# Hypothetical Syllogism

## 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

$p \implies q, q \implies r \vdash p \implies r$

### Formulation 2

$p \implies q, q \implies r, p \vdash r$

### Formulation 3

$\vdash \left({\left({p \implies q}\right) \land \left({q \implies r}\right)}\right) \implies \left({p \implies r}\right)$

### Formulation 4

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

### Formulation 5

$\vdash \left({q \implies r}\right) \implies \left({\left({p \implies q}\right) \implies \left({p \implies r}\right)}\right)$

## 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.