prove or disprove if $\Gamma,A\models^{v}B$ and $\Gamma,\neg A\models^{v}B$ then $\Gamma\models^{v}B$

Question

prove or disprove if $\Gamma,A\models^{v}B $ and $\Gamma,\neg A\models^{v}B$ then $\Gamma\models^{v}B$

I believe this is a generalization of the Dichotomy theorem. The problem I am not quite sure if the same proof will work here.

Definition:

Answer 1

The statement is false.

Consider a language with constant symbol $c$. Let $x$ be a variable.

Then let $A :\equiv (x = c)$.

Let $B :\equiv \forall x (x = c)$.

Then $A \models^v B$. For suppose $M, c_m \in M$ is a structure and for all $\alpha$, $M, \alpha \models A$. Then consider $m \in M$. Then let $\alpha$ be the variable assignment sending all variables to $m$. Then $M, \alpha \models A$. Then $m = c_m$. So $M = \{c_m\}$. Then clearly $M \models B$.

And $\neg A \models^v B$. For suppose that $M, c_m \in M$ is a structure and for all $\alpha$, $M, \alpha \models \neg A$. Then let $\alpha$ be the variable assignment sending all variables to $c_m$. Then $M, \alpha \models A$. Contradiction. Then $M \models B$.

But it is not the case that $\models^v B$. For we can take the structure $M = \{0, 1\}$, $c_m = 0$. Then $M$ does not model $B$.