Biconditional Introduction/Sequent Form

Theorem

Biconditional Introduction can be symbolised by the sequent:

$p \implies q, q \implies p \vdash p \iff q$


Proof 1

By the tableau method of natural deduction:

$p \implies q, q \implies p \vdash p \iff q$
Line Pool Formula Rule Depends upon Notes
1 1 $p \implies q$ Premise (None)
2 2 $q \implies p$ Premise (None)
3 1, 2 $p \iff q$ Biconditional Introduction: $\iff \II$ 1, 2

$\blacksquare$


Proof 2

We apply the Method of Truth Tables.

$\begin{array}{|ccccccc||ccc|} \hline (p & \implies & q) & \land & (q & \implies & p) & p & \iff & q\\ \hline F & T & F & T & F & T & F & F & T & F \\ F & T & T & F & T & F & F & F & F & T \\ T & F & F & F & F & T & T & T & F & F \\ T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$

As can be seen, only when both $p \implies q$ and $q \implies p$ are true, then so is $p \iff q$.

$\blacksquare$


Sources

  • 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction: Exercises $1.6: \ 7$
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