Axiom System: Lukasiewicz In the lecture notes of Bilaniuk 2003  the following axiom system is chosen:


*

*A1: $(\alpha\to(\beta\to\alpha))$,

*A2: $((\alpha\to(\beta \to\gamma))\to ((\alpha\to\beta)\to(\alpha\to\gamma)))$,

*A3: $(((\lnot\beta)\to(\lnot\alpha))\to(((\lnot\beta)\to\alpha)\to\beta))$.


Somewhere I read that the original 3rd axiom of Lukasiewicz was
$$
A3^* (((\lnot\alpha)\to(\lnot\beta))\to(\beta\to\alpha)).
$$
I assume that (A1,A2,A3) and (A1,A2,A3*) are equivalent.


*

*Are $A3$ and $A3^*$ an historical development by Lukasiewicz?

*Is there a simple explanation why a mathematician would prefer A3 to A3*, or the other way around?

 A: "Are A3 and A3∗ an historical development by Lukasiewicz?"
The corresponding set of expressions in Lukasiewicz's notation to (A1, A2, A3*) is dated to 1930 by Lukasiewicz by A. N. Prior in the appendix of his Formal Logic.  (A1, A2, A3) isn't stated as an axiom set by Lukasiewicz, even if you had used the corresponding wffs in Lukasiewicz's notation.  The formula, I don't know of appearing until Elliot Mendelsohn's Introduction to Mathematical Logic, though there's no doubt that it was known earlier as an axiom set and by Lukasiewicz.  However, perhaps Lukasiewicz's student St. Jaskowski first knew about that as an axiom set.
In St. Jaskowski's paper On the Rules of Suppositions in Formal Logic Rule IV states:
"Rule   IV. Given   a   domain   D   of   a   supposition   composed
successively  of  a  symbol  "N"  and  of  a  proposition α, if  two  pro-
positions  ß  and γ are  valid  in  D  such  that γ is  composed  succes-
sively  of  a  symbol  "N"  and  of  a  proposition  equiform  with  ß,  it
is  allowed  to  subjoin  a  proposition  equiform  with α to  that  do-
main whereof D is an immediate subdomain."
We might rewrite the idea, where the domain D gets kept in mind, as:
{N$\alpha$$\vdash$$\beta$, N$\alpha$$\vdash$N$\beta$}$\vdash$$\alpha$.
But, just using A1 and A2 we can end up with:
CCN$\alpha$N$\beta$CCN$\alpha$$\beta$$\alpha$ (this works as the corresponding wff to A3 in Polish notation).
So, at the very least, one might maintain that the idea of (A1, A2, A3) as an axiom set seems implicit in Jaskowski.
"Is there a simple explanation why a mathematician would prefer A3 to A3*, or the other way around?"
At least to my mind, (A1, A2, A3) more easily correspond to a natural deduction system with proof by contradiction then (A1, A2, A3*).  If one uses a natural deduction system to help set up writing proofs by substitution and detachment, or by condensed detachment, then I expect (A1, A2, A3) would end up easier to use.  I also have found it easier to derive (A1, A2, A3*) from (A1, A2, A3) than to derive (A1, A2, A3) from (A1, A2, A3*).  I suggest that you try your hand at it and see which you find easier.
A: Yes, $(A1,A2,A3)$ and $(A1,A2,A3*)$ are equivalent. 
Without too much effort you can show that $A3$ can be derived from $(A1,A2,A3*)$ and that $A3*$ can be derived from $(A1,A2,A3)$.
And, once you can derive the one from the other, any advantages of one system over the other disappear, so I don't see any real preference of one system over the other.
And finally, sorry, I'm not good with the historical details here ...
