Thursday, December 11, 2008

A Cantorian Argument for Open Theism?

I'v just read an interesting paper on Enigman's website entitled "Omniscience and the Odyssey Theodicy". At one point in the paper, he employs Patrick Grim's well-known Cantorian argument against omniscience to argue for open theism over against an essentially epistemically static (EES) deity (my term, not his). The argument is intriguing.

According to set theory, anything can be a member of a set, including other sets. Hence, there can be no such thing as the set of all sets, for reasons pointed out by Cantor. Take any non-empty set (e.g., {1,2}) and form the set of all subsets (the power set) of the original (e.g., {{∅}, {1}, {2}, {1,2}}). That set will always have more members than the original set. And since this operation can be carried out for any set, there can be no such thing as the set of all sets.

This creates a problem for omniscience if that notion is defined set-theoretically, e.g., believing every member of the set of all truths. One can argue along Cantorian lines that there can be no such thing as the set of all truths. That is, for every set of truths, one can construct new truths that are not already members of the set.

As Enigman points out, if this is right, then it follows that an essentially epistemically static (EES) God - i.e., a God who cannot either acquire or lose beliefs - cannot be omniscient (in a set-theoretical sense). Such a God cannot know all truths. Moreover, there would have to exist truths that are forever outside the ken of an EES God. In contrast, according to open theism God is not epistemically static. He can acquire new beliefs. Indeed, his knowledge is, as Enigman puts it, "indefinitely extensible". As a result, there are no higher-order set-theoretical truths that God cannot eventually come to know. And this constitutes an advantage for open theism. On neither account can God know all truths, but, unlike EES theism, open theism is compatible with the idea that there are no truths that God cannot come to know.

Now, I think this is a very interesting argument for open theism, one that I have not heard of or considered before. I'm not sure that it's sound in its present formulation, though, because I'm not sure that omniscience is best understood in a set-theoretical fashion.

I would suggest that omniscience, in its primary sense, is best construed as a kind of knowledge by acquaintance. On this view, "omniscient" means being fully acquainted with all that is, where "all that is" need not be conceived as a set of discrete constituents, but rather as a continuous field. Here's an analogy. Consider a continuous plane surface. On that surface one may analytically isolate or pick out individual points and lines. In so doing, one brings those points or lines to the foreground, so to speak, but only over against a background, the continuous surface. Since the surface is continuous, no analysis of it can bring all of it into the foreground. There are always more points one could identify, more lines one could draw, etc. If this is right, then an omniscient God's knowledge doesn't come in presliced propositional packets; rather, it exists as a plenum of intelligibility. (I take the term "plenum" from Richard Creel's book Divine Impassibility.) Any part of that plenum can be analyzed out of it as a proposition, but no set of propositions can exhaust the plenum.

Now, even if that way of thinking about omniscience is on the right track, it may still be possible for an Enigman-style argument in favor of open theism to get off the ground. For if the plenum cannot be exhaustively analyzed in propositional terms, then no deity can have exhaustive propositional knowledge. However, an EES deity, on the one hand, is permanently stuck with whatever propositional knowledge he starts out with, whereas an open theist deity's propositional knowledge is indefinitely extensible. There is no point at which he can bring the entirety of the plenum into the foreground, but at the same time there is no part of the plenum that cannot be brought into the foreground. Hence, even on this analysis of omniscience an open theist God turns out to have a higher-quality sort of omniscience than an EES God.

By the way, while I don't endorse Enigman's "Odyssey theodicy" I think that's a really cool name. Kudos to Enigman for a stimulating paper.


At 12/11/2008 9:51 PM, Blogger Paul said...

It is an interesting idea, and it raises all sorts of questions.

How many new propositions can the open God familiarize Himself per second? If a finite number, then there are real numbers between 0 and 1 that God will never learn though He live forever. If an infite number (e.g. aleph-null), then there are members of infinite sets of a still higher cardinality that He will likewise never learn though He live forever. Perhaps God learns new propositions at an ever increasing rate, each second increasing the number of propositions that He learns by a new cardinality of infinity.

How many propositions does God know right now? An infinite number? A finite number? If a finite number, then how can He ever learn an infinite number of sets in a second, let alone an increasingly infinite number? If an infinity of a fixed cardinality, then can one work backwards to find out when God began conceiving sets?

I don't think open theism solves this problem so elegantly as the Enigman thought.

At 12/12/2008 11:40 AM, Blogger Heather said...

I'm just wondering...

Why must we think of the # of sets out there as ever expanding rather than already expanded? I'm no set theorist, but it appears that the # of sets existing is infinite. And since omniscience is knowing all that can be known, and assuming that the # of sets existing is also knowable, why can't God, who has infinite knowledge, already have knowledge of this infinite number.

Enigman's argument (haven't read it.. only the summary here) also doesn't account for God's being eternal (ie, always existed and will always exist).

For God to "come to know" these things, who invents the sets for God to come to know them? I propose the sets have always existed in the mind of God. And with God being immediately transparent to Godself, God's knowledge of these sets has also been there for the same duration.

I hope what I'm writing here makes sense. No time to edit...

At 12/13/2008 2:30 AM, Blogger Enigman said...

Hi Alan,

I'm glad you liked the theodicy's name, although I've changed it again (it was originally the Theodolite theodicy) to a Jump theodicy (by comparison with Fall theodicies). I came to think that the alliterative name was a bit distracting... But I'd like to know what you don't like about the theodicy. I'm pretty sure that it needs tidying up in all sorts of ways, but I've got as far as I can with my own objections (to its previous versions).

Hi Paul,

Elegance is a subjective thing, though. Prima facie alchemy was pretty elegant, compared to all the elements scientists knew about a hundred years ago. But then we got Bohr's atom, and chemistry turned out to be much more elegant than alchemy. Most crucially, it is also true, whereas the elegance of alchemy was forced. Ultimately, the truth is more elegant, or so it seems.

If God has been doing arithmetic forever, in some beginningless sequence, I don't know. I think that is not the true way of things. But if all his changes must be finite, then he knows a finite amount of arithmetic, and he began doing arithmetic some definite time ago. The elegance of that idea only really emerges when you think through the transfinite option. But one thing to note early on is that the finite gets unimaginably huge. It's not as though he only knows a finite amount of arithmetic.

Hi Heather,

One can think of the sets as all there. There is then a biggest number, the number of all sets, usually denoted by a capital omega. My problem with that is that it seems that all of the sub-collections of that biggest collection should also be there, just because they are parts of that biggest collection. And they should be distinct. And so they should have a number. Cantor's diagonal argument shows, it seems to me, that that would be a bigger number. Cantor himself thought that there was a biggest collection (such as God would know) but that we could only see it as inconsistent because of our limited epistemic powers. I guess that was quite Kantian of him. As a natural theologian I can't do anything with that idea, but it does make sense. Graham Priest is the guy to read on paraconsistency (true contradictions).

The sets exist as structural possibilities, I think. If a thing (of some kind) is possible, then it is possible (mathematically) that there is another thing, and then there would be two of them. So we have the arithmetic 1 + 1 = 2. So arithmetic is implicit in the concept of a thing, I think. By doing arithmetic God would be drawing out the consequences of the original concept. God would have a complete grasp of the concept of a thing, I think, but would only be able to draw out the arithmetical consequences indefinitely. That is not really a limitation, just a fact about such consequences, I think (Cantor would disagree).

I suspect that the intuition that God should know all arithemtic is in that sense a bit like the intuition that God should know all that we would do in any possible situation, just because he would know (having made us) all about us.

At 12/17/2008 8:53 PM, Blogger Alan Rhoda said...

Hi Enigman,

"Odyssey theodicy" had a nice ring to it. I'm sorry to see it go.

Anyway, the main reason why I'm reluctant to endorse your theodicy is that it's predicated on the idea that God might not be able to know for sure whether he is the only deity. To my mind, that issue could only arise for an essentially finite God, not a fully omnipotent, omniscient being. Now, perhaps you're right that omniscience, as usually understood, has been refuted by Cantorian arguments. I'm not convinced that that's right, but even if it was, divine omnipotence would still be enough to rule out other deities. Perhaps you think that attribute is also impossible as it is usually understood. Again, I'm not convinced that's right. So, in the end it is for broadly Anselmian reasons that I can't endorse your theodicy. It seems to me to place unnecessary limitations on the greatest possible being.

At 12/23/2008 2:31 AM, Blogger Enigman said...

divine omnipotence would still be enough to rule out other deities

One of my anonymous reviewers said something along those lines. What s/he said was actually a Straw Man fallacy though (I've just posted on he/r objections on my blog btw). Could you say a little more about why it would rule them out in an apposite way? I know that if there are other deities, then God is not omnipotent, but I don't deny that God is omnipotent. I thought that I said lots about that difference in my sketch of my theodicy, and I don't know what I failed to say, or where it was unclear. Any help you can give me?

At 12/29/2008 7:07 AM, Blogger Enigman said...

Incidentally, I've just read A New Defense of Anselmian Theism, by Nagasawa (in The Philosophical Quarterly), which is good I think, and there he claims that there has been no good philosophical argument from Anselmian theism to God being omnipotent. In fact, my theodicy requires a much weaker limitation. E.g. I think that God has the greatest possible amount of knowledge, and is omnipotent. But I was wondering if you have the argument that Nagasawa claims is lacking?

At 12/31/2008 6:53 AM, Blogger Enigman said...

That last comment was me trying to play the professional game of philosophy. Sadly, I'm just an amateur scientist (literally, a philosopher). So it occurs to me to note that even I don't endorse my theodicy.

When I realised that I had to take the God hypothesis seriously, I just felt like answering the problem of evil, and my Jump theodicy was the only possibility that I could find that seemed even remotely plausible to me. But I don't completely understand everything that is said about the existing theodicies. And I've yet to be seriously challenged on my theodicy. It may fall apart. I can't see how, but that's not saying much until it's been seriously challenged by a number of professional philosophers... (hint hint:)

At 1/05/2009 6:20 AM, Blogger Enigman said...

Not wishing to be buggy or anything, but I've just tidied up the presentation of my Jump theodicy slightly, which is now here.

At 1/06/2009 8:06 PM, Blogger Alan Rhoda said...

Hi Enigman,

I've been out of town most of the past three weeks. I hope your new year is off to a great start.

Anyway, you asked on 12/23 about the possibility of there being two (or more) omnipotent beings. Now, I don't have an airtight refutation of this idea, but it does confront an obvious difficulty, namely, what could prevent two omnipotent beings from being at odds with each other? If they could be odds, then we face something like the paradox regarding what happens when an irresistible force meets and immovable object.

To avoid the paradox, the defender of the idea of multiple omnipotent beings has to explain why it is not possible for two omnipotent beings to be at odds. For all I know, there might be a good explanation. Perhaps such beings would also be so good that they would always get along or so smart that they would realize that fighting isn't in either of their interests.

At 1/06/2009 8:22 PM, Blogger Alan Rhoda said...

Regarding Nagasawa's argument, I haven't seen his paper, but it seems to me that a cogent argument that a greatest possible being would be omnipotent is easy to sketch.

1. Having power is a great-making property.
2. All other things being equal, having more power is better than having less.
3. Having power has an intrinsic maximum (i.e., there is a point beyond which it doesn't make sense to suppose that a being could have more power).
4. As the greatest possible being, God possesses a maximal consistent set of great-making properties.
5. Having maximal power doesn't conflict with any other great-making properties.
6. Therefore, God has maximal power (i.e., is omnipotent).

I grant that premises 3 and 5 of this argument stand in need of further defense. In particular, we need a plausible analysis of the notion of "having power" that makes all of the premises true. In the absence of such an analysis (which I don't have to offer), the above should not count as a proof.

At 1/08/2009 2:33 AM, Blogger Enigman said...

Hi, I've had worse new years. I hope town is good (is that Las Vegas? If so then I'd guess 'good' was the wrong word:)

Regardng two omnipotent beings, maybe they are impossible, necessarily impossible. I presume they are, because I presume that God is metaphysically necessarily unique. God is also conceptually necessarily unique under some definitions (e.g. of 'divine') of course. But that latter necessity is clearly rather stipulative. The substantial sense is the former one. But then there are the epistemic necessities. And those are the ones that my theodicy is based around.

So to defend that possibility, I only need that the two possible divinities either are not omnipotent, or that their omnipotencies would not contradict each other. I don't presume the latter, as I think of omnipotence as involving perfect freedom, and take a Divine Command view of metaethics. But the latter seems possible. Maybe we sin because of our finiteness, and two perfect beings would always be in harmony. But as I say, I take the former option. If God was not unique, he would not be omnipotent, I think. But the epistemic possibility relevant to my theodicy is only that God does not know for absolute certain that he is unique, that he is omnipotent, not that he is not unique, or that he is not omnipotent.

Regarding Nagasawa's argument, it involves - so far as I can recall without looking at it (since I access the internet away from my own computer)- the idea that there might be a being with more power than God but less of the other attributes. I think he says that no-one has proved that that's impossible, something like that. That may be to question your (5). Also, his argument may be vulnerable to the fact that there is a simple divine essence.

At 1/08/2009 4:28 AM, Blogger Enigman said...

Re that epistemic possibility, I do presume that God has perfect self-awareness. He's therefore infallibly immune to Cartesian sceptical scenarios. And he knows that his range of possible creations is perfectly symmetrical. But could he know that it was perfectly complete, via his perfect self-awareness? The analogy I use (in section 3 of my essay), which suggests otherwise is that of one who knows that she can move freely anywhere within an infinitely dimensional space. Does she know that she has complete freedom of motion? The problem is that such a space is isomorphic to a hyperspace containing such a space as a mere slice.

I've been racking my brains trying to think of how God could be fully justified in being completely certain that he was unique (perhaps triunely), but as yet nothing has occurred to me.


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