Axiom:Rule of Addition
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Context
The rule of addition is one of the axioms of natural deduction.
The rule
This is two axioms in one.
- If we can conclude $p$, then we may infer $p \lor q$: $p \vdash p \lor q$
- 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.
It can be written:
- $\displaystyle {p \over p \lor q} \lor_{i_1} \qquad \qquad {q \over p \lor q} \lor_{i_2}$
- 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."
Demonstration by Truth Table
$\begin{array}{|c|c||ccc|} \hline p & q & p & \lor & q\\ \hline F & F & F & F & F \\ F & T & F & T & T \\ T & F & T & T & F \\ T & T & T & T & T \\ \hline \end{array}$
As can be seen, whenever either $p$ or $q$ (or both) are true, then so is $p \lor q$.
$\blacksquare$
Sources
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 3$
- E.J. Lemmon: Beginning Logic (1965): $\S 1.3$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1$
