Conjunction introduction

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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 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 \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.


  1. Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. pp. 346–51.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  2. Copi and Cohen
  3. Moore and Parker