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

Question

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.

Thanks in advance!

Answer 1

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 $\mathbb N, v \nvDash \lnot (x=0)$.