On Jan 20, 2016, at 8:05 AM, James Propp <jamespropp@gmail.com> wrote:
. . .
Writing to Smullyan put me in mind of an observation my friend the mathematician Michael Larsen made back when we were in college: the sentence "If M is invertible, then M^{-1} is invertible" is a true proposition in linear algebra, whereas its contrapositive "If M^{-1} is not invertible, then M is not invertible" is just plain silly.
I was also reminded of a quip of Alan McKay's: "Like a ski resort full of girls looking for husbands and husbands looking for girls, the situation is not as symmetrical as it might seem."
Combining the two, one might consider the true sentence "If a woman is married, then her husband is married"; its contrapositive is "If a woman's husband is single, then she's single too" --- which seems not only true but eminently fair. :-)
The last of these reminds me of the truism: "If your parents didn't have any children, then you won't have any children and your children won't have any children, either." Though the contrapositive of this seems plainly true also. —Dan