Using context free grammar to work with propositional logic symbols

i'm currently trying to use a context free grammar for propositional logic.

Im imagining that the set of terminals will looks like this:

T=(q,p,¬,∧,∨,→,⟷).

Now what i need is to define a set of productions, that can help me in achieving all legal compound propositions. Can anyone help me? I really dont know where to start, the high amount of terminals is kinda throwing me off

It might help to start by reducing the scope of your problem. Specifically, can you write a CFG for expressions when the only allowed symbols are p, q, and ∧? In that case, every expression is either

• a single propositional variable, or
• a conjunction (∧) of two expressions.

That would give something like this:

E → p | q | E ∧ E

Now, how would you add in the ability to use ∨? How about the other symbols? See if you can take it from here.