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

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

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Answer 1 · 2018-08-12T08:29:16

See 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#