I have found conflicting lists of axioms in propositional calculus in Kleene, $2002$, and on Wikipedia. From what I can tell, carefully reasoning through each of the statements reveals that are tautologies, and since the only real understanding I have surroundings logical axioms is that they are tautologies, I can't quite decide which source to trust.
Wikipedia's axioms:
Kleene's axioms:
Any help is appreciated, thank you.
Best Answer
There are many different (but equivalent) axiomatizations of propositional calculus. See e.g. List of Hilbert systems.
The fist nine are the same.
Kleene's version does not use the $\Leftrightarrow$ connectives; thus, the last three axioms of Wiki's list are not needed in Kleene's version.
The only real difference regards the axioms fo $\lnot$.
Kleene's axiom 10 is Double Negation elimination : it is equivalent to LEM (axiom not-3).