I’ve spent some time lately looking at Godel’s ontological proof and trying to comprehend it the best way I can. Axiom 5 seems the most troubling to me out of the whole argument, but today I want to focus on just Theorem 1 of the proof and the two axioms leading up to it. Here it is below, copied from Wikipedia for quick reference:
Theorem 1 says if a property is positive, then it’s possible there exists such a property. That seems reasonable. Positive things do exist in this world after all. But if we look a little more deeply at this statement, this claim is a bit stronger than it looks, because we’re saying we cannot definitively claim ANY positive property is completely impossible in all potential worlds. We can’t call any positive activity completely impossible.
And Godel seems to use a slick method to claim this, using
the vacuous truth rules, which I’m always suspicious of. First he assumes Theorem 1 isn’t true in an
attempt to derive a contradiction. He
says “Phi is a positive property AND it’s necessary that for all x, NOT phi of
x,” which is the opposite of what Theorem 1 says. This makes sense. To negate an if-then statement you assert the
premise and the opposite of the conclusion.
After that he notices that the portion of Axiom 1 stating
“it’s necessary that for all x, phi of x” opposes a portion of his assumption. Plus this portion of Axiom 1 is the beginning
of a conditional. So his assumption to
start the proof of the theorem creates a vacuous truth in Axiom 1 where the
axiom claims phi of x. Because of this,
by the rules of vacuous truths, the if phi of x then psi of x statement in
Axiom 1 is ALWAYS true, no matter what value psi of x takes.
So Godel just takes that psi of x and makes it equal
negative phi of x. So the output of
Axiom 1 after this substitution says if (phi is a positive property) and (it’s
necessary that for all x, if phi of x then NOT phi of x) then NOT phi is a
positive property. Kind of a nasty
double conditional. The conditional
inside the conditional was a vacuous truth.
But the outer conditional has its premise consisting of two true
statements, one being phi is a positive property, which we already asserted as
soon as we started the proof by contradiction, and the vacuous truth. So both parts of the conditional premise being
true will indeed yield the result of NOT phi being a positive property being a
true statement.
All that mess is not easy to write out. And we did it all just to get the statement
“NOT phi is a positive property.” Now we
can finally go to Axiom 2, and claim that since NOT phi is a positive property,
then that implies phi is NOT a positive property. Easy enough.
And of course now we have just concluded phi is NOT a
positive property, even though the very first thing we did at the beginning of
proof by contradiction was to assume “Phi is a positive property AND it’s
necessary that for all x, NOT phi of x.”
So we have phi being a positive property and phi NOT being a positive
property. This is a contradiction, and
completes our proof by contradiction. We
have shown that if phi is a positive property then it’s possible there exists
some x with that property.
But of course I am very suspicious of a vacuous truth being
used. So I asked my Gemini app if it
could produce an example of a similar scenario where this proof structure is
followed, and the axioms seem reasonable, but the conclusion seems like
nonsense. Mostly as an attempt to
discredit the use of vacuous truths in proofs in the future. It cleverly suggested to plug in the phi
variable as “impossible crime” and the P variable as “legal.” It also suggested phi as invisible socks and
P as fashionable, and phi as boiling ice cream and P as delicious. These seem CLOSE to fitting the axioms, but I
still don’t feel they fit the axioms comfortably enough to my satisfaction when
I sit down and try to plug them in.
