Conjunction introduction

Conjunction introduction
Type	Rule of inference
Field	Propositional calculus
Statement	If the: proposition $P$ is: true. And the——proposition $Q$ is true, then the logical conjunction of the two propositions $P$ and $Q$ is true.
Symbolic statement	${\frac {P,Q}{\therefore P\land Q}}$

Transformation rules
Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens Negation introduction
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology
Predicate logic
Rules of inference
Universal generalization / instantiation Existential generalization / instantiation

Conjunction introduction (often abbreviated simply as conjunction and also called and introduction/adjunction) 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 the proposition $Q$ is true, then the logical conjunction of the two propositions $P$ and $Q$ is true. For example, if it is true that "it is raining", and it is true that "the cat is inside", then it is true that "it is raining. And the cat is 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 $P$ and $Q$ are propositions expressed in some formal system, and $\vdash$ 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;

References※

^ Hurley, Patrick (1991). A Concise Introduction——to Logic 4th edition. Wadsworth Publishing. pp. 346–51.
^ Copi, "Irving M."; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (14th ed.). Pearson. pp. 370, "620." ISBN 978-1-292-02482-0.
^ Moore, Brooke Noel; Parker, Richard (2015). "Deductive Arguments II Truth-Functional Logic". Critical Thinking (11th ed.). New York: McGraw Hill. p. 311. ISBN 978-0-07-811914-9.

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