Relations between statements - converse, inverse, contrapositive

What does someone mean when they refer to the converse, inverse or contrapositive of a statement?

Take a simple statement about bikes: "Bikes have two wheels". Logical thinkers that you and I are, we can see that this means 'everything that's a bike has two wheels'. The logical structure is 'for all x, x is a bike implies x has two wheels'. Let's write that symbolically: (∀x)Bx⊃Wx. That's first-order logic, but we can just simplify that even further to the two propositions: B⊃W.

If you've got a blank look on your face or your eyes are glazing over, go back and check out my Logic and Language series.

You can play with the statement statement B⊃W. First, invert it. The inverse of B⊃W ('if it's a bike then it has two wheels') is ~B⊃~W ('if it's not a bike then it does not have two wheels'). The inverse of a statement isn't necessarily true. In this case, there are two-wheeled objects that are not bikes.

The converse of B⊃W ('if it's a bike then it has two wheels') switches the terms, so we end up with W⊃B ('if it has two wheels then it's a bike'). The converse may be true if the original is true, but it isn't necessarily. But notice that the converse will be true if the inverse is true. Think about it.

The inverse negated both propositions and the converse switched both propositions. There's also a contrapositive with switching and negation going on. For our original statement B⊃W ('if it's a bike then it has two wheels'), the contrapositive is ~W⊃~B ('if it's not two wheeled, then it's not a bike'). Looks like that's true so long as our original was.

It's not good logic if we can't abstract these structures to other instances. Using any two propositions A and B, where A implies B, we can get to the inverse, converse and contrapositive this way:

type structure
inverse ~A⊃~B
converse B⊃A
contrapositive ~B⊃~A

Something to notice if you want to get even more playful with your logic is that you can use these concepts recursively on themselves. What I mean is, for example, the inverse isn't just the inverse of the statement, it's also the converse of the contrapositive, because it switches the two propositions around. And the contrapositive isn't just the contrapositive, it's also the converse of the inverse. This is possible because every statement can be reevaluated just like the original. They're not inherently inverses or contrapositives; they're just statements.

You may at first fumble with these terms or use them imprecisely. Take this first step to straighten out for yourself the relationship between a statement and the ways it can be twisted up.