Truth of sentences

Question

Let say I have a signature $\{ P_1^1 ,P_2 ^2 , F^2 \}$ with the interpretation $$U= \{ 0,1 \}$$ $$R_1= \{ 0 \}$$ $$R_2= \{ (0,1),(1,0) \}$$ $$F={(0,0,1),(1,0,0),(0,1,0),(1,1,1)}$$

I have been given a sentence $$\exists x.P_1 (x)$$

Why is this sentence true?

Answer 1

Assuming that the interpretation of $P_1$ is $R_1$ and of $P_2$ is $R_2$, we have that in the domain $U$ there is an element, $0$, such that it satisfy the interpretation of the predicate symbol $P_1$, that is $R_1$, due to the fact that : $0 \in R_1$.

The inductive definition of the satisfaction relation : $\mathcal M \vDash \varphi$ (read : the structure $\mathcal M$ satisfy the formula $\varphi$) gives us :

let $\varphi$ a formula with a free variable $x$ and let $D$ the domain of the structure $\mathcal M$.

We define :

if $\varphi$ is $P_1(x)$, then $\mathcal M \vDash \varphi(a)$ iff $a \in P_1^M$, where $P_1^M$ is the interpretation of $P_1$ in $\mathcal M$ [in our case $R_1$]

And :

$\mathcal M \vDash ∃x \varphi$ iff there is some $a \in D$ such that : $\mathcal M \vDash \varphi(a)$.