Converse, Inverse, Contrapositive

In the video Geometry, Part I, by the award winning Standard Deviants, we are told the following:

If a statement is true, the inverse is also logically true. Likewise, when the converse is true, the contrapositive is also logically true.

They are mistaken. What they should have said is:

If a statement is true, the contrapositive is also logically true. Likewise, when the converse is true, the inverse is also logically true.

We start with a simple statement of fact, like, (1) A triangle is a polygon, or (2) An even number is divisible by two. Each of these is actually an implication, an if-then statement: (1) If an object is a triangle then it is a polygon. (2) If a number is even then it is divisible by two. Here is a quick definition of "converse," "inverse," and "contrapositive:"

• statement: if p then q
• converse: if q then p
• inverse: if not p then not q
• contrapositive: if not q then not p

I will now show the converse, inverse, and contrapositive of our examples involving triangles and even numbers:

Converse: (1) If an object is a polygon then it is a triangle (false). A square is a polygon but not a triangle. (2) If a number is divisible by two then it is even (true). Of course the first one is false because not all polygons are triangles.

Inverse: (1) If an object is not a triangle then it is not a polygon (false). A square is not a triangle, but is a polygon. (2) If a number is not even, then it is not divisible by two (true).

Contrapositive: (1) If an object is not a polygon, then it is not a triangle (true). (2) If a number is not divisible by two then it is not even (true). The first one, being true, cannot be equivalent to the converse, which is false.

The Standard Deviants' error does not matter, for their purposes, as they proceed to ignore the inverse and contrapositive, because they are equivalent to the other two (statement and converse, even though they are equivalent in the wrong order). Most of the time, logicians and mathematicians also ignore the inverse and contrapositive, although sometimes it may be easier to prove the contrapositive, and then remark that that is equivalent to proving the original statement. Often the converse of a mathematical statement is true. Then it is handy to prove both the statement and its converse.

The above can also be examined with the use of a boolean algeba (symbolic logic) or with truth tables. If we do that, then we must note that in logic an implication (as an "if" is called) is a statement which can be true or false. And when the condition is false, then the entire statement is true. "If my shirt is red, then I am wearing blue jeans," is true when I wear a yellow shirt, regardless of whether I am wearing blue jeans or not. This is important so that our statements will be well defined, we can build a truth table.

My contrapositive road sign: If you aren't home already, then you don't live here.