Friday, September 18, 2015

Improving on Gödel

This is the final post in my series on debugging the ontological argument.

In this series, I've already rebutted every form of the ontological argument that can be expressed logically, but of course that was never really the point of the series.  The point is to dive into technical details and learn a bit about logic and philosophy.  Keeping with that spirit, this, the final post of the series, will go off on a complete tangent.

The central question I want to discuss is, what is necessary existence?

Sameness across modalities

In modal logic, necessary existence can be tricky to define, because we don't really have a sense of "sameness" across possible worlds.  Suppose there's an alternate timeline where I become a mathematician instead of a physicist.  Is that alternate version of me the "same" as me?  Could you say that I exist across both of these possible worlds?

By Gödel's definition of necessary existence, I do not exist across both worlds.  An alternate version of me is only considered the "same" if it has all the same properties as me.  Symbollically, the definition is: $$\forall x ~NE(x) \Leftrightarrow \forall Z~[Z~ess~x \Rightarrow \square\exists y~ Z(y)]\tag{1}\label{1}$$ In English this says "x has necessary existence if given any essential property of x, there necessarily exists an object with that same property."  Next we have to define what an essential property is.  $$\forall Z \forall x ~ \{Z~ess~x \Leftrightarrow Z(x) \wedge \forall Y ~ [ Y(x) \Rightarrow (Z \rightarrow Y) ] \}\tag{2}\label{2}$$ In English, this is "Z is an essential property of x if Z is a property of x, and also entails all properties of x."

A few things immediately follow from the definition of essential properties.  First, we can prove that each object has exactly one unique essential property.  The essential property, or "essence" is simply the conjunction of all properties that the object has.  Thus if two objects have the same essence, then they have all of their properties in common.  In order for something to have necessary existence, then there must exist in every possible world an object which has all the same properties.

Pathological sameness

In my opinion, this is not a very good definition of necessary existence.  The idea of two objects sharing all properties is far too strong.

You can come up with a lot of pathological properties.  For example, in an earlier post in this series, I defined H(x) to mean "There exists a monk singing chants in tight leather pants."  H(x) does not say anything about whether x is a monk singing chants in tight leather pants, it merely says that x is in the same world as said monk.  If x has Gödel's necessary existence, then we would conclude that monks singing chants in tight leather pants exist in all possible worlds, or none of them.

Or consider another kind of pathological property, PS which is the property that object x belongs to set S.  The set S can be chosen arbitrarily.  In fact, why not just choose S to include exactly one object x in one possible world?  No other objects in any other possible world will share property PS, so we can conclude that x does not necessarily exist.1

Now suppose we have an object which is a candidate for God.  As above, we prove that this object does not necessarily exist, and is therefore not God.  How's that for a reverse ontological argument?

I am convinced, however, that this is a purely technical problem in Gödel's argument, and that it could be fixed, although not in any elegant way.  Rather than considering two objects to be equal if they share all properties, we can consider them equal if they share all "relevant" properties.  "Relevance" can be a second-order predicate, just like "positivity", only it obeys different axioms: $$\forall Z \forall Y~ (R(Z) \wedge (Z \rightarrow Y) ) \Rightarrow R(Y)\tag{R1}\label{R1}$$ $$R(\lnot Z) \Leftrightarrow R(Z)\tag{R2}\label{R2}$$ $$\forall Z~R (Z) \Rightarrow \square R(Z)\tag{R3}\label{R3}$$ $$\lnot R(H)\tag{R4}\label{R4}$$ $$R(B)\tag{R5}\label{R5}$$ H is the property of existing in a world that has a singing monk, and B is the property of being a singing monk.2

I'm sure you could use the axiom of choice to construct an acceptable second-order predicate somehow.  Etc. etc., therefore God exists.  Now you're convinced, right?


1. I'm assuming that there is more than one possible world.  If there is only one possible world, then every object in it necessarily exists.

2. Fun exercise: prove that the axioms I chose are inconsistent.