I'm thinking about part (a) of the following exercise in Just/Weese page 77:
Here is the definition of valuation:
For example, say we have a model of the language of group theory, $( \mathbb Z/ 2 \mathbb Z, +, 0)$. Let $\varphi = \forall v_0,v_1: v_0 + v_1 = v_1 + v_0$ and let $s: \omega \to \mathbb Z / 2 \mathbb Z$ be the map $s(n) = 0$ for all $n$. Then we should have $( \mathbb Z / 2 \mathbb Z, +, 0) \models_s \varphi$ but I am confused about what happens to variables under a given valuation. For the valuation I defined above the formula becomes $\varphi = \forall 0,0: 0 + 0 = 0 + 0$. Which is true but what is "$\forall 0,0:$" supposed to mean? Am I misunderstanding what a valuation is? If yes, would someone correct my example? Thanks for your help.