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:enter image description here

Kleene's axioms: enter image description here

Any help is appreciated, thank you.


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).

  • $\begingroup$ Thank you for the clarification, I had a skim over the cited list; interesting stuff. $\endgroup$ – joshuaheckroodt Jan 19 at 15:57
  • $\begingroup$ @joshuaheckroodt - regarding $\lnot$ and different equivalent sets of rules (or axioms) you can see the answer to this post. $\endgroup$ – Mauro ALLEGRANZA Jan 19 at 16:36
  • $\begingroup$ The way this gets phrased suggested that the above consist of axioms. None of what gets listed though makes for an axiom since none of those are well-formed, and by definition, an axiom has to be well-formed. $\endgroup$ – Doug Spoonwood Jan 19 at 18:56

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.

  • $\begingroup$ The assumption in both cases is that the symbols that are used to stand in for statements are well-formed. $\endgroup$ – joshuaheckroodt Jan 19 at 18:58
  • $\begingroup$ @joshuaheckroodt If some sequences of symbols A gets used to stand in for a well-formed statement, then A is not well-formed. If B is an axiom, then it it is well-formed. So, A cannot be an axiom without a contradiction. $\endgroup$ – Doug Spoonwood Jan 19 at 19:09
  • $\begingroup$ How does this assumption "Let $\phi$, $\mathcal{X}$, and $\psi$ stand for well-formed formulas. (The well-formed formulas themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows" aid the composition of axioms, in which case? $\endgroup$ – joshuaheckroodt Jan 19 at 19:13
  • $\begingroup$ @joshuaheckroodt I don't understand what you're asking. $\endgroup$ – Doug Spoonwood Jan 21 at 4:37

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