Question

How can i show that if `Γ∪{φ}⇒ψ` and `Γ∪{φ}⇒¬ψ` then `Γ⊨¬φ`?

how can i show that if `Γ∪{φ}⇒ψ` and `Γ∪{φ}⇒¬ψ` then `Γ⊨¬φ`?

Answer 1

See explanation...

Explanation:

Given:

`Gamma uu { varphi } => psi" "` and `" "Gamma uu { varphi } => not psi`

we can deduce:

`Gamma uu { varphi } => (psi ^^ not psi) => ⊥`

So:

`Gamma |== (varphi => ⊥)`

From which we can deduce:

`Gamma |== not varphi`