Rule of Idempotence/Conjunction/Formulation 2
Theorem
The conjunction operator is idempotent:
- $\vdash p \iff \paren {p \land p}$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p$ | Assumption | (None) | ||
| 2 | 1 | $p \land p$ | Rule of Conjunction: $\land \II$ | 1, 1 | ||
| 3 | $p \implies \paren {p \land p}$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
| 4 | 4 | $p \land p$ | Assumption | (None) | ||
| 5 | 4 | $p$ | Rule of Simplification: $\land \EE_1$ | 4 | ||
| 6 | $\paren {p \land p} \implies p$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
| 7 | $p \iff \paren {p \land p}$ | Biconditional Introduction: $\iff \II$ | 3, 6 |
$\blacksquare$
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
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T41}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $19.$
- 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: $18$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(i)}$