# Simplification

In propositional logic, simplification[1][2][3] (equivalent to conjunction elimination) 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 can be expressed in formal language as:

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

or as

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

where the rule is that whenever instances of "$P \land Q$" appear on lines of a proof, either "$P$" or "$Q$" can be placed on a subsequent line by itself.

## Formal notation

The simplification rule may be written in sequent notation:

$(P \land Q) \vdash P$

or as

$(P \land Q) \vdash Q$

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 a truth-functional tautology or theorem of propositional logic:

$(P \land Q) \to P$

and

$(P \land Q) \to Q$

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

## References

1. ^ Copi and Cohen
2. ^ Moore and Parker
3. ^ Hurley