In my course notes I have a statement like this one:
Let $F_1, ..., F_n \in FOL$. We say that $F$ is a semantic consequence from $F_1, ..., F_n$ and we denote that by $F_1, ..., F_n \vDash F$ if for every structure $S$ for which $S(F_1) = ... = S(F_n) = 1$ we have that $S(F) = 1$.
And there ar some examples:
$(\forall)P(x) \vDash P(c)$
$P(c) \rightarrow Q(x), P(c) \vDash Q(x)$
As I understand, for every structure in which $P(c)$ is true, then $Q(x)$ is also going to be true. But how does that happen? For example, if I interpret $P(x)$ as $x < 10$, $c$ as $4$ and $Q(x)$ as $x > 0$ and $x$ as $-1$. Then, in this structure, $P(c)$ will be true, but $Q(c)$ will be false. What am I getting wrong?
NO; for every structure in which $P(c)$ and $P(c) \to Q(x)$ are true, then $Q(x)$ is also true.
Thus, if $P(c)$ is true, by truth-table for the conditional, the only possibility left when $P(c) \to Q(x)$ is true is that $Q(x)$ is true also.
If you interpret $P(x)$ as $x < 10$, $c$ as $4$ and $Q(x)$ as $x>0$ $x$ as $−1$, in this structure, $P(c)$ will be true, but $P(c) \to Q(x)$ will be false :
is $TRUE \to FALSE$, i.e. $FALSE$, and thus the definition of semantic consequence is not violated.