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Given $Γ$ is semantically inconsistent, need to prove: for all propositions $θ$, we have $Γ⊨θ$.

"$Γ$ is semantically inconsistent", then there is no truth assignment $A$ such that $A(Γ)=1$.

But the definition of entail states:

A set of sentences $Δ$ logically entails a sentence $φ$ (written $Δ⊨φ$ ) if and only if every truth assignment that satisfies $Δ$ also satisfies $φ$.

My question is: Since there is not truth assignment $A$ satisfies $A(Γ)=1$, how can we find a truth assignment satisfies $φ$?

yashirq
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    Look up "Vacuously true". – Sean English Dec 18 '16 at 23:56
  • "how can we find a truth assignment that satisfies $φ$ ?" But in order to assert that $\Gamma \nvDash \theta$ we have to find a truth assignment $A$ such that $A(\Gamma)=1$ and $A(\theta)=0$ and this is impossible, because "there is no truth assignment $A$ such that $A(Γ)=1$". – Mauro ALLEGRANZA Dec 19 '16 at 07:19

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You don't need to find a truth-assignment that satisfies $\varphi$. What you need to prove is that if a truth-assignment satisfies $\Gamma$, then it satisfies $\varphi$. And to prove that, it suffices to point out that there is no truth-assignment that satisfies $\Gamma$: with the antecedent part false, the whole conditional is (vaciously) true.

Bram28
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