Tuesday, June 9, 2009

Gödel's ontological argument, step by step

By request, I am going to explain Gödel's ontological argument step by step, for your reference. Here's how it will work. I am following the symbolic logic proof shown on the Wikipedia article.* In the first section, I will explain everything in English. In the second section, I will write out the proof symbolically, expanding out steps as much as I feel necessary.

Please skip to the section of your preferred level of precision.

*Please note that there are several different variations of Gö
del's ontological argument with different sets of axioms and different steps. C. G. Small has another explanation, using a different variation of Gödel's argument.

Section 1: English

All capital letters represent any given property. All small letters represent objects. I use the word "negative" to merely mean non-positive.
  • Entailment: V "entails" W if and only if in all possible worlds, all objects with property V also have property W. Note that if in all possible worlds, there is not a single object with property V, then V automatically entails W.

  • Axiom 1: If Z entails Y, and if Z is positive, then Y is also positive.

  • Axiom 2: For all properties Z, either Z is positive or not-Z is positive, but not both.

  • Theorem 1: If a property Q is positive, then in some possible world there exists an object with property Q.
    • Proof by contradiction:
    • Suppose that Q is positive, and that there is not any possible world where there exists an object with property Q.
    • Then in all possible worlds, there are no objects with property Q.
    • Therefore Q entails any given property. Q entails R and Q entails not-R.
    • Therefore R and not-R are both positive. This contradicts Axiom 2.

  • God-like: An object is "god-like" if and only if the object has every property which is positive. Note that a god-like object cannot have any negative properties. If a god-like object had negative property V, then it would fail to have the positive property not-V.

  • Axiom 3: The property of being god-like is a positive property.

  • Theorem 2: In some possible world, there exists an object which is god-like.
    • Since god-like is a positive property, Theorem 1 states that in some possible world there exists an object with the god-like property.

  • Essence: Property V is an "essence" of x if and only if the following conditions hold:
    • The object x has the property V.
    • If x has any property U, then V entails U.

  • Axiom 4: If a property is positive, then it is positive in all possible worlds.

  • Theorem 3: If an object x is god-like, then the god-like property is the essence of x.
    • Suppose x has any property Q. x cannot have any negative properties, so Q must be positive.
    • Therefore, in all possible worlds, Q is a positive property.
    • Therefore, in all possible worlds, any god-like object must have property Q.
    • Therefore the god-like property entails Q.

  • Necessary existence: Object x is "necessarily existing" if and only if the following condition holds: For any property V, if V is an essence of x, then in all possible worlds there exists an object with property V.

  • Axiom 5: Necessary existence is a positive property.

  • Theorem 4: In all possible worlds, there exists a god-like object.
    • We know that in some possible world, there exists a god-like object.
    • Since necessary existence is a positive property in all possible worlds, that god-like object must be necessarily existing.
    • That object has the god-like property as its essence.
    • By the definition of necessary existence, there must, in all possible worlds, exist an object which has the god-like property.

  • Corrolary 1: If there are two god-like objects, then they cannot have any properties which are different.
    • Proof by contradiction:
    • Suppose we have two god-like objects, and some property Q which applies to one object, but not the other.
    • Since god-like objects cannot have negative properties, Q must be positive.
    • Similarly, not-Q must be positive. This contradicts Axiom 2.
Section 2: Symbolic Logic

P represents the property of being "positive". G represents the property of being "god-like". NE represents the property of necessary existence. All other capital letters represent any given property. All small letters represent objects. The letters "ess" represent "is the essence of".

[Update April 2015: I finally converted all these equations to LaTeX. Hope that makes it more readable! Javascript must be enabled.]
  • Axiom 1: $\{P(Z) \wedge \square\forall x[Z(x) \Rightarrow Y(x)]\} \Rightarrow P(Y)$

  • Axiom 2: $P(\lnot Z) \Leftrightarrow \lnot P(Z)$

  • Theorem 1: $P(Q) \Rightarrow \Diamond \exists x[Q(x)]$
    • Proof (by contradiction):
    • Suppose $P(Q) \wedge \lnot \Diamond \exists x[Q(x)]$
    • $\square\forall x[\lnot Q(x)]$
    • $\square\forall x[Q(x) \Rightarrow \lnot R(x)]$
    • $P(\lnot R)$ (By Axiom 1)
    • $\lnot P(R)$ (By Axiom 2)
    • $\square\forall x[Q(x) \Rightarrow R(x)]$
    • $P(R)$ (By Axiom 1)
    • $\lnot P(R) \wedge P(R)$ (Contradiction)

  • Definition of G: $G(x) \Leftrightarrow \forall V[P(V) \Rightarrow V(x)]$

  • Axiom 3: $P(G)$

  • Theorem 2: $\Diamond \exists x[G(x)]$
    • Proof: Combine Axiom 3 and Theorem 1

  • Definition of ess: $V~ess~x \Leftrightarrow V(x) \wedge \forall U\{U(x) \Rightarrow \square\forall y[V(y) \Rightarrow U(y)]\}$

  • Axiom 4: $P(Z) \Rightarrow \square P(Z)$

  • Theorem 3: $G(x) \Rightarrow G~ess~x$
    • Proof: Suppose $G(x) \wedge Q(x)$
    • $P(\lnot Q) \Rightarrow \lnot Q(x)$ (By definition of G)
    • $Q(x) \Rightarrow \lnot P(\lnot Q)$
    • $\lnot P(\lnot Q)$
    • $P(Q)$ (By Axiom 2)
    • $\square P(Q)$ (By Axiom 4)
      • Suppose (for contradiction) that $\lnot \square\forall y[G(y) \Rightarrow Q(y)]$
      • $\Diamond \exists y[G(y) \wedge \lnot Q(y)]$
      • $\Diamond \exists y[[P(Q) \Rightarrow Q(y)] \wedge \lnot Q(y)]$ (By definition of G)
      • $\Diamond \exists y[\lnot P(Q)]$
      • $\Diamond \lnot P(Q)$
      • $\lnot \square P(Q)$
      • $\square P(Q) \wedge \lnot \square P(Q)$ (contradiction)
    • $\forall Q\{ Q(x) \Rightarrow \square\forall y[G(y) \Rightarrow Q(y)] \}$
    • $G~ess~x$ (By definition of ess)

  • Definition of NE: $NE(x) \Leftrightarrow \forall V[V~ess~x \Rightarrow \square \exists y V(y)]$

  • Axiom 5: $P(NE)$

  • Theorem 4: $\square \exists x G(x)$
    • Proof:
    • $\Diamond \exists y[G(y)]$ (Theorem 2)
    • $\square P(NE)$ (By Axiom 4)
    • $\Diamond \exists y[G(y) \wedge NE(y)]$ (By definition of G)
    • $\Diamond \exists y[G(y) \wedge ( G~ess~y \Rightarrow \square \exists x G(x) )]$ (By definition of NE)
    • $\Diamond \exists y[G~ess~y \wedge ( G~ess~y \Rightarrow \square \exists x G(x) )]$ (By Theorem 3)
    • $\Diamond \exists y[\square \exists x G(x) ]$
    • $\Diamond \square \exists x G(x)$
      • Lemma: $\Diamond \square S \Rightarrow \square S$ for any statement S
      • $\Diamond \lnot S \Rightarrow \square\Diamond \lnot S$ (an axiom of modal logic)
      • $\lnot \square\Diamond \lnot S \Rightarrow \lnot \Diamond \lnot S$
      • $\Diamond \square S \Rightarrow \square S$
    • $\square \exists x G(x)$

  • Corrolary 1: $G(x) \wedge G(y) \Rightarrow \forall Q[Q(x) \Leftrightarrow Q(y)]$
    • Proof (by contradiction):
    • Suppose $G(x) \wedge G(y) \wedge \exists Q[ Q(x) \wedge \lnot Q(y) ]$
    • $P(\lnot Q) \Rightarrow \lnot Q(x)$ (By definition of G)
    • $\lnot P(\lnot Q)$
    • $P(Q)$ (By Axiom 2)
    • $P(Q) \Rightarrow Q(y)$ (By definition of G)
    • $\lnot P(Q)$
    • $\lnot P(Q) \wedge P(Q)$ (Contradiction)
Questions, corrections welcome.

I realize that in the comments, it's hard to use symbols which don't appear on a keyboard. The following symbols will be understood:
~ or $\lnot$ is "not"
[] or $\square$ is "in all possible worlds" or "it is necessary that"
<> or $\Diamond$ is "in some possible world" or "it is possible that"
=> or $\Rightarrow$ is "if ..., then..."
<=> or $\Leftrightarrow$ is "if and only if"
^ or $\wedge$ is "and"
v or $\vee$ is "or"
E or $\exists$ is "there exists"
A or $\forall$ is "for every" or "for all"


Scott said...

A minor correction: am I right in guessing that in your proof of Th. 3, "by definition of G" should read "by definition of P"?

Scott said...

Oops, no I was misreading that line. Sorry!

The Barefoot Bum said...

Many thanks!

I'll have to look at this carefully later on.

Mike Flynn said...

Looks a lot like Duns Scotus.

Keith Brian Johnson said...

The problem, as with Hartshorne's and Plantinga's versions of the ontological argument, lies not with any single statement but with an illegitimate combination of premisses. Without assuming that property A is instantiated in some possible world (and without assuming that it isn't), one may say that if property A is a positive property, and if property A entails property B, then property B is also positive. But without assuming that property A is instantiated in some possible world, it is then possible that property A isn't instantiated in any possible world, in which case it entails all properties--in which case all properties are positive, and one cannot then take as axiomatic that of B and not-B, only one can be positive. To make that assumption as well requires assuming that property A is instantiated in some possible world, at which point it is hardly surprising that one can "prove" that there is a possible world in which property A is instantiated. So, we should not be surprised to see, what has been "proven" has really been assumed.

miller said...

thanks for plain explanation!

miller said...

Hi Keith, just saw your post and got interested. I think the issue you raise is resolved by looking at Godel's definition of a positive property. I think his definition of it goes something like: a positive property is one whose negation would be non-positive. If we were to accept this definition as a working one, the issue you bring up is immediately resolves, because it then becomes impossible for B and not-B to both be positive. From that perspective, therefore, I think the proof holds. However, I do see significant merit to your argument about the degree of acceptability of the combination of premises (although I could be wrong here). For one thing, I for the life of me can't see how "the absence of A from all objects in all worlds automatically means that A entails B". Well, what if property A is absent from all objects in all worlds, while B does exist in one/some of them? Doesn't this simply mean that the preceding parenthetical statement is unverifiable? If anything, doesn't it only mean that the presence of property B doesn't necessitate that of A (as an entailing/entailed property)?

I just learnt about Godel's ontological argument by doing a bit of self-reading a couple of days earlier, so I'm in all likelihood missing something important here. Will try to look for an answer to it somewhere else as well. Thanks.

miller said...

Check the timestamp.

miller said...

In any case, I'm still here so I can respond. Keith was exactly correct. This argument corresponds to Theorem 1 above.

The critical thing you are missing is that entailment is defined using the "material conditional". (That's a search term, if you'd like to learn more.) For example, the statement "If A, then B" is true whenever A is false. And it doesn't really matter whether A and B have any physical connection, and it doesn't matter whether B is true or false.

For example, here is a true statement: "If the moon is made of cheese, then I'm having lunch right now."