Rule of Addition/Explanation

Proof Rule

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.