September 2, 2010

Modal Collapse: Sobel's Objection to Gödel's Ontological Argument

The last ontological argument Sobel discusses is the Leibniz-inspired argument put forward by the famous logician Kurt Gödel. Gödel sets up a formal system in third-order quantified modal logic with equality and abstraction (!) and proves within that system the theorem:


Where the predicate G is defined as follows:
Gx ↔ ∀φ[P(φ) → φ(x)]

Where P is primitive. (Sobel includes the complete source texts for Gödel's proof on pp. 144-146.)

Now, unsurprisingly, given that the proof was originated by Gödel, everyone agrees that the proof is valid in the formal system. The question is whether there are any interpretations of the formal system such that (1) the axioms are true, and (2) the theorem means that God necessarily exists.

On Gödel's intended interpretation, P(φ) means that φ is a positive property. It is an axiom of the system that, for any property φ, either φ or ~φ is positive, but not both. It is also an axiom that the conjunction of any two positive properties is positive. Gödel remarks, rather cryptically:

Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world). Only then the ax. true. It may also mean pure "attribution" as opposed to "privation" (or containing privation). This interpret. simpler proof. (Sobel, p. 145; for discussion of this remark, see pp. 118-119)

Gx is now defined as having all positive properties (because the 'negative' properties are all and only the negations of the positive properties, Gx also entails that x has no negative properties), and this, Gödel thinks, amounts to being 'God-like'. (This term is used in Dana Scott's notes from a conversation with Gödel about the argument - see Sobel, p. 145.) Sobel challenges this assumption, but I will hold off on addressing that challenge until the next post, because Sobel thinks it applies to ontological arguments generally, and not only to Gödel's.

Sobel believes that Gödel's system must be flawed (on the intended interpretation) because it leads to necessitarianism. This is why: the logic Gödel uses includes abstraction which basically is a mechanism for defining new properties from formulas. That is, we can define a property as the property of being an x such that φ(x), where φ(x) is a formula in which x may or may not have free occurrences. In his proof, Gödel uses abstraction to define the properties of self-identity and self-non-identity, so we can't just remove it from the language. Now, in the system it is proved that necessarily, God has all and only the positive properties. Furthermore, a property's being positive or negative is a matter of necessity. Since we have abstraction we can say that God has the property being creator of Kenny (this would have a free occurrence of x: being an x such that x creates Kenny) and the property being such that Obama wins the 2008 election (this one wouldn't have a free occurrence of x). Now, if each of these is a positive property, then God has each of these properties necessarily, which implies that, necessarily, I exist and, necessarily, Obama won the 2008 election. But this is false: I might never have existed, and McCain might have won the election. Sobel concludes: "The necessity that Leibniz was so concerned to avoid obtains in Gödel's system" (p. 134).

Sobel is wrong about this. While it is true that Gödel's system entails that all propositions are what most of us would call 'necessary' and, indeed, we can, within the formalism, apply the box operator to any true proposition, nevertheless, Gödel does not end up with "the necessity Leibniz was so concerned to avoid." In fact, Leibniz embraced this consequence; he just refused to call it necessity.

Leibniz believed that divine omnipotence and benevolence entailed that God would create the unique best possible world. The relation here is entailment, hence necessity, in the ordinary sense of the word. However, in Leibniz-speak, a proposition is necessary iff there is a finite demonstration of it, and contingent just in case there is not. Call these L-necessity and L-contingency. Now, while it is L-necessary that God create the best possible world, it is L-contingent that in the best possible world I exist and Obama wins. This is because the problem of determining whether it is best for me to exist is infinite along two dimensions: first, the effect of my existence on every one of the infinitely many other substances and events must be considered and, second, the resulting world must be compared with each of the infinitely many other possible worlds, to see if it is best. As a result, that I exist is not finitely demonstrable, and so is L-contingent. Nevertheless, my existence is necessary in the ordinary (non-Leibnizian) sense of the word, since it is necessary that God create the best possible world, and the best possible world in fact includes me, and which world is best is not the sort of thing that varies from world to world.

Now, this result still seems a bit unfortunate, but I'm not sure how bad it is, especially if we can combine it with the view that, even though it is necessary that God create the best possible world, he is still free not to. To my mind, this falls short of being a full reductio of Gödel's theory, just because it is hard for me to determine how bad it is.

Posted by Kenny at September 2, 2010 11:22 PM
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Explanatory Principles and Infinite Propositions
Excerpt: In the course of his discussion of cosmological arguments, Sobel argues against the Principle of Sufficient Reason and similar strong explanatory principles. In particular, he argues that even a weak principle like "there is a deductive explanation tha...
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