Wednesday, September 29, 2010

Some Further Points About The Stone Paradox

In the "Analytic Philosophy" class I'm teaching in Korea this semester, this week we're covering the two chapters of Bertrand Russell's book My Philosophical Development concerning his and Whitehead's work writing Principia Mathematica. I always find the following fact fairly striking:

Following the discovery of Russell's Paradox, Russell got around it by means of his somewhat awkward and complicated theory of types. ZFC and other orthodox set theories got around it by the "hierarchical conception of sets" and similar means. In more recent decades, a few logicians on the radical fringe have argued for rehabilitating naive set theory at the expense of the Law of Non-Contradiction. Practically no one, though, seems to have thought of responding to the paradox by simply amending Frege's Basic Law V to something like Basic Law 5.1:

"Wherever it is logically possible for there to be a set of all and only the objects matching some description, there actually is such a set."

On a slightly different but closely related note, here's a true story about a friend of mine, D.:

One day in Hebrew school when D. was 12 or 13, the Rabbi was talking about how God can do anything. D. responded by pointing to the door of the classroom, which was always sticking and never quite closed. D. (who grew up to be a lawyer, and fondly refers to this as his "first cross-examination") pointed at the door and asked the Rabbi, "could God make that door closed all the way?" The Rabbi said "of course." Then D. asked, "could God make that door closed so that no one would open it?" Finally, D. asked, "could God make that door closed so that even God couldn't open it?" The Rabbi hemmed and hawed and never gave D. a good answer, and D. promptly gave up on belief in God.*

If, however, D. had gone into a Philosophy PhD program instead of law school, he might have discovered, through the writings of folks like Richard Swinburne, the standard theistic response to his point from Hebrew School, and that--again, the standard theistic response, indeed, almost the only response one ever encounters from philosophically trained theists to the worry--is, transparently, just a theological version of Basic Law 5.1 above. The previously mentioned Professor Swinburne tries to argue, in The Coherence of Theism, that God can perform any action, but that things like "create a stone an omnipotent being could not cause to rise" don't count as "actions." Other rather more intellectually honest theists just clarify that by "all-powerful" they mean that God can do anything that's logically possible given His omnipotence, not that God can do anything at all.

So, what, on earth, is the difference between the two cases?

Well, in terms of concern about ad-hocness, Basic Law 5.1 actually fares rather better than its theological counterpart, as it would represent a reversion to Cantor's original definition of "set," at the very dawn of naive set theory**, whereas the level of intellectual sophistication necessary for subtle caveats about "anything that's logically possible" comes rather late in the overall history of the Judeo-Christian religious tradition.

Other than that, I really have no idea. The uncharitable suggestion that rather forces itself on one after thinking about it for too long is that, by and large, set theorists are simply a bit more scrupulous than theistic philosophers when it comes to paying attention to epistemic standards like "general principles that need to have 'except when this produces a contradiction' caveats at the end of them are a lot less likely to be true than those that organically fail to produce contradictions." If one wanted to follow this thought to its natural conclusion, the sociological and psychological factors involved in filling out this story (even when it comes to very bright and otherwise epistemically scrupulous theists) aren't terribly hard to come up with.....bluntly, no one is, as a small child, indoctrinated by all the adult authority figures in their life to believe in naive set theory, so that, as an adult, they have an enormous antecedent emotional need to preserve their belief in Basic Law V.

Alternately, here's a (much more charitable) explanation:

The fact that the move from Basic Law V to Basic Law 5.1 is a bit ad hoc, that general principles are more plausible when they lack such epicycles, gives us some reason to think that naive set theory so amended gets things wrong. Similarly for the move from "naive omnipotence" to "God can do anything as long as no contradiction would follow from an omnipotent being performing the action." Still, the fact that the need for the epicycle at the end of that last sentence gives us some reason to believe that the theory in question is false isn't decisive if we have some tremendous independent reason to believe that the theory is true. Put crudely, you can get a point deducted for ad-hocness but still get more points overall than the alternative views.

If a theist had this view--they believed they had some excellent argument for the existence of God in their back pocket*** and that this justified belief in omnipotence-with-an-epicycle--that would be fair enough. Of course, that would involve acknowledging that the Stone Paradox gives us some reason to disbelieve in the existence of God, even if not a decisive one, and, at least anecdotally, that kind of attitude seems surprisingly rare.

In any case, though, it might be objected (indeed, commenters on previous posts where I've brought this sort of thing up have objected) that this whole discussion only applies to theists who take God to be omnipotent in all possible worlds, whereas a viable alternate version (we can call it Omnipotence 3.0) holds that God is omnipotent-without-epicycles in the actual case, and that He could indeed create an unliftable stone. If he chose to do so, He would, at that point, no longer be omnipotent--having just limited his power by creating a stone he couldn't lift--but that this counterfactual point doesn't bear on His actual omnipotence.

For a while, I thought this was a pretty good defense--certainly a lot more promising than the near-universal Standard Theistic Response--but, on second thought, I don't actually find it very plausible.

Here's why:

Talk of "abilities" or "powers" is counterfactual through and through.

Even if a man is part of a distant rainforest tribe that has never heard of baseballs, it can still be true of them that they have a powerful enough throwing arm to toss a baseball across a large field. This is an *actual* attribute of his, despite the fact that it doesn't come up in the circumstances of their life. If the tribesman in question was flown to America, shown a baseball, and he did indeed toss is across a large field, this would decisively confirm that he had the ability, but his having the ability is unaffected by the absence of the test.

Another example:

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.












*With D.'s permission, I used this incident in my short story Dark Coffee, Bright Light and the Paradoxes of Omnipotence, which is going to be reprinted this winter in the Prime Books' anthology People of the Book. (I'll admit to being pretty excited about that, since I get to share a Table of Contents with the likes of Neil Gaiman and Michael Chabon.) At the time, I asked him, "is it OK with you if I steal some of your life story, and portray it as part of the life story of a character who's (a) gay and (b) a terrorist?" His response was, "geez, Ben, how do you know I'm not either of those things? Also, yeah, sure."

**Cantor distinguished between "sets" whose members could be consistently jointly thought of as one thing, and "inconsistent multiplicities" that could not. I suspect that part of the reason that people didn't respond to Russell's Paradox with, "oh well, I guess Frege's wrong, but Cantor's right," is that, while he was certainly a brilliant mathematician who contributed many still-interesting proofs, Cantor's foundational ideas about the nature of set theory were never clearly and systematically laid out the axiomatic way that Frege's were, and a lot of his extant writings that touch on it (essays, letters to Dadekind, etc.) are full of unclear assumptions, weird religious baggage, quasi-mystical beliefs about "true infinity" and so on. For anyone interested in plunging into those waters, though, a good place to start is Michael Hallett's book Cantorian set theory and limitation of size.

***Given the large flaws in the standard contenders (cosmological, teleological and so on), I'm extremely skeptical that such an argument exists, but that, of course, is quite outside the subject matter of this post. At any rate, my (largely hypothetical) problem-acknowledging theist merely has to sincerely take themselves to have such a good argument, not to actually have one.

13 comments:

Daniel Lindquist said...

Basic Law V was supposed to be a law of logic. The laws of logic delimit logical possibility. Your Basic Law 5.1 requires knowing what is logically possible prior to one of the laws of logic.

The standard response to the "paradox" of the stone presumes a prior understanding of possibility, but does not itself purport to delimit possibilities, so there's no circularity to it.

MichaelPJ said...

Well, the thing is that strictly speaking that isn't Basic Law V. Basic Law V was all about concepts and when they're the same (F= G iff Fx <-> Gx). Basic Law V does look like a plausible law of logic.

What Ben is (actually) talking about is the naive Comprehension axiom, which says that for any statement in the language of set theory phi, {x|phi(x)} is a set. Now, in set theory where we're actually talking about sets rather than concepts the problem is more obviously caused by naive comprehension, and so most modern set theories limit it in some way (in ZFC, the domain from which the xs come has to be a pre-existing set).

As for Ben's proposed variant, I have to say that when I first learnt about Russell's paradox it seemed like the obvious solution to me! I think there are two main problems: firstly, previously were just using any sentence in TLOST we wanted to define a set. Now we can only draw from some difficult to define class of "sentences that don't cause a contradiction when every member of a set satisfies them". This leads into the second problem, which is that it would be really hard to use! You'd be obliged to show that your sentence didn't lead to a contradiction every time you wanted to use comprehension!

Plus, of course, the epicycle complaint.

My favourite answer to the paradox of the stone is that God could create a stone that he couldn't lift. He could then also lift that stone. After all, he can do anything :P

Emil said...

The standard argument against omnipotence is fallacious as it commits the modal fallacy.

http://www.sfu.ca/philosophy/swartz/modal_fallacy.htm#omnipotence

Whether or not some other, more advanced argument is sound, I don't know, but I don't know of any such argument. But see Sobel (2004) for some discussion of the logical coherence of omnipotence.

Download link:
http://torrentmybooks.com/details.php?id=473

Review by T. Drange:
http://www.infidels.org/library/modern/theodore_drange/sobel.html

Ben said...

Daniel,

Not buyin' it.

For one thing, even if Frege thought of Basic Law V as a law of "logic", it was never anything of the kind. It was about the extension of concepts--or, equivalently in naive terms, set-theoretic comprehension--which is pretty clearly an extra-logical matter. For another thing, I'd think that the basic principles of one's modal logic of choice, the kind that are written out with boxes and diamonds and whatnot, would have more claim to be called "laws of logic" than any principle about sets or concepts. And, as they're *about possibility*, it seems to me that the "requirement of prior knowledge" objection would at least equally apply to them. Finally, we could trivially re-phrase Basic Law 5.1--Basic Law 5.1.1--to talk about contradiction-avoidance rather than the offending term "possibility", and even if we(a) for whatever reason insist on calling rules about sets/extensions "laws of logic", and (b) don't get to use possibility talk in "laws of logic" (other than modal ones?), plenty of actual unambiguous base-level laws of logic do use contradiction-talk.

Michael,

Basic Law V is about the extensions of concepts. Given naive, pre-Russell'S Paradox assumptions, "set" and "extension of a concept" are more or less equivalent--hence a paradox about sets being fatal to a principle about the extensions of concepts--and I take it that the reason we care about Basic Law V in historical retrospect is set-theoretic, which is why my playfully-suggested revision just talked about sets.

Hmm...

OK, making a quick run to the Stanford Encyclopedia to see whether I'm out of line with received wisdom in that last paragraph, it looks like the entry there kind of fudges the difference:

"The principle that undermined Frege's system (Basic Law V) was one that attempted to systematize the notions ‘course-of-values of a function’ and ‘extension of a concept’...The extension of a concept is something like the set of all objects that fall under the concept."

http://plato.stanford.edu/entries/frege-logic/#S2

Later on, though, the entry does (following Frege's discussion) carefully distinguish between Basic Law V and its entailment for set-theoretic comprehension--http://plato.stanford.edu/entries/frege-logic/#2.6--and you're probably right that I shouldn't have run those two together quite so casually. I just really like the sound of "Basic Law 5.1."

Emil,

In the link you give me, Swartz simply makes *both* of the moves that I devoted the post to arguing against. In fact, he seems to combine them in a slightly confused way.

The standard theistic response is to add a logical possibility epicycle to the formulation of omnipotence and use it to reject Swartz's first premise (while implicitly accepting the second premise). Swartz, despite assuming the epicycle, asserts w/o argument that the first premise is true, then (with argument) he says that the second premise is false.

His reasoning for why the second premise is false is exactly the move that the entire second half of my post was devoted to criticizing--that God *could* create an unliftable-even-by-Him stone, but that this mere possibility doesn't threaten His in-the-actual-case omnipotence. So my reply to Swartz's point is already there, in the last bunch of paragraphs of my post. Do you have an objection? Or take an objection to be implicit in anything that Swartz says?

MichaelPJ said...

Yeah. The thing is that in Frege's system you have "every concept has an extension" (= comprehension), and Basic Law V (= extensionality). Since the philosophical background in concepts pretty much requires every concept to have an extension, that wasn't really introduced as an explicit principle (notoriously, Frege gives it a brief mention in a footnote). So the source of the problem was seen to be Basic Law V. However, in modern set theories they're both explicitly mentioned, so you can modify comprehension instead.

But it doesn't really matter that much. I was actually trying to defend you from Daniel's point. Set comprehension (or it's Fregean variant) was not so clearly intended to be a law of logic.

Ben said...

I see. Fair enough.

Emil said...

I can't connect your analogy to the God case in a way that seems relevant, can you clarify how it is supposed to work?

I'm very much in agreement with Swartz about the common argument against omnipotence. Tho such an argument with respect to an essentially omnipotent god does work.

1. □(∀xGx→Ox)
2. □(∀xOx→◊Cx)
3. □(Cx→¬Ox)
⊢ 4. ∃xGx

But without the first premise, it doesn't work.

Proof trees here:
http://img541.imageshack.us/i/screenhunter11oct010543.jpg/

http://img819.imageshack.us/i/screenhunter10oct010543.jpg/

Ben said...

I think you're missing a diamond in Step 3, and, I'm guessing, a negation sign in 4?

In any case, the point of the analogy is this:

Talk of "powers" or "abilities" is counterfactual through and through. The fact that the right sort of thing that would need to exist in order to *test* some ability is absent doesn't mean that the ability or inability to relate to that thing in the right way doesn't exist. It can be true that someone *could* climb a few flights of stairs but *couldn't* climb a hundred flights of stairs, even in a world that contingently just doesn't happen to have any stairs in it at all.

Emil said...

For some reason my last reply didn't show.

First, yes, I am missing a negation in the conclusion. I did not forget this in my proofs linked to.

Secondly, no, I am not missing a star in the premises.

Thirdly, I agree with what you wrote about power and ability being counter-factual concepts that do not require them to be instantiated in the actual world. What then?

Ben said...

Emil,

You're sure you aren't missing a "possibility" symbol next to the Cx in Step 3 (like the one you have next to the Cx in Step 2)? Because with the diamond in 2 but not 3, the two don't connect up and I don't see how the proof works.

"Thirdly, I agree with what you wrote about power and ability being counter-factual concepts that do not require them to be instantiated in the actual world. What then?"

Then the fact that God (according to the sort of theistic response under consideration) contingently hasn't happened to create the right sort of stone doesn't seem relevant to the Stone Paradox--just as John's contingent failure to order the construction of hundred-floor buildings is irrelevant to the limits on his stair-climbing abilities. That, at least, is the point of the analogy.

Emil said...

Ben,

"You're sure you aren't missing a "possibility" symbol next to the Cx in Step 3 (like the one you have next to the Cx in Step 2)? Because with the diamond in 2 but not 3, the two don't connect up and I don't see how the proof works."

Yes, I'm sure. And that's right, the argument is invalid unless we add a premise about God being necessarily omnipotent. That's my point. I did supply proofs of my points above. The traditional stone-paradox argument is invalid.

"Then the fact that God (according to the sort of theistic response under consideration) contingently hasn't happened to create the right sort of stone doesn't seem relevant to the Stone Paradox--just as John's contingent failure to order the construction of hundred-floor buildings is irrelevant to the limits on his stair-climbing abilities. That, at least, is the point of the analogy."

But I still don't understand what the analogy is supposed to show. The trouble with analogy-arguments is that they are tricky or impossible to formalize and thus test for validity (or whatever corresponding notion, aptness perhaps?). For this reason, I am always super careful with analogies.

Ben said...

Actually, that's not my point at all. I'm not seeing how the proof works even with the necessary omnipotence premise in place. (That's Premise 1, right?) If you either stuck a possibility symbol next to the Cx in Premise 3 or took out the possibility symbol next to the Cx in Premise 2, the proof would be short and straightforward that, if He exists, He both is an is not omnipotent. Without that, though, I'm not seeing it. It's possible I'm just missing something about the modal rules here. Could you spell out for me how the steps of the proof go, given the mismatch?

In any case, to make things simpler:

Take the formulation of the argument from the Stone Paradox in Swartz's piece that you initially linked me too. I endorse that formulation.

Swartz thinks that it's valid but not sound, because he thinks that the second premise is false. He doesn't of course symbolize his reasons for thinking that the second premise is false--it's not the kind of thing that's productively conducive to clarification through symbolization. He does, however, gesture at his reasons:

"God, thus, remains omnipotent provided that God does nothing, e.g. making an immovable stone, which destroys His/Her omnipotence."

In my post, I similarly explained (in a much lengthier form, since I don't take these matters to be as obvious as Swartz does) why I think his second premise is true after all.

Neither of us symbolized our reasons for finding the premise true or false, but in neither case does this strike me as a problem.

Also:

See today's post for some more on this.

Emil said...

Ben,

"Actually, that's not my point at all. I'm not seeing how the proof works even with the necessary omnipotence premise in place. (That's Premise 1, right?) If you either stuck a possibility symbol next to the Cx in Premise 3 or took out the possibility symbol next to the Cx in Premise 2, the proof would be short and straightforward that, if He exists, He both is an is not omnipotent. Without that, though, I'm not seeing it. It's possible I'm just missing something about the modal rules here. Could you spell out for me how the steps of the proof go, given the mismatch?"

Yes, premise one is the one about necessary omnipotence.

No, you are right. It is not valid. I made a mistake in my proof of it (see link above, the mistake is in the last step where I accidentally add a negation to the right branch of an implication). I did the proof again and it comes out as invalid (and infinite). I can't figure out what the counter-example is though. So either I'm too dumb or there is something wrong with my proof.