Axioms that are not evident

Question

How can we give a meaning to the three axioms of prepositional logic? As axioms, is not it supposed to be obvious? For example, how is

$(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))$

evident?

Answer 1

It is a way (in addition to Modus Ponens) to "encode" the crucial properties of the conditional needed to prove the Deduction Theorem.

The fact can be more "obvious" (less obscure...) if we write it as a sequent:

$p → (q → r), p →q \vdash p → r$.

In Natural deduction, it is an easy consequence of the rules for $\to$:

if we have a derivation $\mathcal D_1: \Gamma_1, p \vdash (q \to r)$ and a derivation $\mathcal D_2: \Gamma_2, p \vdash q$, we apply $(\to$-Elim) followed by $(\to$-Intro) to get a derivation:

$\mathcal D_3 : \Gamma_1, \Gamma_2 \vdash (p \to r)$.

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The axiom $\phi \to (\psi \to \phi)$ expresses the Weakening rule:

we can add an additional premises in a derivation.

Answer 2

How is

$~~~~(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))$

evident?

Don't despair. The axiom you cite is actually a trivial theorem in natural deduction, the axioms of which are much more intuitive IMHO. I'm sure there is some good historical reason that it was chosen to be an axiom in an early formulation of formal propositional logic, but it eludes me.

(Screenshot from my proof checker)