Definition:Parenthesis

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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 $)$.


Parenthesis in Natural Language

When parenthesis is needed in natural language, it is usual to employ a number of different techniques.

It is often the case that ambiguity is avoided by taking care with the word order.


Also denoted as

There is no universal convention as to exactly what shaped brackets are used for parentheses, but (usually) round brackets $\paren \;$ are used.

A notable counterexample is the elegantly-presented 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability, which uses square ones: $\sqbrk \;$.

Some sources use the vinculum, that is, a line over the aggregated terms:

$3 \times \overline {6 - 4}$


Some authors, when writing complicated statements with nested parentheses, use differently shaped brackets, either square brackets $\sqbrk \;$ or braces $\set \;$ for each different parentheses, 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ round brackets are used throughout for parenthesis.


Also known as

A parenthesis is also known as an aggregation, but this is not recommended as there are already a number of other interpretations of that word.


Examples

Example $1$

Consider the following this formula of propositional logic:

$p \land q \lor r$

This can mean either:

the conjunction of $p$ with $q \lor r$

or:

the disjunction of $p \land q$ with $r$.

Using parenthesis, the ambiguity is removed by presenting what is required either as:

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

or:

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


Example $2$

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.


Example $3$

In the expression:

$3 \paren {6 - 4}$

the $6 - 4$ part is in parenthesis, or aggregated, before being multiplied by $3$.


Also see

  • Results about parenthesis can be found here.


Historical Note

Round brackets $\paren \;$ first appeared in $1544$.

Square brackets $\sqbrk \;$ and braces $\set \;$ were used by François Viète in around $1593$.

1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica idiosyncratically use dots "$.$" for no immediate discernible benefit. The rules governing their use are complex and clumsy.


Linguistic Note

The plural of parenthesis is parentheses.

The correct pronunciation of parenthesis is par-en-te-sis (or par-en-the-sis), while parentheses is pronounced par-en-te-sees (or par-en-the-sees).


It also needs to be pointed out that US English uses the term parentheses to mean the brackets $\paren \ldots$ themselves (specifically the round ones), rather than their content.

The word brackets is generally reserved for square $\sqbrk \ldots$ and curly $\set \ldots$ versions (although the technical term for the latter is braces).

While this is common in natural language, such usage is discouraged in $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is more useful to have a word which can be specifically used to unambiguously refer to the content.

It is also worth pointing out that use of parentheses to mean the brackets is considered by much of the rest of the world as ignorant.


Sources

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