Rule of Conjunction/Sequent Form/Formulation 1
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Theorem
The Rule of Conjunction can be symbolised by the sequent:
| \(\ds p\) | \(\) | \(\ds \) | ||||||||||||
| \(\ds q\) | \(\) | \(\ds \) | ||||||||||||
| \(\ds \vdash \ \ \) | \(\ds p \land q\) | \(\) | \(\ds \) |
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p$ | Premise | (None) | ||
| 2 | 2 | $q$ | Premise | (None) | ||
| 3 | 1, 2 | $p \land q$ | Rule of Conjunction: $\land \II$ | 1, 2 |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
$\begin{array}{|c|c||ccc|} \hline p & q & p & \land & q\\ \hline \F & \F & \F & \F & \F \\ \F & \T & \F & \F & \T \\ \T & \F & \T & \F & \F \\ \T & \T & \T & \T & \T \\ \hline \end{array}$
As can be seen, only when both $p$ and $q$ are true, then so is $p \land q$.
$\blacksquare$
Sources
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables: Exercise $\text{I} \ \mathbf d.$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $8.$