Conjunction introduction

Conjunction introduction (often abbreviated simply as conjunction) is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. It is the inference that if the proposition p is true, and proposition q is true, then the logical conjunction of the two propositions p and q is true. For example, if it's true that it's raining, and it's true that I'm inside, then it's true that "it's raining and I'm inside". The rule can be stated:

${\frac {P,Q}{\therefore P\land Q}}$

where the rule is that wherever an instance of "$P$" and "$Q$" appear on lines of a proof, a "$P\land Q$" can be placed on a subsequent line.

Formal notation

The conjunction introduction rule may be written in sequent notation:

$P,Q\vdash P\land Q$

where $\vdash$ is a metalogical symbol meaning that $P\land Q$ is a syntactic consequence if $P$ and $Q$ are each on lines of a proof in some logical system;

or as the statement of a truth-functional tautology or theorem of propositional logic:

$P\land Q\leftrightarrow (P\land Q)$

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

Proof

Proposition	Derivation
$P\,\!$	Given
$Q\,\!$	Given
$\neg P\lor Q$	Addition
$P\rightarrow Q$	Material implication
$P\rightarrow (P\land Q)$	Absorption
$P\land Q$	Modus ponens