Confusion - exercise - satisfiable or valid I've the following exercise:

Determine if the following formula is satisfiable  or valid:
$P(x_{0}) \rightarrow \forall x_{1}P(x_{1})$


I've no idea what $P$ means, no information is given and my first thought is that if we define $P(x_{0}) := \ x_{0} \wedge \neg x_{0}$ then the formula is satisfiable, but if $P(x_{0}) := x_{0} \doteq c$ where $c$ is a constant then this is not true in general.
So from my point of view the answer depends on $P$.

How do I solve it? What is my problem?
 A: $P$ is an arbitrary property. The sentence says that if $x_0$ has property $P$ then everything has property $P$. This is not universally true for all choices $P$ and $x_0$. But it is true for some choices of $P$ and $x_0$.
A: This is the drinker's paradox.
Edit: this was longer than I expected so I'll summarize:  the statement is always satisfiable (there is always a choice of $x_0$ that makes it true no matter what $P$ is), but $x_0$ can be defined in a way to make the statement invalid.  If $x_0$ is existentially quantified, then the statement is valid.
Consider 3 cases for $P$:
1) $P$ is universally false.  Witness $x_0$ to be anything, the implication becomes:
$$P(x_0) \rightarrow \forall x_1\,P(x_1)$$
$$\text{false} \rightarrow \text{false}$$
$$\text{true}$$
2) $P$ is universally true.   Witness $x_0$ to be anything, the implication becomes:
$$P(x_0) \rightarrow \forall x_1\,P(x_1)$$
$$\text{true} \rightarrow \text{true}$$
$$\text{true}$$
3) $P$ is sometimes true and sometimes false.   Witness $x_0$ to be a value that makes $P(x_0)$ false:
$$P(x_0) \rightarrow \forall x_1\,P(x_1)$$
$$\text{false} \rightarrow \text{false}$$
$$\text{true}$$
So you see, no matter the choice of $P$, the statement (let's call it $T$) is satisfiable.  To make it a valid tautology, you have to consider$x_0$, is it already defined, or is it quantified:
1) $T$ can be invalid if $P$ is sometimes true and sometimes false:
$$\begin{cases}
\text{Define } x_0 \text{ as something that makes } P(x_0) = \text{true} \\
\text{Define } T \text{ as } P(x_0) \rightarrow \forall x_1\, P(x_1) \\
\end{cases}
$$
2) $T$ can be valid if $P$ is sometimes true and sometimes false:
$$\begin{cases}
\text{Define } x_0 \text{ as something that makes } P(x_0) = \text{false} \\
\text{Define } T \text{ as } P(x_0) \rightarrow \forall x_1\, P(x_1) \\
\end{cases}
$$
3) $T$ is valid if $P$ is always true or always false
$$\begin{cases}
\text{Define } x_0 \text{ as anything } \\
\text{Define } T \text{ as } P(x_0) \rightarrow \forall x_1\, P(x_1) \\
\end{cases}
$$
4) $T$ is valid for any definition of $P$ if $T$ is existentially quantified:
$$\begin{cases}
\text{Define } T \text{ as } \exists x_0\,\bigg(P(x_0) \rightarrow \forall x_1\, P(x_1) \bigg)\\
\end{cases}
$$
Some of the above statements are only true if the domain of discourse is non-empty.  Most logics assume that the universe is not nonempty.
