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.
On
No, what gets listed are not axioms. None of those are well-formed, and thus to call them axioms is not correct. That said, it probably isn't difficult for many who know an appropriate definition of a well-formed formula to write the intended axioms from the symbols given.
On
There are many equivalent axiomatizations of propositional logic; the one that gets used is up to you. It is possible to prove that a theory is complete by showing that any tautology in the the theory is provable. For all of these axiom systems, their completeness has been proven. Some axiom systems prefer to be incomplete, such as theories which do not accept the law of excluded middle. Different axiom systems have different strengths and weaknesses, depending on what you're trying to prove.
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).