# Hypothetical Syllogism

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## Theorem

The **(rule of the) hypothetical syllogism** is a valid deduction sequent in propositional logic:

- If we can conclude that $p$ implies $q$, and if we can also conclude that $q$ implies $r$, then we may infer that $p$ implies $r$.

### Formulation 1

\(\ds p\) | \(\implies\) | \(\ds q\) | ||||||||||||

\(\ds q\) | \(\implies\) | \(\ds r\) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds p\) | \(\implies\) | \(\ds r\) |

### Formulation 2

\(\ds p\) | \(\implies\) | \(\ds q\) | ||||||||||||

\(\ds q\) | \(\implies\) | \(\ds r\) | ||||||||||||

\(\ds p\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds r\) | \(\) | \(\ds \) |

### Formulation 3

- $\vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$

### Formulation 4

- $\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$

### Formulation 5

- $\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$

## Also known as

It is referred to by some authors as the **principle of syllogism**

It is also known as the **transitivity law**.

Its abbreviation in a tableau proof is $\textrm{HS}$.

## Examples

### Ancient Chinese Proverb

*If there is light in the soul,**then there will be beauty in the person.*

*If there is beauty in the person,**then there will be harmony in the house.*

*If there is harmony in the house,**then there will be order in the nation.*

*If there is order in the nation,**then there will be peace in the world.*

The conclusion is:

*If there is light in the soul, then there will be peace in the world.*

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.12$: Laws of sentential calculus - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 3.9$: Derivation by Substitution - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**hypothetical syllogism**