# Law of Syllogism

In mathematical logic, the Law of Syllogism says that if the following two statements are true:

(1) If $p$, then $q$.

(2) If $q$, then $r$.

Then we can derive a third true statement:

(3) If $p$, then $r$.

Example:

If the following statements are true, use the Law of Syllogism to derive a new true statement.

1) If it snows today, then I will wear my gloves.

2) If I wear my gloves, my fingers will get itchy.

Let $p$ be the statement "it snows today", let $q$ be the statement "I wear my gloves", and let $r$ be the statement "my fingers get itchy".

Then (1) and (2) can be written

1) If $p$, then $q$.

2) If $q$, then $r$.

So, by the Law of Syllogism, we can deduce

3) If $p$, then $r$

or

If it snows today, my fingers will get itchy.