If Φ⊢¬(ϕ→ψ) , show that Φ,¬ϕ and Φ, ψ are both inconsistent.

Question

So far I have shown that Φ, ψ is inconsistent:

If Φ⊢¬(ϕ→ψ) then Φ, ψ⊢¬(ϕ→ψ)

By the axiom ψ→ϕ→ψ and Modus Ponens, Φ, ψ⊢ϕ→ψ.

So Φ, ψ is inconsistent.

Could anyone help me to prove that Φ,¬ϕ is inconsistent?

Answer 1

From the same reasoning you used for the first part, $$ \Phi,\lnot\phi \vdash \lnot \psi \to \lnot \phi.$$ By the contrapositive axiom (theorem? you haven't told us which system you are using... it seems clear that it's Hilbert style from context, but there are several commonly used versions of Axiom 3), $$ \Phi,\lnot \phi\vdash (\lnot\psi\to \lnot \phi)\to(\phi\to \psi), $$ so by MP $$\Phi,\lnot \phi\vdash \phi\to\psi $$