Rule of Exportation/Formulation 2
Theorem
- $\vdash \paren {\paren {p \land q} \implies r} \iff \paren {p \implies \paren {q \implies r} }$
This can be expressed as two separate theorems:
Forward Implication
- $\vdash \left({\left ({p \land q}\right) \implies r}\right) \implies \left({p \implies \left ({q \implies r}\right)}\right)$
Reverse Implication
- $\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \land q} \implies r}$
Proof
Proof of Forward Implication
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\paren {p \land q} \implies r$ | Assumption | (None) | ||
| 2 | 1 | $p \implies \paren {q \implies r}$ | Sequent Introduction | 1 | Rule of Exportation: Forward Implication: Formulation 1 | |
| 3 | $\paren {\paren {p \land q} \implies r} \implies \paren {p \implies \paren {q \implies r} }$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Proof of Reverse Implication
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \implies \paren {q \implies r}$ | Assumption | (None) | ||
| 2 | 1 | $\paren {p \land q} \implies r$ | Sequent Introduction | 1 | Rule of Exportation: Reverse Implication: Formulation 1 | |
| 3 | $\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \land q} \implies r}$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |
$\blacksquare$
Sources
- 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T27}$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms: Exercise $\text{II}. \ 3$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $18.$
- 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: $19$