# Parenthesis/Examples/Arbitrary Example 2

## Examples of Parenthesis

Consider the following this formula of propositional logic:

$p \lor q \implies \neg \, r \implies p \land q$

This can be interpreted in several different ways:

If either $p$ or $q$ is true, then it is not the case that the truth of $r$ implies the truth of both $p$ and $q$.
Either $p$ is true, or if $q$ is true, then it is not the case that the truth of $r$ implies the truth of both $p$ and $q$.
and so on.

So we need a way, for such a formula, to determine which of these interpretations is the one intended.

In the example above, the two different interpretations will be written in the style we have chosen as:

$\paren {p \lor q} \implies \paren {\neg \paren {r \implies \paren {p \land q} } }$
$p \lor \paren {q \implies \paren {\neg \paren {r \implies \paren {p \land q} } } }$

In these expressions, $\paren {p \lor q}$ and $\paren {\neg \paren {r \implies \paren {p \land q} } }$ are examples of formulas in parenthesis.

Note that while the latter expressions may in fact be WFFs of propositional logic, the ambiguous expression they were derived from is not.