5/15/2012

A note on omnipotence

There are people who think the following argument proves that an omnipotent being is logically impossible:

Argument A:
(A1)  For any x, if x is capable of creating a stone that x cannot lift, then x is not omnipotent (because it is incapable of lifting such a stone).
(A2)  For any x, if x is incapable of creating a stone that x cannot lift, then x is not omnipotent (because there is something that it is incapable of doing).
(A3)   For any x, either x is capable of creating a stone that x cannot lift, or x is incapable of doing so.
(A4)   Therefore, an omnipotent being is logically impossible.

This is an appealing argument, but (A2) is problematic. Consider the following argument:

Argument B:
(B1)  In any possible world in which x is omnipotent, there is no stone that x cannot lift.
(B2)  Therefore, there is no possible world in which x is omnipotent and in which there is a stone that x cannot lift.
(B3)  Therefore, “x is omnipotent and there is a stone that x cannot lift” (S) is impossible.
(B4)  For any x, if x is omnipotent, then what it is for x to create a stone that x cannot lift is for x to actualize (S).
(B5)  Therefore, for any x, if x is omnipotent, then what it is for x to create a stone that x cannot lift is for x to actualize something impossible.
(B6)  Even if a being is omnipotent, it is incapable of actualizing the impossible.
(B7)  Therefore, for any x, if x is omnipotent, then it is incapable of creating a stone that x cannot lift.

Let us symbolize (B7) as “("x)(Ox ® Ix)”. Premise (A2) can then be symbolized as “("x)(Ix ® ~Ox)”, which is equivalent to “("x)(Ox ® ~Ix)”.

If argument B is sound, which I think it is, then “("x)(Ox ® Ix)” is proven true. Will argument B then prove that (A2) (i.e. “("x)(Ox ® ~Ix)”) is false? Not quite, for if there are no omnipotent beings, then both “("x)(Ox ® Ix)” and “("x)(Ox ® ~Ix)” are true.

But that doesn’t mean (A2) is not problematic as a premise of argument A. Here is how I see the dialectic: “("x)(Ox ® Ix)” is true whether there is an omnipotent being, while “("x)(Ox ® ~Ix)” is true only if there are no omnipotent beings. For this reason, (A2) should not be taken as obviously true and needs to be defended.

A defense of (A2) may begin with arguing that there are no omnipotent beings. Such an argument has to be independent of argument A, for otherwise it would be question-begging. It is, however, not clear how this can be done.

Another way is to argue for (A2) without first arguing that there are no omnipotent beings. If there are independent grounds for arguing for (A2), then those can be used, and the argument for (A2) will also be an argument for the non-existence of omnipotent beings. Argument A will then be used to argue for a stronger conclusion, namely, that an omnipotent being is logically impossible.

In any case, (A2) can’t just be assumed without argument.

18 comments:

  1. With no formal training in philosophy and logic (especially all those symbols and notations), I look at this from a rather layman and naive angle : what is the use of these 2 arguments ???

    It appears that argument B assumes omnipotence from the very beginning (B1 : x is omnipotent), thus drives B7 which resonates again the premise that x is omnipotent; whereas argument A questions what is the exact meaning of omnipotence ...

    Argument A may not succeed at all in refuting omnipotence but it at least throws doubt against such concept, whereas argument B informs me of no new news as it starts with omnipotence and ends with omnipotence ...

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    1. Argument A, like argument B, also assumes an intuitive understanding of the meaning of "omnipotent".

      Argument B does not assume that there is an omnipotent being. (B1) only states that the following is impossible: a being is omnipotent and there is a stone it cannot lift.

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  2. i guess it is all 觀點與角度 so no point to argue further ...

    maybe it is only me; i read A with no whatsoever intuitive pre-shadowing of omnipotence at all ... i strictly treat it as some sort of Q&A's to delineate definition of omnipotence ...

    and i read B as there is already an omnipotent being because in your starting argument B1 you clearly states that "In any possible world in which x is omnipotent" ... so, there is x and x is omnipotent ...

    anyway, just my reading only :-) ...

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    1. //i read A with no whatsoever intuitive pre-shadowing of omnipotence at all//

      - If you had no intuitive understanding of "omnipotent", how would you understand (A1) and (A2) (and probably accept them as true)?

      //i read B as there is already an omnipotent being because in your starting argument B1 you clearly states that "In any possible world in which x is omnipotent" ... so, there is x and x is omnipotent ...//

      - If I say "in any possible world in which x is a unicorn, x has only one horn", am I saying there is a unicorn (in the actual world) and it has only one honr?

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    2. hahahaha, you really like to argue ... didn't i just say no point to argue further :-) ...

      is this some kind of 明人之迷 :-) ???

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    3. I wasn't arguing; I was only clarifying...

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    4. hahahaha, couldn't help but to write this very last reply ... to a layman like me, your clarifying sounds very much like 迷人之迷 ...

      before your clarifying, i could still understand A and B ... now, after your clarifying, i am not sure if i understand them at all ...

      hahahaha ...

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  3. oh by the way, hope you don't mind ...

    i think there is difference between the following 2 objects :

    object A : 4-sided triangle
    object B : a stone so heavy that one would find it difficult to lift

    i think object A is by itself not possible, no matter who is going to draw it ...

    and object B is by all means possible ... it is not possible only when the lifter is omnipotent ...

    sorry for my layman language ... hope i did make myself clear to communicate my point :-) ...

    thx ...

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    1. //object B : a stone so heavy that one would find it difficult to lift
      and object B is by all means possible ... it is not possible only when the lifter is omnipotent ... //

      - Right. So the problem lies in the word "one", which, like "here", "now", "he", can refer to different things depending on the contexts.

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    2. hahahaha, you remind me clinton once said "It depends on what the meaning of the word 'is' is." ...

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    3. Anonymous, you remind me Zhuangzi once said "You can't explain to a summer bug what is ice." ...

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    4. //object B : a stone so heavy that one would find it difficult to lift

      and object B is by all means possible ... it is not possible only when the lifter is omnipotent ... //

      So an omnipotent being is impossible?

      object C : a man who is a bachelor

      and object C is by all means possible ... it is not possible only when the man is married ...

      So a married man is impossible!

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  4. meshi, you are such a great philosopher, much greater than wong !!! ... you got to live long enough to see ice and see the omnipotent being !!! ... may god bless you :-) ...

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    1. no way, you have to die in order to see the omnipotent being (if any). This is so-called "eschatological verifiability".

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  5. don't be so pessimistic, meshi ...

    (c1) in any possible world in which x is omnipotent, there ain't nothing that x cannot do, which includes letting x be seen by meshi before meshi is dead

    (c2) meshi you are living in one of these possible worlds

    (c3) so, meshi, always be hopeful with your life :-) ...

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  6. Note that the contra-positives of A1 and A2 are such that satisfying their antecedents (same) would discharge seemingly contradictory claims.

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    1. Yes, that's right. Indeed, (B7) is just the contrapositive of (A1). Both (A1) and (A2) are assumed to be obviously true in argument A; I try to prove that (B7), and hence (A1), is true, and question (A2).

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  7. // A defense of (A2) may begin with arguing that there are no omnipotent beings...

    Another way is to argue for (A2) without first arguing that there are no omnipotent beings...//

    I really hate so much arguing. (Yet, thanks to J. Hwang's mentioning of the two contra-positives. It makes your argument very clear to me.) To me, the existence of omnipotent being is something that one cannot (to date? in foreseeable future? maybe ever?) prove or disprove empirically. Resorting to argument is probably futile.
    --zpdrmn (yep, it's me with the same bias again.)

    Oh, I think I argued against it using some energy argument in your Chinese blog before. Shoot. I am slapping myself in the face about arguing. Sue me.

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