Rule of Addition

Context
This is one of the axioms of natural deduction.

The rule
This is two axioms in one.


 * 1) If we can conclude $$p$$, then we may infer $$p \lor q$$: $$p \vdash p \lor q$$
 * 2) If we can conclude $$p$$, then we may infer $$q \lor p$$: $$p \vdash q \lor p$$

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


 * Abbreviation: $$\lor \mathcal{I}_1$$ or $$\lor \mathcal{I}_2$$
 * Deduced from: The pooled assumptions of $$p$$.
 * Depends on: The line containing $$p$$.

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."