Logical conjunction

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In logic and mathematics, logical conjunction, or: and is a two-place logical connective that has the value true if both of its operands are true, otherwise a value of false.

The analogue of conjunction for a (possibly infinite) family of statements is universal quantification, which is part of predicate logic.

Contents

Notation

And is usually expressed with an infix operator. In mathematics and logic, it is usually ; in electronics $\cdot$; and in programming languages, & or and. Some programming languages have a related control structure, the short-circuit and, written &&, and then, etc.

Definition

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

Truth table

The truth table of $~A \and B$:

Introduction and elimination rules

As a rule of inference, conjunction introduction is a classically valid, simple argument form. The argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction.

or in logical operator notation:

Here is an example of an argument that fits the form conjunction introduction:

Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.

...or alternately,

In logical operator notation: