Classical Propositional Logic and Axioms

Question

We define a new proof system N over the connectors: {∨,¬} For every α and β-

1: ( ∨ ( ∨ (¬))) (axiom)

Deductions:

1: if we have , then we can deduce (¬(¬(α∨β)))

2: if we have ((¬α)∨(¬β)) then we can deduce (α∨β)

Prove/refute: For every α if ⊢α in the new system (defined above) then ⊢CPLα (classical propositional logic)

(I have asked the question before, but did not get any answer - and so far I know that it must be refuted)

Answer 1

Classical Propositional Logic calculus is sound and complete for classical semantics, i.e.

$\vdash_{ \text {CPL} } \alpha \text { iff } \alpha \text { is a tautology.}$

Thus, the problem amounts to prove that :

$\text { if } \vdash_{ \text N } \alpha, \text { then } \alpha \text { is a tautology.}$

To do this, we have to answer to the following questions :

1) is $\text A_1$ a tautology ? [Yes: check it with truth table.]

2) are $\text {MP}_1$ and $\text {MP}_2$ sound ? i.e. do they produce TRUE conclusions from TRUE premises ?

$\text {MP}_1$ is sound : when $$ and $$ are both TRUE, also $¬(¬(α∨β))$ is.

The issue is with $\text {MP}_2$ : what happens when both $$ and $$ are FALSE ?