Definition:Parenthesis

Definition
Parenthesis is a syntactical technique to disambiguate the meaning of a statement form.

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

Such a statement form 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 statement forms are in parenthesis.

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

Often, these brackets are referred to as parentheses, identifying them with their meaning.

Example
For example, this statement form 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 statement form, 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:


 * $\left({p \lor q}\right) \implies \left({\neg \left({r \implies \left({p \land q}\right)}\right)}\right)$


 * $p \lor \left({q \implies \left({\neg \left({r \implies \left({p \land q}\right)}\right)}\right)}\right)$

In these expressions, $\left({p \lor q}\right)$ and $\left({\neg \left({r \implies \left({p \land q}\right)}\right)}\right)$ are exemplars of statement forms in parenthesis.

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

Also defined as
There is no universal convention as to exactly what shaped brackets are used for parentheses, but (usually) round brackets "$\left({\;}\right)$" are used. The elegantly-presented Keisler and Robbin for example, uses square ones: "$[\;]$".

Some authors, when writing complicated statements with nested parentheses, use differently shaped brackets for each different parenthesis, in an attempt to make it clearer which brackets go with which substatements. However, some have the opinion that this does not actually aid comprehension and can add unnecessary confusion - especially when particular bracket styles are being used for particular mathematical tasks, as they frequently are.

It also happens, unfortunately, that square brackets do not render well in all browsers when they have been automatically scaled by our rendering software.

Therefore it is recommended that on ProofWiki round brackets are used throughout for parenthesis.

Also see

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

Linguistic Note
The plural of parenthesis is parentheses.

The correct pronunciation of parenthesis is pa-ren-ti-sis, while parentheses is pronounced pa-ren-ti-sees.