Definition:Parenthesis

Symbolic Logic
Parenthesis is the operation of encompassing a compound substatement of a given compound statement such that it is treated as a single entity. Such a statement is referred to as being in parenthesis.

Brackets are used to identify the substatements of a compound statement that are in parenthesis.

The brackets that are usually used are round ones:


 * $\left({\textrm {This} \ \textrm {statement} \ \textrm {is} \ \textrm {in} \ \textrm {parenthesis}}\right)$

Differently shaped brackets are used in different contexts. In the context of a (logical) statement, they are sometimes referred to as parentheses (the plural of the word parenthesis, which is of Greek origin).

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: "$[\;]$".

An obvious definition: "$($" is a left bracket, while "$)$" is a right bracket.

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.

Parenthesis
For example, this compound statement:


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

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 compound statement, to determine which of these interpretations is the one intended.

One way of doing this is to group substatements together so they are treated as one entity. We do this by using the concept of parenthesis, in which brackets are used to identify the substatements of a compound statement that are to be treated as one.

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


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


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

Binding Priorities
However, complicated statements may require a large number of parentheses to write. This can make it difficult to read, whatever style is used to write it. In order to limit the number of brackets that are used, some connectives are usually understood as binding more strongly to its substatements than others.

The binding priority, or precedence, is the convention defining the order of binding strength of the individual connectives.