claim that I do know that; it’s an assumption in this discussion that knowledge doesn’t require certainty.

So, if it’s not enough for certainty that S knows P, and that P is true in all possible worlds, what does give certainty? Is it that S knows P, and P is true in all conceivable worlds? That seems plausible. Perhaps equivalently, the criteria could be that S knows P, and that S is not required, as a responsible epistemic agent, to entertain any suggestion that P might be false. S would be reasonable, to put it another way, to risk absolutely anything at all on the bet that P is true.

To say that we are never certain of anything (and I don’t think we are), is to say that it’s never responsible to completely ignore any possibility that any given belief is false. We can certainly ignore contrary evidence in some circumstances; if I get spam about exciting new evidence that the Earth is flat, I can go phht and ignore it, because there are so many false claims like that around, and comparatively little rides on the issue. But there are other circumstances where this would be irresponsible; if some being asks me to bet a dollar against the lives of every person in Maine, that the Earth is not flat, it would be irresponsible of me to accept. That is, perhaps equivalently, my Bayesian prior on the Earth being flat is not quite 100%; if it was, it would be irrational not to take that bet and win the dollar.

For simpler necessary truths, like “two plus two is four”, it seems that one might claim certainty. But would it be responsible to bet a dollar against 1.3 million human lives, on the assumption that one is not somehow mistaken, or has been hypnotized into using the wrong names for numbers, or something so peculiar that one hasn’t thought of it? I think it’s pretty clear that the answer is No; weird stuff happens all the time, and risking lives that there is no sufficiently weird stuff in this case isn’t warranted. So, basically, we can never be certain.

(Another possible reading of “S knows P with certainty” or “S is certain that P” would be something like “S knows that P for reasons R, and in every possible / conceivable world in which S knows P for R, R is true, and furthermore S knows those preceding things”. I think it’s relatively clear that the “something weird might still be going on” argument applies in this case as well, and since something really weird might be going on for all we know, and mere knowledge is not certainty, we also never have certainty under this modified definition.)

Having more or less established what certainty is, and that we ordinary mortals don’t have it, we can now ask whether an omniscient, omnipotent (and optionally omnibenevolent) being G can be certain of anything. At first blush is seems that the answer must be Yes, because being omnipotent G can do anything, and “being certain” is a thing. But this is also like “creating a boulder so heavy that G cannot lift it”, so we ought to think a bit more about how that would work.

G is omniscient, which we can safely take to mean that for all P, G believes P if and only if P is true. I think we can also safely grant that for all P, G knows P if and only if P is true. That is, for all and only true propositions P, G believes P, and in every reasonably nearby conceivable world in which G believes P, P is true. (Note that I’m mostly thinking these thoughts as I go along, and have just noticed that I have nothing in particular to say about the reasons R for which G believes P. We’ll see if we need to think about that as we proceed.)

Since G knows P, and G is omniscient, G also knows that G knows P, and so on. But can G be certain of one or more propositions P? Is God’s Bayesian prior 100% for all, or any, true propositions?

Just what is G’s evidence for any given P? How does God’s knowledge work? (Ah, apparently R is coming up almost immediately; that’s good!) G knows “I am omniscient,” so that’s a start. It seems that, given that, G could go from “I believe P” to “P” more or less directly. That feels rather like cheating, but let’s let G have it for the moment.

If G’s evidence for P is typically “I believe P” and “I am omniscient”, can G get certainty from that? We are often somewhat willing to grant “incorrigibility” to beliefs about one’s own beliefs, and while normally I’d say that only gets one to the level of knowledge, let’s stipulate for the moment that G can be certain “I believe P” for any P for which that’s true.

But what about “I am omniscient”? How does G know that? What is G’s evidence? Can they truly be certain of it?

There is a set of possible beings Q, each of which believes “I am omniscient”, but is mistaken about that. Some of them are quasi-omniscient, and know everything except for some one tiny unimportant detail D of their universe, of which they are unaware, and for the fact that they are not omnipotent due to not knowing D. They believe two false propositions: not-D, and “I am omniscient”. They also believe an infinite number of other false propositions, including those of the form “not-D and 3 > 2” and so on, and perhaps some that are more interesting (depending among other things on just what D is).

Other members of Q are simply deluded, and believe “I am omniscient” even though their knowledge of their universe is in fact extremely spotty, even less than the average human’s, and they have just acquired the belief “I am omniscient” through trauma or inattention or that sort of thing.

Now, how does G know, how is G certain, that G is not in the set Q? This seems like a hard question! For any reasoning that might lead G to believe that G is not in Q, it seems that we can imagine a member of Q who could reason in the same way, at least in broad outline. G can do various experiments to establish that a vast number of G’s beliefs are in fact true, but so can a member of Q. The quasi-omniscient in Q can do experiments not involving D, and the deluded in Q can do experiments but then be mistaken about or misinterpret the results, or unconsciously choose experiments which involve only members of their comparatively small set of true beliefs. G can follow various chains of logic that imply that G is not in Q, but so can many members of Q. The latter chains of logic are invalid, but how can G be certain that the former aren’t?

One conventional sort of being G is one that not only knows everything there is to know about the universe, but also created it in the first place, and in some cases resides outside of it, sub specie aeternitatis, observing it infallibly from the outside. Such a G has a comparatively easy time being omniscient, but can even this G be certain that they are? Again, we can look at various members of Q who also believe that they created the universe, exist outside of it able to see and comprehend it all at once, etc. They have, at least in broad terms, the same evidence for “I am omniscient” that G does, but they are mistaken. The quasi-omniscient created D by accident, perhaps, or created it and then forgot about it, and when looking in from the outside fail to notice it because it is behind a bush or equivalent. The deluded in Q are simply mistaken in more ordinary ways, stuck in a particular sort of psychedelic state, and so on. Those in Q who are in-between, are mistaken for in-between sorts of reasons.

There may be many ways that G can determine “I am not in Q” with relatively high reliability. One can imagine various checks that G can do to confirm “I am not like that” for various subsets of Q, and we can easily grant I think that G can come to know “I am not in Q” and “I am omniscient.” But the question remains of how G can become certain of either of those. It seems that any method will fall victim to some sort of circularity; even if G attempts to become certain of a P by exercising omnipotence, we can imagine a member of Q (or an enhanced Q-prime) who believes “I am omnipotent and have just exercised my omnipotence to become certain of P”, but who is mistaken about that. And then we can ask how G will determine “I am not like that” in a way that confers not merely belief or knowledge, but certainty.

We can imagine a skeptic’s conversation with G going from a discussion of G’s omniscience, to questions about the certainty of G’s knowledge of G’s omniscience, to questions about how G knows that G is not in some particular subset of Q, and ultimately to how G knows that G is not in some subset of Q that the discussants simply haven’t thought of, and to how G can be certain of that. It would seem that every “since I am omniscient, I know that there is no part of Q that we haven’t covered” can be called out as circular, and that “I know that I have certain knowledge of P” can always be countered by observing that knowledge does not imply certainty.

Another consideration (pointed out by a colleague who asked if I wasn’t anthropomorphizing God) is that I am assuming here that G forms beliefs and qualifies for certainty in roughly the same way(s) that we humans do. But shouldn’t we expect G’s ways of knowing and being certain to be very different from our mortal ways? Why should these human-based considerations apply to G at all? The response, again, is that members of Q, and in particular deluded human members of Q, could make exactly that same claim: that they come to know things with certainty through special means that aren’t accessible or understandable to mere humans, and to which none of this logic applies. And, again, if a deluded human member of Q could make this claim without in fact having certainty of anything, then G’s making this claim has no useful traction either, unless G can separately demonstrate not being in Q, which is exactly the point at issue.

To put it roughly and briefly, perhaps, the question comes down to how even G can be certain that G is not just a deeply deluded mortal human, who (in addition to lots of ignorance and false beliefs) is very good at self-deception. Even if G truly is omniscient, can G be certain of that? If so, how? Is it perhaps fundamental to the nature of knowledge and evidence, that no knowledge is ever certain knowledge? Is certainty, more or less by definition, a property that no knowledge can ever in fact have?

And is there anything more to this question than the perhaps rather casual and vaguely-defined (and/or perhaps very important) idea that however confident some being might be in their evidence for a belief, there might always be something weird going on?

That was fun!