# Definition:Logical Connective

## Contents

## Definition

A **logical connective** is an object which either modifies a statement, or combines existing statements into a new statement, called a compound statement.

It is almost universal to identify a **logical connective** with the symbol representing it.

Thus, **logical connective** may also, particularly in symbolic logic, be used to refer to that symbol, rather than speaking of a **connective symbol** separately.

In mathematics, **logical connectives** are considered to be **truth-functional**.

That is, the truth value of a compound statement formed using the **connective** is assumed to depend *only* on the truth value of the comprising statements.

Thus, as far as the **connective** is concerned, it does not matter what the comprising statements precisely *are*.

As a consequence of this truth-functionality, a **connective** has a corresponding truth function, which goes by the same name as the **connective** itself.

The arity of this truth function is the number of statements the **logical connective** combines into a single compound statement.

### Unary Logical Connective

A **unary logical connective** (or **one-place connective**) is a connective whose effect on its compound statement is determined by the truth value of *one* substatement.

In standard Aristotelian logic, there are four of these.

The only non-trivial one is logical not, as shown on Unary Truth Functions.

### Binary Logical Connective

A **binary logical connective** (or **two-place connective**) is a connective whose effect on its compound statement is determined by the truth value of *two* substatements.

In standard Aristotelian logic, there are 16 **binary logical connectives**, cf. Binary Truth Functions.

In the field of symbolic logic, the following four (symbols for) **binary logical connectives** are commonly used:

- Conjunction: the
**And**connective $p \land q$:**$p$ is true**.*and*$q$ is true

- Disjunction: the
**Or**connective $p \lor q$:**$p$ is true**.*or*$q$ is true,*or possibly both*

- The conditional connective $p \implies q$:
.*If*$p$ is true,*then*$q$ is true

- The biconditional connective $p \iff q$:
**$p$ is true**, or*if and only if*$q$ is true**$p$**.*is equivalent to*$q$

## Also defined as

Some sources reserve the term **logical connective** for what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is defined as a binary logical connective, on the grounds that a unary logical connective does not actually "connect" anything. However, this is a trivial distinction which can serve only to confuse.

## Also known as

Other terms for **logical connective** which may be encountered include:

**Connective****Truth-functional connective****Propositional connective****Sentential connective****Logical constant****Logical operator****Sentence-forming operator****Boolean operator**(in the context of mathematical logic)**Conjunction**(as used in natural language - mathematics has a more specialised use for the term conjunction, however)

## Also see

## Sources

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 2.3$: Basic Truth-Tables of the Propositional Calculus - 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): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 1$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 1.2$: Conditionals and Negation - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Connectives - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 2$: Logical Constants $(1)$ - 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $1$: Introduction: $\S 1.2$: Propositional and predicate calculus - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.1$: Introduction - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if... - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**truth function**