# What is the difference between $Γ⊭Φ$ and $Γ⊭¬Φ$?

Did I understand this correctly?

1. $Γ⊨Φ$ ($Φ$ is considered true)
2. $Γ⊨¬Φ$ ($Φ$ is considered false)
3. $Γ⊭Φ$ ($Φ$ is considered neither true nor false)
4. $Γ⊭¬Φ$ ???

Please help me understand. How can we know (Γ⊭Φ) → (Γ⊨¬Φ) ? Is Church's theorem on the form Γ⊢¬Φ (Proving that there is no algorithm..)

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As for the last part, we can't. Take $\Gamma$ to be a single sentence $\gamma$. And take $\delta$ to be a completely "disjoint" sentence (i.e. there is no variable appearing in both sentences), then $\Gamma\not\models\delta$ but also $\Gamma\not\models\lnot\delta$. – Asaf Karagila Mar 20 '13 at 11:45
@AsafKaragila Of course. What happens if δ is the empty sentence {} ? – Dac Saunders Mar 20 '13 at 11:57
There is no empty sentence. – Asaf Karagila Mar 20 '13 at 12:01

The notation $\models$, as opposed to $\vdash$, is about truth, not about provability.
$\Gamma\models\Phi$ means $\Phi$ is true in every structure in which every member of $\Gamma$ is true.
$\Gamma\not\models\Phi$ means it is not the case that $\Phi$ is true in every structure in which every member of $\Gamma$ is true. That means there is at least one structure in which every member of $\Gamma$ is true and $\Phi$ is not true.
$\Gamma\models¬\Phi$ means $¬\Phi$ is true in every structure in which every member of $\Gamma$ is true. The difference between this and the statement preceding it is the difference between "at least one" and "every".
"That means there is at least one structure in which every member of $\Gamma$ is true and $\Phi$ is not true." -- Is it possible that there are no structures in which every member of $\Gamma$ is true? (it would imply that there are no structures in which every member of $\Gamma$ is true and $\Phi$ is not true) – Vladimir Reshetnikov May 4 '15 at 19:00