Conjunction elimination

Conjunction elimination

Type	Rule of inference
Field	Propositional calculus
Statement	If the conjunction $A$ and $B$ is true, then $A$ is true, and $B$ is true.
Symbolic statement	$\frac{P\wedge Q}{\therefore P},\frac{P\wedge Q}{\therefore Q}$ $(P\wedge Q)⊢P,(P\wedge Q)⊢Q$ $(P\wedge Q)\to P,(P\wedge Q)\to Q$

In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) 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:

$\frac{P\wedge Q}{\therefore P}$

and

$\frac{P\wedge Q}{\therefore Q}$

The two sub-rules together mean that, whenever an instance of "$P\wedge Q$" 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:

$(P\wedge Q)⊢P$

and

$(P\wedge Q)⊢Q$

where $⊢$ is a metalogical symbol meaning that $P$ is a syntactic consequence of $P\wedge Q$ and $Q$ is also a syntactic consequence of $P\wedge Q$ in logical system;

and expressed as truth-functional tautologies or theorems of propositional logic:

$(P\wedge Q)\to P$

and

$(P\wedge Q)\to Q$

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