## Wednesday, October 6, 2010

### Once More On The Stone Paradox (This Time With Symbols)

A quick summary of that post:

I think the reductio argument against theism goes through, that the standard response (watering down the definition omnipotence) is ad hoc and unconvincing, and that the initially-more-promising-sounding response one ocassionally hears (that God could create such a stone, and if He did so, He wouldn't be omnipotent any more, but so long as he happens to contingently continue to choose not to do so, He's still omnipotent) is completely hopeless on closer examination.

In any case, for the ease of quick reference, we can refer to the former move as the Standard Defense (SD) and the latter as the Mere Possibility Defense (MPD).

In the comments, Emil pointed me towards this formulation of the Mere Possibility Defense, from Norman Swartz. At first, I was a bit annoyed at reading it, since he simply asserts that the MPD works, without bothering to argue for it.

On second thought, I actually find Swartz's formulation useful, since his symbolization of versions of the argument helps clarify nicely where the philosophical fault lines are. Here's his symbolization (where "God is omnipotent" and "M" = "God makes an immovable stone"), re-formatted slightly because of the limits of easy symolization on the Korean computer I'm typing this on at the moment:

1: G → ◊M
2: ◊M → ~G
-----------------------
3 (from 1 & 2): G → ~G
4 (from 3): ~G

Swartz admits, of course, that this is a valid argument, but he thinks that it's unsound, because he denies Premise 2. I think both premises are true, so I take the argument to be sound.

This is a nice, useful symbolic formulation, though, for several reasons. It certainly captures the way I think of the anti-theistic argument from the Stone Paradox, and also nicely demonstrates why the Frankfurt-type response ("God could create such a stone, and He could lift it! After all, if His omnipotence lets him do one impossible thing, why not another?") that seems to strike so many people as being so clever is actually quite silly and irrelevant. The whole point is that the notion of an omnipotent being existing is inconsistent. Frankfurt's recommendation that the theist joyfully embrace the inconsistency simply underlines the point. One can do that--just as one can respond to Russell's Pardox by continuing to embrace naive set theory but paraconsistentizing one's logic--but what of it? As an argument (assuming the principle of non-contradiction) against theism, it goes through.

Formulated in these terms, the Standard Defense revolves around denying the first premise. This requires adding an inconsistency-avoidance epicycle to the definition of omnipotence. In last week's post, I argued that there's no particular reason why such a move should be more plausible here than in naive set theory, or any other instance of a general principle producing contradictions. (Indeed, I argued that it would have been far less ad hoc if naive set theorists had done this than it is for theists do so.) Of course, as I said then, if a theist has sufficiently compelling external reasons to think that God exists, this move might still result in the overall best explanation of the data, despite the ad-hocness. That's fine. But the argument does give us a reason (albeit a defeasible one) to reject theism.

The Mere Possibility Defense revolves around rejecting Step 2.

(It looks like, if one is both a theist and a classical logician, and thus unwilling to follow Frankfurt's advice and simply accept the contradictions, one had better make at least one of these two moves.)

In last week's post, I argued that, if one thinks that God contingently hasn't happened to create an unliftable stone, His inability to lift it is still a limit on his powers. After all, "powers" are always and everywhere counterfactual. I gave two analogies, one quite long and developed, but the point was this:

If Object X contingently doesn't happen to exist, that's quite irrelevant to whether Agent Y could perform Action Z to Object X. If Object X exists, that's epistemically relevant--we get to test whether Agent Y has the ability to perform Action Z--but the non-existence of the test doesn't normally entail anything one way or the other about the power.

For anyone who doesn't feel like clicking through to last week's post , here's the relevant passage:

*******************

Take an obese chain-smoking alcoholic named John. Despite his many health problems, he has the ability to climb a few flights of stairs without having a heart attack and dying. It seems fair, though, to say that John's stair-climbing abilities are limited. He could not, for example, climb a hundred flights of stairs without having a heart attack. Whether or not either of these situations will ever actually come up--e.g. whether John ever climbs stairs or he exclusively frequents buildings with elevators and escalators, whether John lives close enough to a city with a hundred-story building in it that he could attempt this feat if he were unwise enough to try it, etc.--seems quite irrelevant to our talk about John's powers. If John is a North Korean whose government will never allow him to travel to a place with hundred-story buildings, that doesn't seem to impact the truth of our statement about the limitations on his stair-climbing abilities. Nor would it, indeed, matter if, as a matter of contingent fact, the tallest building in the world happened to be ninety-eight floors tall.

To brings things closer to the God case, imagine a possible world where John--still an obese chain-smoking alcoholic--is the undisputed absolute ruler of the planet. Nothing can get built without his say-so, and he refuses to allow any building on earth to be constructed higher than four stories.

One day, two of his subjects--Jim and Jerry--are having a quiet conversation, perhaps in a quiet stairwell in one of the many four-story buildings where, as far as they know, John's secret police hasn't bothered to install any CCTV cameras or listening devices. They like to go there sometimes to hold the kind of private conversations that Winston Smith and Julia enjoyed in the early parts of Nineteen Eighty-Four.

At one point, Jim boldly speculates that, based on how pudgy and red-faced and out-of-breath Emperor John looks in the newsreels, the reason why he never allows buildings to be built over four feet high is that he doesn't have the ability to climb more flights of stairs than that and he wants to avoid the embarrassment. Jerry responds that, well, he could imagine John climbing as many as five or six flights of stairs without having heart attack, but there's no way he's healthy enough to climb, say, a hundred flights of stairs without collapsing.

At this point, of course, just like the capture scene in Nineteen Eighty-Four, Jim and Jerry find out that the secret police was listening all along, and both are tortured with rats in Room 101 until they admit that two and three make six if the Party says they do, and that Emperor John has the power to climb thousands of flights of stairs without physical setback.

Now, how would we evaluate Jerry's original claim about the limit's on John's stair-climbing abilities?

Given the innately counterfactual nature of all ability/power/powerfulness talk, the fact that John hasn't happened to create any such stairs, and has thus deprived himself of the opportunity to expose this particular limitation on his stair-climbing powers, seems quite irrelevant to the truth of Jerry's claim. Just so for God and unliftable stones.

************

So....for anyone who wants to argue that the argument symbolized above valid but unsound, here's the challenge:

Either (a) explain the relevant disanalogy between the move made by theists who employ the SD and put a logical-consistency epicycle in their new, watered-down definition of omnipotence, and other cases where a general principle generates contradictions, and we all think that the rational response is to reject the general principle rather than stick in a consistency epicycle, (b) provide an argument for theism so devestatingly convincing that it justifies the SD as the overall best explanation even in the face of the ad hocness, (c) explain the relevant disanalogy between the contingent absence of hundred-floor buildings in the John-ruled world (which seems irrelevant to the limits on John's stair-climbing powers) and the contingent absence of an unliftable stone in the God-ruled universe (which, according to partisans of the MPD, is relevant to the limits of God's powers), or (d) provide an alternative way out, thus showing that (a)-(c) aren't jointly exhaustive of the options for defenders of the doctrine of divine omnipotence who want their beliefs to be closed under some sort of (non-paraconsistent) logical consequence relation.

Emil said...

First, see the comment on the other post.

Second, I can't of any good reason to think 2 is true. It seems obviously false to both me and Swartz and many other people that I explained the stone argument to.

Why do you think that it is true?

Since I can't make sense of your analogy, it does not act as a justification for me. Did I mention that I find analogies unsuitable for good justification because that they are not formalizable or otherwise able to be validated by logic, AFAIK.

Using possible world semantics, we can see why 2 is false. 2 says that if there is a possible world, w, where God does create the stone he cannot lift, then God is not omnipotent in the actual world. Surely this does not follow.
But, if we also have as a premise that God is a necessary being, then the conclusion does. Consider these 4 propositions:

1. For any possible world, w, there is a being, x, such that x is God in that world.
∀w∃xGxw

2. For any possible world, w, and for any being, x, if x is God in that world, then x is omnipotent in that world.

∀w∀xGxw→Oxw

3. For any possible world, w, and for any being, x, if x is omnipotent in that world, then there is a possible world, w', such that in that world x creates a stone that he cannot lift in that world.

∀w∀xOxw→∃w'Cxw'

4. For any possible world, w, and for any being, x, if x creates a stone he cannot lift in that world, then x is not omnipotent in that world.

∀w∀xCxw→¬Oxw

These 4 are inconsistent.

Ben said...

Emil,

Did you see my reply to your comment on the previous post?

To re-cap:

In the piece you linked to, Swartz certainly didn't symbolize his reasons for finding 2 false, anymore that I symbolized mine for finding it true. It's not the kind of thing that's conducive to clarification through symbolization, since it's a question of providing conceptual justification for the symbolized claims.

To go back to the analogy:

You've said you can't make sense of it, but you have yet to either bite the bullet or explain a relevant disanalogy. So: Would you agree that someone who made the parallel move, and said that the mere fact that John could order the creation of a building with more floors that he could climb in some possible world doesn't mean that there are limits on John's stair-climbing abilities in the actual world was confused? If so, what's the difference?

Ben said...

Or, to try again, from a slightly different angle:

Whether some Agent A has or lacks Power P in World W does not depend on contingent happenstance of whether or not A has chosen in W to give themselves a test of whether or not they can P. If, in world W*, which is relevantly similar in every other way, and in which A has decided to create a test of their P prowess, A fails that test, A lacks the ability to do P in the actual world.

This reasoning seems intuitively clear (at least to me....if you disagree, I'd be interested in hearing it) when it comes to non-God analogies, making me wonder (if you do agree in that case) what relevant disanalogy you think sinks it in the God case.

You say:

"2 says that if there is a possible world, w, where God does create the stone he cannot lift, then God is not omnipotent in the actual world. Surely this does not follow."

I think I see how you're looking at this--in some other possible world, there's no omnipotent being (because there's an unliftable stone), but that doesn't rule out an omnipotent being in the actual world.

Is that a fair summary of your point?

If so, it usefully isolates the disagreement. I don't think God would be omnipotent in the contingent absence of a divine decision to create an unliftable stone, any more than John has the ability to climb a hundred flight of stairs in the contingent absence of a decision on his part to order a building that tall built.

Emil said...

This platform is wholly inadequate for discussion. Do you have a recommendation for some other way we can discuss this?