I have the following grammar, but I'm not sure what exactly it is that it does:
$\qquad\begin{align} S &\to S \vee T \mid T \\ T &\to T \wedge F \mid F \\ F &\to p \mid \thicksim p \end{align}$
How can I go about converting it to a Push Down Automata?
There is an algorithm for converting a context free grammar to pushdown automaton. It only has one state and the stack symbols are $\{terminals\}\cup\{nonterminals\}$ see for example:
http://www.cs.uiuc.edu/class/fa05/cs475/Lectures/new/bwlec12.pdf
(Don't really know if that is good source, I just googled and glanced at it, (where I learned this stuff isn't in English :D)
Looking at this grammar, it seems to generate logical statements. The symbols are $\vee =$ or, $\wedge =$ and, $\thicksim$ $= $ no and $p$ is a statement.
By these grammar rules, way you can form a statement is by connecting two statements with an $\vee$ or with an $\wedge$. Also the statements $p$ and $\thicksim p$ are ok by themselves.