A logic $L$ has the finite model property if for every formula $\phi$ of $L$, $\phi$ is an $L$-theorem iff $\phi$ is a theorem of the theory of finite models of $L$.
A structure $M$ is called pseudofinite if $M\models \phi$, then there a finite structure $M_0$ such that $M_0\models \phi$.
Question: What is the different between the finite model property and pseudo-finiteness? I'm confused by their definitions!