Rule of Commutation/Disjunction/Formulation 2
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Theorem
- $\vdash \paren {p \lor q} \iff \paren {q \lor p}$
Forward Implication
- $\vdash \left({p \lor q}\right) \implies \left({q \lor p}\right)$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \lor q$ | Assumption | (None) | ||
| 2 | 1 | $q \lor p$ | Sequent Introduction | 1 | Disjunction is Commutative | |
| 3 | $\paren {p \lor q} \implies \paren {q \lor p}$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
| 4 | 4 | $q \lor p$ | Assumption | (None) | ||
| 5 | 4 | $p \lor q$ | Sequent Introduction | 4 | Disjunction is Commutative | |
| 6 | $\paren {q \lor p} \implies \paren {p \lor q}$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
| 7 | $\paren {p \lor q} \iff \paren {q \lor p}$ | 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 match for all boolean interpretations.
$\begin{array}{|ccc|c|ccc|} \hline (p & \lor & q) & \iff & (q & \lor & p) \\ \hline \F & \F & \F & \T & \F & \F & \F \\ \F & \T & \T & \T & \T & \T & \F \\ \T & \T & \F & \T & \F & \T & \T \\ \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 9$
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.7$: The Derivation of Formulae: $D \, 14$
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T53}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $11.$
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $10$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(iv)}$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.14$: Exercise $12 \ (5)$