Board Thread:General Discussion/@comment-32182236-20190721003717/@comment-26907577-20190724192005

And now, an explanation for inverses, converses, and contrapositives.

Let's use, for an example, types of quadrilaterals.
 * An object which is a shape and has 4 sides is a quadrilateral.
 * A shape which is a quadrilateral and has 2 sets of parallel sides is a rectangle.
 * A shape which is a quadrilateral and has 4 congruent sides (and 4 congruent angles) is a rhombus.
 * A shape which is a rhombus and is a rectangle is a square.

This is a statement: If a shape is a square, it is a rectangle. This is that statement's converse: If it is a rectangle, that shape is a square. (Note that, in this example, the statement is true but the converse is not. As stated above, assuming that the statement's truth implies the converse's truth is affirming the consequent, and is a fallacy.)

This is the inverse of the original statement: If a shape is not a square, it is not a rectangle. (Again, assuming the truth of the inverse based on the truth of the statement is a fallacy of denying the antecedent.)

Note the reason that these are fallacies: there are several shapes which are rectangles but not squares.

And now, the contrapositive: If a shape is not a rectangle, it is not a square. In this case, the contrapositive is true, because rectangularity is a requirement of being a square- precisely the same reason the original statement is true.

Furthermore, the same relationship here shown between rectangles and squares exists between quadrilaterals/rectangles, rhombi/squares, quadrilaterals/rhombi, quadrilaterals/squares, and shapes/[any of those], all for the same reason: the set of the latter is contained entirely within the former, but does not entirely fill it. Although there are sets for which this does not apply, one should not assume whether it is, as such assumption is a fallacy!