How to make a parse tree for the following propositional logic formula?

Question

I have a formula $\neg (( q \rightarrow \neg q) \vee p \vee ( \neg q \rightarrow ( r \wedge p)))$.

As it contains 3 subformulas between the $\vee$'s, how can I put it into a parse tree. Would it be just the one branch, the node being the $\vee$?

Answer 1

I believe that there are (at least) a couple of options.

1. It is possible that $\vee$ is taken to have arbitray (finite) arity, meaning it could take any finite number of disjuncts. In this case the root of the parse tree would be a single $\vee$ having 3 children.
2. If $\vee$ is certainly binary, then
   • it could be that $p \vee q \vee r$ is an abbreviation for a specific formula, say $( p \vee q ) \vee r$, and then take the parse tree of this un-abbreviated version; or
   • as $p \vee ( q \vee r )$ and $( p \vee q ) \vee r$ are easily seen to be logically equivalent, if no specific formula have been designated the un-abbreviation of $p \vee q \vee r$, then you can just choose one of these, and parse it accordingly.