Sunday, September 6, 2015

Gödel's positive predicates

This is part of my series on debugging the ontological argument.

As explained in the previous post, the basic structure of Gödel's Ontological Argument (GOA) is to first prove the "consistency" of God, and then it follows that God must exist.  In this post, I will how we prove God's consistency.

The GOA introduces a second order predicate, that is, a predicate that is applied to other predicates.  This second order predicate is called "positivity", but the name is more suggestive than meaningful.  For example, we could say that the property of being red is a "positive" property, or we could say that being red is not a positive property.  Symbolically, we would write this as $$P(R)\tag{1}\label{1}$$ where R means "is red" and P means "is positive".

GOA at first appears to stipulate a definition of "positivity", and of course it is fine to stipulate a new definition for a new concept.  But my objection is that the definition is poorly formed.

Too many premises

The GOA takes the following premises about "positivity":

$\ref{P1}$: If positive predicate Z entails1 predicate Y, then Y is also positive.
$\ref{P2}$: Given any predicate and its negation, exactly one of them is positive.
$\ref{P3}$: The conjunction of all positive predicates (called "God-like") is itself a positive predicate.
$\ref{P4}$: If a predicate is positive, then it is necessarily positive.
$\ref{P5}$: Necessary existence is positive.  Necessary existence basically means that in every possible world there is a copy of the given object.

And in symbolic logic:2 $$\forall Z \forall Y~ (P(Z) \wedge (Z \rightarrow Y) ) \Rightarrow P(Y)\tag{P1}\label{P1}$$ $$\forall Z~ P(\lnot Z) \Leftrightarrow \lnot P(Z)\tag{P2}\label{P2}$$ $$P(G);\qquad \text{Definition of G:}~ \forall Z~ P(Z) \Rightarrow (G \rightarrow Z)\tag{P3}\label{P3}$$ $$\forall Z~ P(Z) \Rightarrow \square P(Z)\tag{P4}\label{P4}$$ $$P(NE)\tag{P5}\label{P5}$$ I leave out the logical definition of necessary existence because it's very technical and not relevant yet.

The GOA is not an especially popular form of the ontological argument, and perhaps now you can see why.  It gets rid of some of the questionable premises in other ontological arguments, but it replaces them with five whole new ones.  Five!  It's easy enough to just say, one of those premises must be wrong.

Indeed, I can't think of any reason we should think that any of the premises are true.  As far as I can tell, the proof is not referring to any natural concept of "positivity".3  In particular, I don't see why every property either needs to be positive or its negation does.  What if being red is positive in some contexts, but in other contexts it's better to be not red?  Or what if it's positive in this world, but there's a possible world where it's not positive?

Indeed, we can stop here, as far as rebutting the GOA is concerned.  It sure is a fancy argument, too bad its premises aren't remotely persuasive.

But the purpose of this series is to dig deeper.  So we turn to the question: exactly which of the premises is wrong?

Stipulative definitions

Since "positivity" obviously doesn't correspond to any real concept of positivity as far as I know, the only way I can interpret the word is simply as a placeholder.  We're creating a whole new concept, and "positivity" is just the name we gave it.  And since it's a whole new concept, we can stipulate whatever definition we like for it.

Indeed, I believe that none of the premises are individually "wrong".  "Positivity" is a meaningless word, and we're allowed to stipulate certain things about it.  Any of the premises, taken individually, I find acceptable to stipulate as a partial definition.  Taken together, however, there might be an issue.

It is not true that you can stipulate just any definition for a word.  The Stanford Encyclopedia of Philosophy lists two criteria that a stipulative definition must follow, the first of which is relevant here:
A stipulative definition should not enable us to establish essentially new claims—call this the Conservativeness criterion. We should not be able to establish, by means of a mere stipulation, new things about, for example, the moon.
The whole project of establishing the existence of God by stipulative definitions, therefore seems rather quixotic.  The very success of the proof only demonstrates that the stipulative definition that we began with was wrong.

What's happening here, mechanically, is that the definition of positivity is overly constraining.  As a second-order predicate, "positivity" has only so many degrees of freedom.  As we give it partial definitions, we are constraining its degrees of freedom, but at some point we begin also to make constraints on the world (or on the set of possible worlds).4

It's quite similar to giving a single word two definitions.  I can define a "foo" as an eight-legged snake, and I can define a "foo" as Socrates, and either of those definitions are fine on their own.  Taken together, they can be used to prove that Socrates is an eight-legged snake, which is a sign the definitions too constraining (even if by a quirk of history it turned out to be true that Socrates was an eight-legged snake).

If I were to pinpoint any of the five premises as particularly problematic, I would say that the $\ref{P1}$ is.  The concept of entailment is based on the material conditional, which as I argued previously is a very counterintuitive concept.  Using this premise, if any impossible predicate is positive then all predicates are positive.  I'm not sure what "positivity" is really intended to signify, but this premise would seem to go against the spirit of it.


1. As explained in the previous post, entailment means necessary implication.  In the case of predicates, if Z entails Y, then it is necessary that for every object which has predicate Z, it also has predicate Y.

2. Here are some details on how the proof follows from these premises.  Using premises $\ref{P1}$ and $\ref{P2}$, we can prove that all positive predicates are strictly consistent.  If a positive predicate were not strictly consistent, then it would entail a both Z and not Z, which would mean that both Z and not Z are positive predicates, which contradicts $\ref{P2}$.  Using the rest of the premises, we show that being God-like is positive, and therefore strictly consistent.

3. According to Gödel, "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)".  Thus it is clear that he intended to capture some sort of natural concept of positivity, although I think the proof fails to live up to these intentions.  I will disregard Gödel's intentions with the hope of exploring the best possible version of the proof.

4. Premises P1, P2, and P5 suffice to make a constraint on the actual world: they prove that something exists.  Just P1 and P2 suffice to constrain the possible worlds: they prove that it is possible that something exists.  Though what they prove is trivial, it is a sign that the definition is already too constrained and cannot be stipulated.