Does Max-SNP hard imply NP-hard
$\newcommand{\maxsnp}{\mathsf{Max}\text{-}\mathsf{SNP}}$The definition of $\maxsnp$ gives us the ability to define:


Universal ($\forall$) and existential ($\exists$) quantifiers over variables
Existential quantifiers over relations


With this definition we can define $\mathsf{Max}$-$\mathsf{SAT}$:

$$\exists x, \forall y \text{ such that } |\psi(y)| \leq |\psi(x)|$$ where $|\psi(x)|$ is the number of clauses satisfied in formula $\psi$ under assignment $x$.

Given that $\mathsf{Max}$-$\mathsf{SAT}$ is $\mathsf{NP}$-$\mathsf{Hard}$, we see that $\mathsf{NP}$-$\mathsf{Hardness}$ is implied from $\maxsnp$.