Rule of Conjunction/Sequent Form/Formulation 1

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Theorem

The Rule of Conjunction can be symbolised by the sequent:

$p, q \vdash p \land q$


Proof 1

By the tableau method of natural deduction:

$p, q \vdash p \land q$
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: $\land \mathcal I$ 1, 2

$\blacksquare$


Proof 2

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