Rule of Simplification/Sequent Form/Formulation 2

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Theorem

The Rule of Simplification can be symbolised by the sequents:

$(1): \quad \vdash p \land q \implies p$
$(2): \quad \vdash p \land q \implies q$


Form 1

$\vdash p \land q \implies p$

Form 2

$\vdash p \land q \implies q$


Proof 1

Form 1

By the tableau method of natural deduction:

$\vdash p \land q \implies p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \land q$ Assumption (None)
2 1 $p$ Rule of Simplification: $\land \mathcal E_1$ 1
3 $p \land q \implies p$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$


Form 2

By the tableau method of natural deduction:

$\vdash p \land q \implies q$
Line Pool Formula Rule Depends upon Notes
1 1 $p \land q$ Assumption (None)
2 1 $q$ Rule of Simplification: $\land \mathcal E_2$ 1
3 $p \land q \implies q$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$


Proof 2

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective are $T$ for all boolean interpretations.

$\begin{array}{|ccc|c|c||c|c|} \hline p & \land & q & p & q & p \land q \implies p & p \land q \implies q \\ \hline F & F & F & F & F & T & T \\ F & F & T & F & T & T & T \\ T & F & F & T & F & T & T \\ T & T & T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$