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 Town are going up this season or I'm a Dutchman."

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

Technical Note
When invoking the Rule of Simplification 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.