Principle of explosion: Other arguments? I've come across a proof-theoretic argument for explosion on Wikipedia, which is as follows:


*

*$A \ \ \wedge\sim A$

*$A$

*$ \sim A$

*$ A \lor B$

*$B$

*$(A \ \ \wedge \sim A) \implies B$
I've thought of another argument, which isn't on the same Wikipedia page as the above. As far as I can see, it is valid but I would like to see your opinions. Perhaps you could provide me with some more (relatively simple) arguments for explosion within classical logic?


*

*$A \ \ \wedge\sim A$

*$A$

*$A \lor B$

*$ \sim A$

*$B$

*$(A \ \ \wedge \sim A) \implies B$
 A: I use Polish/Lukasiewicz notation.  The rule of Negation elimination that I use says that from N$\beta$ having the same scope as an instance of K$\alpha$N$\alpha$ we can infer $\beta$.
assumption                   1 | KaNa
assumption                   2 || Nb
2, 1 Negation Elimination    3 | b
1-3 Conditional Introduction 4 CKaNab.

As a more axiomatic proof, I'll assume that we have the following three axioms:
A1 CpCqp.
A2 CCpCqrCCpqCpr.
A3 CCNpKqNqp.

Then, with Dx.y indicating the condensed detachment with x as the major premise and as the minor premise we can proceed as follows:
assumption                   1 | KaNa
assumption                   2 || Nb
D[A1].1                      3 || CpKaNa
D3.3                         4 || KaNa
2-4 Conditional Introduction 5 | CNbKaNa
D[A3].5                      6 | b

Or as follows:
assumption                   1 | KaNa
D[A1].1                      2 | CpKaNa
D[A3].2                      3 | p

Thus, a fully axiomatic proof can proceed as follows:
axiom 1 CpCqp.
axiom 2 CCpCqrCCpqCpr.
axiom 3 CCNpKqNqp.
D1.3  4 CpCCNqKrNrq.
D2.4  5 CCpCNqKrNrCpq.
D5.1  6 CKpNpq.

