A Proof That God Cannot Create a Rock So Heavy He Cannot Lift It

Note. These proofs use the law of the excluded middle
https://ncatlab.org/nlab/show/computational+trinitarianism
Placeholder description for https://ncatlab.org/nlab/show/computational+trinitarianismlaw of the excluded middle (and double-negation elimination, which is equivalent to the law of the excluded middle) [nCatLab: Classical Logic
https://ncatlab.org/nlab/show/computational+trinitarianism
Placeholder description for https://ncatlab.org/nlab/show/computational+trinitarianismnCatLab: Classical Logic].

Theorem. God cannot create contradictions.

Proof. By the law of the excluded middle, either God can create contradictions or He cannot.

• First, suppose that God can create contradictions. It is a contradiction for a contradiction (i.e. an impossibility) to be possible, so by the principle of explosion
  https://en.wikipedia.org/wiki/Escrow
  Placeholder description for https://en.wikipedia.org/wiki/Escrowprinciple of explosion it follows that God cannot create contradictions.
• Second, suppose that God cannot create contradictions. Then the result follows immediately.

Since the result holds in both cases, the result holds.

Corollary. God cannot create a rock He cannot lift.

Proof. Suppose for the sake of contradiction that God can create a rock He cannot lift. God is omnipotent (by definition), which implies that He is able to lift any rock. It is contradictory for there to be a rock that God cannot lift while God has the ability to lift any rock. So the supposition introduced a contradiction, and by the previous theorem God cannot create contradictions, hence the supposition must be false. The result follows by double-negation elimination.

Corollary. God cannot keep all of His promises while also breaking one of his promises.

Proof. Suppose for the sake of contradiction that God can keep all of His promises while also breaking one of his promises. This is a contradiction, and by the previous theorem God cannot create contradictions, hence the supposition must be false. The result follows by double-negation elimination.

Remark. The corrolaries also follow fairly directly from the principle of explosion, since they both have the form "suppose a contradiction, then derive the result." They are presented as corollaries for the sake of demonstrating specific instances of applying the derivation of the theorem.

§ References

• law of the excluded middle
• nCatLab: Classical Logic
• principle of explosion

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