May 4, 2009

The Ontological Argument for the Non-Existence of God

Posted in Atheism tagged , , , , at 10:07 am by Andrew

Via strongathiesm.net:

1. g=Df (the x such that Px)

  • God is defined as a perfect being. (premise)

2. N(Eg->Pg)

  • We reformulate (1) by saying that God’s existence necessarily entails its perfection. All we did here was explain in terms of existence what (1) means. (from 1)

3. N(x)(Px->NEx)

  • Let us assume, as the Ontological arguments do, that the perfection of x necessarily implies the existence of x, for all x. (premise)

4. N(Pg->NEg)

  • Instantiating the principle in (3) for God. (from 3)

5. N(Eg->NEg)

  • We now see that (2) and (4) can be combined into one proposition. If Eg implies Pg, and Pg implies NEg, then Eg implies NEg – the existence of God implies the necessary existence of God. (from 2 and 4)

We have to take a break here. As Pollock explains in his development, this is the furthest that we can take (1) by logical means. Even assuming the truth of the premise of the ontological arguments in (3), it is impossible to arrive at Eg, the proposition that God exists.

Rather, the best we can do is the proposition that IF God exists, then NEg necessarily obtains. This is important for two reasons: one because it shows that we cannot arrive at Eg, and two because we will use this conclusion again at the end of our argument.

6. (g=Df the x such that Px) -> N(Eg->NEg)

  • Here we simplify the first half of our argument in one proposition. (from 1 to 5)

7. ~ [(g=Df the x such that Px) -> Eg]

  • We can explain this proposition in two ways. The first is to remember, as I discussed before, that a definition cannot entail actual existence. The other is to point out that we already showed that we cannot logically obtain Eg from (1). Either way, it is a fact that Eg is unattainable from the definition alone. (premise)

8. NEg iif [(g=Df the x such that Px) -> Eg]

  • This is obtained from the definition of logical necessity. Something is logically necessary iif it follows logically from its definition. (premise)

9. ~NEg

  • If something is only logically necessary iif it follows logically from its definition, and God’s existence does not follow logically from its definition, then God’s existence is not logically necessary. (from 7 and 8 )

10. N(~Eg)

  • But we saw in (5) that it is necessary that if God exists, he exists necessarily: N(Eg->NEg). Since it is not the case that NEg, it is logically necessary that God does not exist. (from 5 and 9)

Our conclusion in (10) proves the strongest form of strong-atheism (“God cannot exist”), but also implies the weaker claim that ~Eg (“God does not exist”).

It seems to me that the only avenue open to the theist is to dispute (7), but perhaps some of the theists on this site can chip in with responses I haven’t seen.

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4 Comments »

  1. Phil said,

    In response to your uncritical advertisement of this poor guy’s mostly uncritical acceptance of an argument Pollock made over 40 years ago, I offer an uncritical advertisement of another philosopher’s rejection of Pollock’s argument:

    —————
    Pollock wants to show that the being described by the proposition (Eg ⊃ □Eg) cannot exist (E means “exists,” and P means “perfect”). The ontological argument is used as fodder for Pollock’s project throughout. He considers the following two versions of it, the second version being somewhat more Kant-proof:

    (1) g =Df (the x such that Px);
    (2) therefore, Pg;
    (3) □(x)(Px ⊃ Ex);
    (4) therefore, □(Pg ⊃ Eg);
    (5) therefore, Eg. (31ff)

    (1) g =Df (the x such that Px);
    (2) therefore, Pg;
    (3′) □(x)(Px ⊃ □Ex);
    (4′) therefore, □(Pg ⊃ □Eg);
    (5′) therefore, □Eg. (32)

    [Pollock] says that the move from (1) to (2) is illegitimate. For what if we let ‘Ax’ in a =Df (the x such that Ax) be ‘Bx & ~Bx’? Then ‘Bg & ~ Bg’ will be true, which is absurd. The most we can get from (1) is

    (2′) □(Eg ⊃ Pg), from which we can derive
    (5”) □(Eg ⊃ □Eg) or “it is a necessary truth that if God exists, then He exists necessarily.”

    Now assuming that God exists necessarily iff the meaning of “God” requires that He exist,

    (8) □Eg ≡ [(g =Df the x such that Px) → Eg].

    But Eg does not follow, because the argument (1) – (5) is compromised; hence

    (9) ~□Eg and, by contraposition from (5”), ~Eg. Therefore God does not exist; moreover, “it is necessarily true that God does not exist” (because if God existed in some non-actual possible world, then He would again exist necessarily, which we have proven He does not).

    Evaluation. There are two problems here. First, (2′) does follow from (1), but it is far too weak. God would be perfect (in the understanding, which is all we need) even if He did not exist or rather existed only as a concept. Thus, we have

    (2*) (Eg ⊃ Pg) & (~Eg ⊃ Pg) which is equivalent to Pg.

    Further, (2) does not follow from (1) logically, but it does follow from it given the interpretation of (1) as “g is a being than which nothing greater can be conceived.” The stronger inference is valid due to the nature of the predicate P, because P is surely not a self-contradiction, unlike ‘Bx & ~Bx’.

    There are two possibilities. Either (5”) does not follow, because the argument equivocates with respect to the word “perfect”: according to Pollock, if God exists, then He is perfect in reality (by definition), and if God is perfect in the understanding, then He exists (by the OA). From this he derives “if God exists, then He exists necessarily.” But, of course, this does not follow. Or Pg = “God is perfect in the understanding,” in which case (2) follows, and (1) – (5) is sound.

    Therefore, if the ontological argument is correct, then Eg follows, that is, (g =Df the x such that Px) → Eg, contrary to what he claims.

    Second, (8) should rather be

    (8*) [(g =Df the x such that Px) → Eg] → □Eg

    in order to accommodate other possible definitions of necessary existence. And if the antecedent is false, then who knows about the consequent.

    Pollock should have realized that proving that God does not exist “by logical means” is perilous business
    ———–

    It seems to me that the only avenue open to the atheist is to dispute one of the points above, but perhaps someone can chime in with something else they heard in somebody’s opinion of what somebody else said that I haven’t seen yet.

  2. Hylomorphic said,

    That notation is really wonky, and it causes some real problems.

    Strictly speaking, one can never say that there are direct, immediate implications arising from an act of definition. To define something is simply to denote a term as shorthand to call up the whole definition. Something like “(g=Df the x such that Px) -> N(Eg->NEg)” is at best needless, and at worst tautological. One only need point out that N(Eg->NEg); you’ve already defined “g”, so there’s no need to bring up the definition again.

    It’s not immediately clear to me how #7 can be reworked. We have to look at the reasoning behind it:

    “* We can explain this proposition in two ways. The first is to remember, as I discussed before, that a definition cannot entail actual existence. The other is to point out that we already showed that we cannot logically obtain Eg from (1). Either way, it is a fact that Eg is unattainable from the definition alone. (premise)”

    I’m not sure what he means by “we already showed that we cannot logically obtain Eg from (1).” He hasn’t shown any such thing.

    If we take his first explanation of the proposition, then it’s not at all clear why we’re engaging in an ontological argument in the first place. If one can never adduce the existence of an entity from its definition, then one must reject all ontological arguments–and any notion of a God whose logical necessity can be ascertained from that notion. (All other notions of God are unchallenged, however.)

    But that doesn’t tell us how to translate #7 into precise notation. The closest I can come to is: ~NEg.

    But to do that is to toss out #7 and #8 (remembering that we already had to toss out #6 as entirely unnecessary), and proceed directly from #5 to #9.

    But without the (apparent) support of 6, 7, and 8, #9 is revealed for what it is–an assumption. And a question-begging assumption, at that. The whole business of ontological arguments is to determine whether indeed God necessarily exists. Rejecting it as a premise in the middle of an ontological argument is irremediably circular.

  3. Nathaniel said,

    Aside from these other criticisms, premise 7 of the argument as it stands, by itself, entails that God does not exist:

    ~ [(g=Df the x such that Px) -> Eg]

    entails

    [(g=Df the x such that Px) & ~Eg]

    whence, by simplification,

    ~Eg.

  4. wissam said,

    @Nathanial,

    Premise 7:

    ~ [(g=Df the x such that Px) -> Eg]

    entails

    ~[~(g=Df the x such that Px) v Eg] by implication.

    which entails

    ~~(g=Df the x such that Px) & ~Eg by De Morgan’s theorem.

    which entails

    (g=Df the x such that Px) & ~Eg by Double Negation.

    which entails

    ~Eg by Simplification.

    So premise 7 entails that God does not exist, which is of course question-begging.

    What you said was true.

    This argument is false. At any rate, I think that all ontological arguments, whether theistic or atheistic, are question-begging. Philosophers should stick to the typical arguments for the existence and non-existence of God.

    Here’s my assessment of the various arguments:

    The Cosmological Arguments:

    The Leibnizian version is not viable because it assumes the necessarily false PSR.
    The kalam version assumes a controversial A-theory of time and has other defects.

    The Teleological Arguments:

    Probabilistically invalid.

    Moral Arguments:

    False premise: if not-God, then not-objective morals.

    Problem of evil:

    sound but may very well be irrelevant given total evidence.

    Humean Objections to Miracles:

    Probabilistically invalid.

    I basically think that all arguments posed in the philosophy of religion suck but there is one thing which is undoubtably true: Platonism and traditional theism are incompatible. Whether or not Platonism is true is an open question.


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