Hypothetical Syllogism/Formulation 3/Proof 1
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Theorem
- $\vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$
Proof
Let us use the following abbreviations
\(\ds \phi\) | \(\text{ for }\) | \(\ds p \implies q\) | ||||||||||||
\(\ds \psi\) | \(\text{ for }\) | \(\ds q \implies r\) | ||||||||||||
\(\ds \chi\) | \(\text{ for }\) | \(\ds p \implies r\) |
By the tableau method of natural deduction:
Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|
1 | 1 | $\phi \land \psi$ | Assumption | (None) | ||
2 | 1 | $\phi$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
3 | 1 | $\psi$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
4 | 1 | $\chi$ | Sequent Introduction | 2, 3 | Hypothetical Syllogism: Formulation 1 | |
5 | $\paren {\phi \land \psi} \implies \chi$ | Rule of Implication: $\implies \II$ | 1 – 4 | Assumption 1 has been discharged |
Expanding the abbreviations leads us back to:
- $\paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$
$\blacksquare$
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 $\text{T26}$