Converse vs. Inverse

Mathematical logic is a way of expressing concepts in a formal and specific way. The "inverse" and "converse" in logic are two ways of rewriting a statement with predefined structures. It is not possible to determine whether or not the converse and inverse of a statement are true if all you have is knowledge of the original statement, although if either the converse or inverse of a statement is true, the other is true as well.

In logic, a basic if-then statement is composed of two parts: the hypothesis and the conclusion. The hypothesis is a fact that is either true or false, and the conclusion is another true or false fact that follows from the hypothesis. For example, the fact that all squares are rectangles can produce the following true if-then statement: "If a polygon is a square, then it is a rectangle." The true fact that the polygon in question is a square is the hypothesis, and the true fact that all squares are rectangles (so this one must be a rectangle as well) combine to make a true statement.

A logical if-then statement can also be a false one. For example, take the statement "If a polygon is a square, then it is a circle." Polygons cannot be both squares and circles, so even though the statement is structured correctly, it is still false. The conclusion does not follow from the hypothesis. The truth of an if-then statement requires that the hypothesis and conclusion both be true and that the hypothesis implies that the conclusion happens.

The inverse of a statement is a way of writing the statement using the opposite of its original hypothesis and conclusion. Using the previous example, the inverse of the statement "If a polygon is a square, then it is a rectangle" is "If a polygon is not a square, then it is not a rectangle." It is impossible to say if the inverse is true without more information. In this example, there are some polygons that are rectangles but not squares, so the inverse statement will be true or false for different polygons.

The converse of a statement is another way of rewriting it, except instead of using the opposite of the hypothesis and conclusion, the converse flips them around. Continuing with the example, the converse of "If a polygon is a square, then it is a rectangle" is "If a polygon is a rectangle, then it is a square." The converse just reverses the order of the hypothesis and conclusion. Again, it is not possible to tell if the converse is true without more information. Some rectangles are squares but not all of them.