We Can’t Say Something Positive is Impossible???

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.

So even though I don’t like vacuous truth arguments, I can’t seem to invalidate them the way I’d like to.  Perhaps we really do have to admit that if something is positive, then it can't be impossible, at least under the two seemingly reasonable axioms Godel used.  I really have to admire his skill in coming up with this trick.  It’s ugly, and questionable, but I can’t find a way to attack it the way I’d like to, try as I may.  Maybe someone else will have better luck.