Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. It only takes a minute to sign up.
Sign up to join this community
As part of Gödel's ontological proof of God's existence he offers a Theorem that all positive properties are possible. In his proof of it he says that impossible properties entail all properties. But why is this? Every explanation of his argument assumes it, so I don't actually know why he believed this.
Why and how does an impossible property entail all properties?
Do impossible properties include all properties?
Why so ? I assume that you are referring to Wiki's exposition of Gödel's proof:
Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
Proof
1) P(φ) --- premise
2) ¬P(ψ) --- form 1) : if φ is positive, by Ax.2, ¬φ is negative, and thus there is a negative property ψ
3) ¬∀x(φ(x) → ψ(x)) --- by Ax.1: a property entailed by a positive one is also positive
4) ¬□∀x(φ(x) → ψ(x)) --- from 3)
5) ⋄∃x(φ(x) ∧ ¬ψ(x)) --- from 4) by properties of modalities, quantifiers and propositional logic
6) ⋄∃xφ(x) --- from 5).
Thus, from 1) and 6), by Deduction Theorem:
P(φ) → ⋄∃xφ(x)
i.e. If a property is positive, then it is possibly exemplified.
