# 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

\(\displaystyle \phi\) | \(\text{ for }\) | \(\displaystyle p \implies q\) | |||||||||||

\(\displaystyle \psi\) | \(\text{ for }\) | \(\displaystyle q \implies r\) | |||||||||||

\(\displaystyle \chi\) | \(\text{ for }\) | \(\displaystyle p \implies r\) |

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: $\land \mathcal E_1$ | 1 | ||

3 | 1 | $\psi$ | Rule of Simplification: $\land \mathcal E_2$ | 1 | ||

4 | 1 | $\chi$ | Sequent Introduction | 2, 3 | Hypothetical Syllogism: Formulation 1 | |

5 | $\paren {\phi \land \psi} \implies \chi$ | Rule of Implication: $\implies \mathcal I$ | 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 $\text{T26}$