Rule of Addition

Proof Rule
The rule of addition is a valid deduction sequent in propositional logic:

This is two proof rules in one:
 * $(1): \quad$ If we can conclude $p$, then we may infer $p \lor q$.
 * $(2): \quad$ If we can conclude $q$, then we may infer $p \lor q$.

It can be written:
 * $\displaystyle {p \over p \lor q} \lor_{i_1} \qquad \qquad {q \over p \lor q} \lor_{i_2}$

Explanation
Note that there are two axioms here in one. The first of the two tells us that, given a statement, we may infer a disjunction where the given statement is the first of the disjuncts, while the second says that, given a statement, we may infer a disjunction where the given statement is the second of the disjuncts.

At this stage, such attention to detail is important.

The statement $q$ being added may be any statement at all. It does not matter what its truth value is. If $p$ is true, then $p \vdash p \lor q$ is 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 Dutchman.

Also known as
This is sometimes known as the rule of or-introduction.

Some sources give this 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, as it is argued that it may cause clarity to suffer.

Also see

 * Rule of Or-Elimination

Technical Note
When invoking the Rule of Addition in a tableau proof, use the Addition template:



or:

where:
 * is the number of the line on the tableau proof where the assumption is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof upon which this line directly depends
 * should hold 1 for Addition_1, and 2 for Addition_2
 * is the (optional) comment that is to be displayed in the Notes column.