# Disjunction introduction

Disjunction introduction or addition (also called or introduction)[1][2][3] is a simple valid argument form, an immediate inference and a rule of inference of propositional logic. The rule makes it possible to introduce disjunctions to logical proofs. It is the inference that if P is true, then P or Q must be true.

Socrates is a man.
Therefore, either Socrates is a man or pigs are flying in formation over the English Channel.

The rule can be expressed as:

$\frac{P}{\therefore P \or Q}$

where the rule is that whenever instances of "$P$" appear on lines of a proof, "$P \or Q$" can be placed on a subsequent line.

Disjunction introduction is controversial in paraconsistent logic because in combination with other rules of logic, it leads to explosion (i.e. everything becomes provable). See Tradeoffs in Paraconsistent logic.

## Formal notation

The disjunction introduction rule may be written in sequent notation:

$P \vdash (P \or Q)$

where $\vdash$ is a metalogical symbol meaning that $P \or Q$ is a syntactic consequence of $P$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

$P \to (P \or Q)$

where $P$ and $Q$ are propositions expressed in some formal system.

## References

1. ^ Hurley[full citation needed]
2. ^ Moore and Parker[full citation needed]
3. ^ Copi and Cohen[full citation needed]