A Criticism of Aristotle’s Syllogism

Consider one of Aristotle’s syllogisms:

All A are B. All A are C. Therefore some B are C.

Why is this not an entirely sound argument? Or, in what case would the argument fail?

Hint: Reducing the argument to simple Venn diagram might help.

If all the elements of set A are elements of set B, and all the elements of set A are also elements of set C, then the elements of set A form the intersect of sets B and C (Aristotle’s conclusion, Some B are C).

The argument fails though if set A is an empty (or null) set. In this case, no element of set B is an element of set C.

Aristotle’s argument would have been deductively valid (or infallible) if he eliminated the possibility of set A being empty. So, his syllogism should have been:

All A are B. All A are C. [A is not an empty set.] Therefore some B are C.

20 July 2003