Rule of Addition/Sequent Form
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Theorem
The Rule of Addition can be symbolised by the sequents:
| \(\text {(1)}: \quad\) | \(\ds p\) | \(\) | \(\ds \) | |||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \lor q\) | \(\) | \(\ds \) |
| \(\text {(2)}: \quad\) | \(\ds q\) | \(\) | \(\ds \) | |||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \lor q\) | \(\) | \(\ds \) |
Formulation 1
| \(\text {(1)}: \quad\) | \(\ds p\) | \(\) | \(\ds \) | |||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \lor q\) | \(\) | \(\ds \) |
| \(\text {(2)}: \quad\) | \(\ds q\) | \(\) | \(\ds \) | |||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \lor q\) | \(\) | \(\ds \) |
Formulation 2
| \(\text {(1)}: \quad\) | \(\ds \vdash p\) | \(\implies\) | \(\ds \paren {p \lor q}\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \vdash q\) | \(\implies\) | \(\ds \paren {p \lor q}\) |
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 |
$\blacksquare$
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 |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
$\begin{array}{|c|c||ccc|} \hline p & q & p & \lor & q\\ \hline \F & \F & \F & \F & \F \\ \F & \T & \F & \T & \T \\ \T & \F & \T & \T & \F \\ \T & \T & \T & \T & \T \\ \hline \end{array}$
As can be seen, whenever either $p$ or $q$ (or both) are true, then so is $p \lor q$.
$\blacksquare$