Definition:Parenthesis

Definition
Parenthesis is a syntactical technique to disambiguate the meaning of a logical formula.

It allows one to specify that a logical formula should (temporarily) be regarded as being a single entity, being on the same level as a statement variable.

Such a formula is referred to as being in parenthesis.

Typically, a formal language, in defining its formal grammar, ensures by means of parenthesis that all of its well-formed words are uniquely readable.

Generally, brackets are used to indicate that certain formulas are in parenthesis.

The brackets that are mostly used are round ones, the left (round) bracket $($ and the right (round) bracket $)$.

Example
For example, this formula of propositional logic:


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

could 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 are in fact WFFs of propositional logic, the ambiguous expression they were derived from is not.

Also see

 * Definition:Binding Priority, a technique to reduce the amount of parenthesis