# 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** $)$.

## 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

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

- 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-**), while

*en*-the-sis**parentheses**is pronounced

**par-**(or

*en*-te-sees**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

- 1910: Alfred North Whitehead and Bertrand Russell:
*Principia Mathematica: Volume $\text { 1 }$*... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations - 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 3.2$: Logical Punctuation and the Scope of Constants - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 2$: The Axiom of Specification - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 1$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation: Implicit: "(we introduce brackets here in an entirely obvious way)". - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.1$: Simple and Compound Statements - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 5$: Using Brackets - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.2$: Propositional formulas - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**aggregation** - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.1$: Declarative sentences - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**aggregation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**infix notation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**brackets**