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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 {} ? –  909 Niklas Mar 20 '13 at 11:57
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There is no empty sentence. –  Asaf Karagila Mar 20 '13 at 12:01

1 Answer 1

up vote 7 down vote accepted

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".

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