Independence of Axioms of Propositional Logic

Chenkai Wang

It is an interesting question that whether some mathematical statement(s) can or cannot be derived from other statement(s). To show a statement is provable, simply present a proof. But is there a way to show a statement is unprovable? For example, the Euclid's fifth postulate turned out to be unprovable from other postulates. Because one can find a geometry where all Euclid's postulates are ture (including the fifth) and another where all but the fifth are true. Since proof preserves truth one can say the fifth is unprovable. I am going to demonstrate a simple technique of showing the three axioms for propositional logic is independent from each other.

The axioms for proposition logic are

1. $ \varphi\rightarrow(\psi\rightarrow\varphi) $

2. $ (\varphi\rightarrow(\psi\rightarrow\chi))\rightarrow((\varphi\rightarrow\psi)\rightarrow(\varphi\rightarrow\chi)) $

3. $ (\lnot\varphi\rightarrow\lnot\psi)\rightarrow(\psi\rightarrow\varphi ) $