Classical Propositional Logic and Axioms

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

𝐴1: (𝛼 ∨ (𝛽 ∨ (¬𝛼))) (axiom)

Deductions:

𝑀𝑃1: if we have 𝛼, 𝛽 then we can deduce (¬(¬(α∨β)))

𝑀𝑃2: 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)

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 $$𝛼$$ and $$𝛽$$ are both TRUE, also $$¬(¬(α∨β))$$ is.

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

• That's why I said it should be refuted rather than proven. However, the problem is that I couldn't find a counterexample. – Kamal Khalaily May 12 at 19:20