# Rule of Conjunction

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## Sequent

The **rule of conjunction** is a valid argument in types of logic dealing with conjunctions $\land$.

This includes propositional logic and predicate logic, and in particular natural deduction.

### Proof Rule

- If we can conclude both $\phi$ and $\psi$, we may infer the compound statement $\phi \land \psi$.

### Sequent Form

The Rule of Conjunction can be symbolised in sequent form as follows:

\(\ds p\) | \(\) | \(\ds \) | ||||||||||||

\(\ds q\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds p \land q\) | \(\) | \(\ds \) |

## Explanation

The Rule of Conjunction can be expressed in natural language as:

- If we can show that two statements are true, then we may build a compound statement expressing this fact, and be certain that this is also true.

Thus a conjunction is added to a sequent.

## Also known as

The Rule of Conjunction can also be referred to as:

- the
**rule of and-introduction** - the
**rule of adjunction**.