Disjunction introduction or Addition is a valid

Validity

In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

, simple argument form in logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

:

A.

Therefore, A or B.

or in logical operator (sequent

Sequent

In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system.-...

) notation:

The argument

Argument

In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

 form has one premise, A, and an unrelated proposition, B. From the premise it can be logically concluded that either A or B is true, or both are true.

Here is an example of such an argument:

Socrates

Socrates

Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...

 is a man.

Therefore (either or both of) Socrates

Socrates

Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...

 is a man, or pigs are flying in formation over the English Channel

English Channel

The English Channel , often referred to simply as the Channel, is an arm of the Atlantic Ocean that separates southern England from northern France, and joins the North Sea to the Atlantic. It is about long and varies in width from at its widest to in the Strait of Dover...

.

Disjunction introduction is controversial in paraconsistent logic

Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

 because in combination with other rules of logic, it leads to explosion

Principle of explosion

The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...

 (i.e. everything becomes provable). See Tradeoffs in Paraconsistent logic.