Simplest Proof of Pierce's Law

Question

I am trying to prove Pierce's Law in the simplest way possible. Many of the proofs I've encountered utilize proof by cases, or distribution laws. I believe it can be done without.

What I am trying to show, formally, is the following: $ \{ ( A \to B ) \to A \} \vdash A $

using only basic rules of inference and the definition of the binary connectives.

The following proof is the best I could surmise, but it uses distribution, which I know can be avoided.

$ ( A \to B ) \to A $

$ \neg( A \to B ) \lor A $

$ ( A \land \neg B ) \lor A $

$ ( A \lor A ) \land ( A \lor \neg B ) ~~~ distribution ~ \text{what I want to avoid} $

$ A \lor A $

$ A $

Thank you!

Answer 1

You are using the rules of propositional calculus.

If you may use the Law of Absorption, you may use it.

$${(A\wedge\neg B)\vee A~~\equiv~~ A}$$