Rule of Simplification/Sequent Form/Formulation 2

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: $\land \EE_1$	1
3			$p \land q \implies p$	Rule of Implication: $\implies \II$	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: $\land \EE_2$	1
3			$p \land q \implies q$	Rule of Implication: $\implies \II$	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.

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

$\blacksquare$