Proof of the Principle of Vacuous Truth (ILIEKMATHPHYSICS)

Here are the following preliminary results used in the proof:

Thanks and enjoy the video!

[iliekmathphysics]

Here is the proof in the video, with details expanded out.



1
4____¬Q → 2–3


4, 6
5, 7
6–8
10____¬P → 5–9

The inference rules in the video come from what is provided in the description. An alternative proof is as follows (though, I didn't do this one because it uses explosion; i try to avoid using the explosion rule whenever possible just as a preference):



1, 2
X 3
5____¬P → 2–4

[methatis]

So this essentially is just that anything follows from a contradiction

[cheese]

1. P⊢(P∨Q) (Disjunction introduction)

2. P↔¬¬P (Double negation Elimination)

3. ∴ (P∨Q) ↔(¬¬P∨Q) (Substitution)

4. ∴ P⊢(¬¬P∨Q) (Substitution)

5. (¬¬P∨Q) ↔ (¬P→Q) (Definition of material implication)

6. ∴ P⊢(¬P→Q) (Substitution)

[mtaur]

"When pigs fly" but sort of formal.