In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, Sect.3.1.2.1, p.46 or simplification)Copi and CohenMoore and ParkerHurley is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction A and B is true, then A is true, and B is true.
The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself.
An example in English:
It's raining and it's pouring.
Therefore it's raining.
The rule consists of two separate sub-rules, which can be expressed in formal language as:
\frac{P \land Q}{\therefore P}
and
\frac{P \land Q}{\therefore Q}
The two sub-rules together mean that, whenever an instance of "P \land Q" appears on a line of a proof, either "P" or "Q" can be placed on a subsequent line by itself.
The above example in English is an application of the first sub-rule.
Formal notation
The conjunction elimination sub-rules may be written in sequent notation:
(P \land Q) \vdash P
and
(P \land Q) \vdash Q
where \vdash is a metalogical symbol meaning that P is a syntactic consequence of P \land Q and Q is also a syntactic consequence of P \land Q in logical system;
and expressed as truth-functional tautologies or theorems of propositional logic:
(P \land Q) \to P
and
(P \land Q) \to Q
where P and Q are propositions expressed in some formal system.
References
Category:Rules of inference Category:Theorems in propositional logic
sv:Matematiskt uttryck#Förenkling
