Definition:Logical Connective

From ProofWiki
Jump to navigation Jump to search


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 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 (or two-place connective) is a logical 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:

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:

Also see

  • Results about logical connectives can be found here.