Rule of Material Equivalence/Formulation 2
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Theorem
- $\vdash \paren {p \iff q} \iff \paren {\paren {p \implies q} \land \paren {q \implies p} }$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \iff q$ | Assumption | (None) | ||
| 2 | 1 | $\paren {\paren {p \implies q} \land \paren {q \implies p} }$ | Sequent Introduction | 1 | Rule of Material Equivalence: Formulation 1 | |
| 3 | $\paren {p \iff q} \implies \paren {\paren {p \implies q} \land \paren {q \implies p} }$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
| 4 | 4 | $\paren {p \implies q} \land \paren {q \implies p}$ | Assumption | (None) | ||
| 5 | 4 | $p \iff q$ | Sequent Introduction | 4 | Rule of Material Equivalence: Formulation 1 | |
| 6 | $\paren {\paren {p \implies q} \land \paren {q \implies p} } \implies \paren {p \iff q}$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
| 7 | $\paren {p \iff q} \iff \paren {\paren {p \implies q} \land \paren {q \implies p} }$ | Biconditional Introduction: $\iff \II$ | 3, 6 |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective are true for all boolean interpretations.
$\begin{array}{|ccc|c|ccccccc|} \hline (p & \iff & q) & \iff & (p & \implies & q) & \land & (q & \implies & p) \\ \hline \F & \T & \F & \T & \F & \T & \F & \T & \F & \T & \F \\ \F & \F & \T & \T & \F & \T & \T & \F & \T & \F & \F \\ \T & \F & \F & \T & \T & \F & \F & \F & \F & \T & \T \\ \T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T81}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $17.$
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2):$: The remaining rules of inference: $17$