Conjunction introduction

Conjunction introduction (often abbreviated simply as conjunction and also called and introduction^[1][2][3]) 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 coming, and it's true that I'm inside, then it's true that "it's coming and I'm inside". The rule can be stated:

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

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

Formal notation

The conjunction introduction rule may be written in sequent notation:

$P, Q \vdash P \and Q$

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

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

References

1. Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. pp. 346–51.
2. Copi and Cohen
3. Moore and Parker