Rule of Addition/Sequent Form/Formulation 2
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Theorem
The Rule of Addition can be symbolised by the sequents:
| \(\text {(1)}: \quad\) | \(\ds \vdash p\) | \(\implies\) | \(\ds \paren {p \lor q}\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \vdash q\) | \(\implies\) | \(\ds \paren {p \lor q}\) |
Form 1
- $\vdash p \implies \paren {p \lor q}$
Form 2
- $\vdash q \implies \left({p \lor q}\right)$
Proof 1
Form 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p$ | Premise | (None) | ||
| 2 | 1 | $p \lor q$ | Rule of Addition: $\lor \II_1$ | 1 | ||
| 3 | $p \implies \paren {p \lor q}$ | Rule of Implication: $\implies \II$ | 1 – 3 | Assumption 1 has been discharged |
$\blacksquare$
Form 2
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $q$ | Premise | (None) | ||
| 2 | 1 | $p \lor q$ | Rule of Addition: $\lor \II_2$ | 1 | ||
| 3 | $q \implies \paren {p \lor q}$ | Rule of Implication: $\implies \II$ | 1 – 3 | Assumption 1 has been discharged |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives are $T$ for all boolean interpretations.
$\begin{array}{|c|c|ccccc|ccccc|} \hline p & q & p & \implies & (p & \lor & q) & q & \implies & (p & \lor & q) \\ \hline \F & \F & \F & \T & \F & \F & \F & \F & \T & \F & \F & \F \\ \F & \T & \F & \T & \F & \T & \T & \T & \T & \F & \T & \T \\ \T & \F & \T & \T & \T & \T & \F & \F & \T & \T & \T & \F \\ \T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$