Definition:Logical Connective

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 logical connective is assumed to depend only on the truth value of the comprising statements.

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

As a consequence of this truth-functionality, a logical connective has a corresponding truth function, which goes by the same name as the logical 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 is a logical 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 is a logical connective whose effect on its compound statement is determined by the truth value of two substatements.

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

• Definition:Truth Function

• Definition:Connective of Propositional Logic

• Results about logical connectives can be found here.

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): Chapter $1$: The Propositional Calculus $1$: $2$ 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): connective
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): constant: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): truth function
• 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): connective
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): constant: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): truth function