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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?

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    What is Goedels ontological argument? Do you mean his formalization of the ontological argument for the existence of God? Please quote the passage about the impossible properties and give a reference; thanks.
    – Jo Wehler
    Feb 4 '16 at 22:28
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    Yes, because of the law of explosion in classical logic en.wikipedia.org/wiki/Principle_of_explosion
    – Conifold
    Feb 4 '16 at 23:28
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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.

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  • No I was referring actually to Robert e. Maydoles discussion of the ontalogical argument in "the Blackwell companion to Natural Theology". Conifold nailed it, thanks!
    – Steve
    Feb 5 '16 at 23:54
  • @Steve - unfortunately, Maydoles call Th.1 what Wiki calls Th.2... Feb 6 '16 at 11:01

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