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Why is "If $Γ ⊨ ¬ψ$, then $Γ ⊭ ψ$" false?

I believed that this is true and Stanford doesn't agree. So I worked at the problem again and here is what I got. I wanted to check my reasoning:

Suppose $Γ ⊨ ¬ψ$ is true. Then for any truth value assignment that makes $Γ$ true, $¬ψ$ is true; that is, for any truth value assignment that makes $Γ$ true, $ψ$ is false. Now, take any assignment that make $Γ$ true. Then $ψ$ is false.

New work: The problem is that there need not be any truth value assignment that makes $Γ$ true. Indeed, take $Γ = \{ p ∧ ¬p \}$.

Is my new reasoning correct?

Squirrel-Power
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    Well done! Indeed, the fact that every model of $\Gamma$ makes $\lnot \psi$ true does not mean that there exists a model of $\Gamma$. – Taroccoesbrocco Dec 19 '18 at 00:13
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    Another way of saying essentially the same thing would be: in the case that $\Gamma$ is inconsistent, then $\Gamma \models \phi$ for every formula $\phi$, so $\Gamma \models \psi$ and $\Gamma \models \lnot \psi$ both hold. – Daniel Schepler Dec 19 '18 at 00:26

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I put it into everyday words.

lets sat the antecent is all components in the set T all equalling 1 "entails" that ¬ψ is true.

Human Words: if all the workers, t_1, t_2, and t_3 are at work (equal 1), then "this tells us for a fact" that the boss being mad (ψ) is not(¬) accurate. so, ¬ψ = 1, which means ψ= 0

Tina, Terry and Tom are at work, therefore we know for a fact that the boss is not mad.

now we move to the consequent. Here we have

then Tina, Terry and Tom being at work (all equal to 1) does not entail that ψ = 1

in better human form:

Tina, terry and Tom are at work, and this does not tell us that the boss is mad.

Well, the lines put together are this:

If Tina, Terry and Tom are at work we know that the boss is not mad,

Then Tina, Terry and Tom are at work and we do not know if the boss is mad.

Clearly this is false. If we know the boss is not mad, then it is impossible to not know if he is mad.

Does that make more sense?

Jason Spyres
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