# not always $A \models \phi$ or $A \models \neg \phi$ example

I have the following definition in my lecture notes:

We say that a formula $\phi$ is truth in the structure $\mathscr A$, $A \models \phi$, if for every valuation $\mathscr v$: $A \models _{\mathscr v} \phi$

Then I have the following statements:

Always $A \models \phi$ or $A \not \models \phi$

Not always $A \models \phi$ or $A \models \neg \phi$

I understand the first statement, but I don't completely grasp the second one. Can somebody give me an example which might clarify it.

• From what I understand it's defined for every valuation v in the structure. It's not for a concrete valuation but for all in the structure. Does that answer your question? Jul 3, 2017 at 11:31

Consider the structure $$\mathbb N$$ of natural numbers and consider the formula $$(x=0)$$.

For a valuation $$v$$ such that $$v(x)=0$$ we have:

$$\mathbb N, v \vDash (x=0)$$

while obviously, for a valuation $$v'$$ such that $$v'(x)=1$$ we have: $$\mathbb N, v' \nvDash (x=0)$$.

Thus, if we define: $$A \vDash \phi$$ as "$$A,v \vDash \phi$$, for every valuation $$v$$", we have that:

neither: $$\mathbb N \vDash (x=0)$$, because we have the valuation $$v'$$ above such that $$\mathbb N, v' \nvDash (x=0)$$,

nor: $$\mathbb N \vDash \lnot (x=0)$$, because we have the valuation $$v$$ above such that $$\mathbb N, v \nvDash \lnot (x=0)$$.