Rule of Association/Conjunction/Formulation 2
Theorem
- $\vdash \paren {p \land \paren {q \land r} } \iff \paren {\paren {p \land q} \land r}$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \land \paren {q \land r}$ | Assumption | (None) | ||
| 2 | 1 | $\paren {p \land q} \land r$ | Sequent Introduction | 1 | Rule of Association: Formulation 1 | |
| 3 | $\paren {p \land \paren {q \land r} } \implies \paren {\paren {p \land q} \land r}$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
| 4 | 4 | $\paren {p \land q} \land r$ | Assumption | (None) | ||
| 5 | 4 | $p \land \paren {q \land r}$ | Sequent Introduction | 4 | Rule of Association: Formulation 1 | |
| 6 | $\paren {\paren {p \land q} \land r} \implies \paren {p \land \paren {q \land r} }$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
| 7 | $\paren {p \land \paren {q \land r} } \iff \paren {\paren {p \land q} \land r}$ | 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
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 10$
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.7$: The Derivation of Formulae: $D \, 27$
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T25}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $12.$
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $11$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(v)}$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.14$: Exercise $12 \ (2)$