# Proof of Peirce's Law in Propositional Calculus

Is it possible to derive Peirce's Law:

⊢∗ [(α → β) → α] → α

in a calculus that has modus ponens, the Deduction Theorem, Cut rule, Inconsistency effect and Principle of Indirect Proof?

• What do you mean by Inconsistency effect and Principle of indirect Proof? Neither are standard labels. – Peter Smith Feb 4 '18 at 7:31
• Sorry about that. Inconsistency Effect: If Φ ⊢ , then Φ ⊢ β for every formula β (if Φ is inconsistent, then any formula can be deduced) . Principle of Indirect Proof: If Φ, ¬ α ⊢ , then Φ ⊢ α (If Φ, ¬ α is inconsistent, then α can be deduced from Φ) . – DiscipleOfKant Feb 4 '18 at 7:44
• What you're calling the "Inconsistency Effect" is (much) more commonly called the Principle of Explosion or "ex falso quodlibet" (often shortened to just "ex falso") or, in some contexts, falsity/bottom elimination though that's often written as $\bot E$. – Derek Elkins Feb 5 '18 at 1:28
• The Wikipedia article on Peirce's law has good background information. – hardmath Feb 5 '18 at 17:27

We refer to: Moshe Machover, Set Theory, Logic and Their Limitations Cambridge UP (1996), page 116-on for the definitions and some results about propositional calculus.

Proof

1) $(ϕ → ψ) → ϕ$ --- premise

2) $\lnot \phi$ --- premise

3) $\vdash^* \lnot \phi \to (\phi \to \psi)$ --- Problem 8.8 [page 125]

4) $\phi \to \psi$ --- from 2) and 3) by mp

5) $\phi$ --- from 1) and 4) by mp.

Up to now we have: $(ϕ → ψ) → ϕ, \lnot \phi \vdash^* \phi$.

Obviously: $(ϕ → ψ) → ϕ, \lnot \phi \vdash^* \lnot \phi$.

Thus, we can use Indirect proof to get:

6) $(ϕ → ψ) → ϕ, \vdash^* \phi$.

7) $\vdash^* ((ϕ → ψ ) → ϕ) → ϕ$ --- from 6) by Deduction Theorem.