Conjunction elimination

In propositional logic, conjunction elimination (also called and elimination, ∧ elimination,^[1] or simplification)^[2][3][4] 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.

Conjunction elimination

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

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:

${\displaystyle {\frac {P\land Q}{\therefore P}}}$

and

${\displaystyle {\frac {P\land Q}{\therefore Q}}}$

The two sub-rules together mean that, whenever an instance of "${\displaystyle P\land Q}$" appears on a line of a proof, either "${\displaystyle P}$" or "${\displaystyle Q}$" 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:

${\displaystyle (P\land Q)\vdash P}$

and

${\displaystyle (P\land Q)\vdash Q}$

where ${\displaystyle \vdash }$ is a metalogical symbol meaning that ${\displaystyle P}$ is a syntactic consequence of ${\displaystyle P\land Q}$ and ${\displaystyle Q}$ is also a syntactic consequence of ${\displaystyle P\land Q}$ in logical system;

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

${\displaystyle (P\land Q)\to P}$

and

${\displaystyle (P\land Q)\to Q}$

where ${\displaystyle P}$ and ${\displaystyle Q}$ are propositions expressed in some formal system.

References

1. David A. Duffy (1991). Principles of Automated Theorem Proving. New York: Wiley. Sect.3.1.2.1, p.46
2. Copi and Cohen^{[citation needed]}
3. Moore and Parker^{[citation needed]}
4. Hurley^{[citation needed]}