Disjunction introduction

Disjunction introduction or addition (also called or introduction) 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.