Positive Implicational Logic can be axiomatized as follows:
P1: $\phi \to \psi \to \phi$
P2: $(\phi \to \psi) \to (\psi \to \chi) \to \phi \to \chi$
P3: $(\phi \to (\phi \to \psi)) \to \phi \to \psi$
Inspired by this post Proving $(p \to (q \to r)) \to ((p \to q) \to (p \to r))$, I decided to try to figure out an axiomatization of similar “complexity” using
$(\phi \to \psi) \to (\chi \to \phi) \to \chi \to \psi$
instead of P2.
I came up with the following:
P1: $\phi \to \psi \to \phi$
P2’: $(\phi \to \psi) \to (\chi \to \phi) \to \chi \to \psi$
P3’: $(\phi \to (\psi \to (\phi \to \chi))) \to \psi \to \phi \to \chi$
Is P3’ the simplest axiom that can be added to P1 and P2’ to yield Positive Implicational Calculus, aside from the Frege/distribution axiom? Also, I’m aware that axioms usually (always?) correspond to some sort of combinator. Is there a known combinator that corresponds to P3’?
