# Rule of Addition

## Sequent

The **rule of addition** is a valid argument in types of logic dealing with disjunctions $\lor$.

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

### Proof Rule

- $(1): \quad$ If we can conclude $\phi$, then we may infer $\phi \lor \psi$.
- $(2): \quad$ If we can conclude $\psi$, then we may infer $\phi \lor \psi$.

### Sequent Form

The Rule of Addition can be symbolised by the sequents:

\(\text {(1)}: \quad\) | \(\ds p\) | \(\) | \(\ds \) | |||||||||||

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

\(\text {(2)}: \quad\) | \(\ds q\) | \(\) | \(\ds \) | |||||||||||

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

## Explanation

The Rule of Addition consists of two proof rules in one.

The first of the two can be expressed in natural language as:

- Given a statement, we may infer a disjunction where the given statement is the
**first**of the disjuncts.

The second of the two can be expressed in natural language as:

- Given a statement, we may infer a disjunction where the given statement is the
**second**of the disjuncts.

The statement being added may be any statement at all.

It does not matter what its truth value is.

That is: if $p$ is true, then $p \lor q$ is likewise true, **whatever** $q$ may be.

This may seem a bewildering and perhaps paradoxical axiom to admit. How can you deduce a valid argument from a statement form that can deliberately be used to include a statement whose truth value can be completely arbitrary? Or even blatantly false?

But consider the common (although admittedly rhetorical) figure of speech which goes:

*Reading Football Club are going up this season or I'm a monkey's uncle.*

## Also known as

The Rule of Addition is sometimes known as the rule of **or-introduction**.

Some sources give is as the **law of simplification for logical addition**.

Such treatments may also refer to the Rule of Simplification as the **law of simplification for logical multiplication**.

This extra level of wordage has not been adopted by $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is argued that it may cause clarity to suffer.