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.