These discussions about properties of a nonexistent object require nontrivial interpretation, which can't be done mathematically. Better, I think, to say that some "propositions" have no truth value (perhaps because they are not actually propositions). The classic example is "The present King of France is bald." Reasoning about the empty set cannot help us here. One can defend the proposition by saying, "Show me one hair from the present King of France's head!" One can attack it by saying, "Show me his shiny scalp!" I think "propositions" like "So-and-so's husband is not married" have the same problem. The statement "Chris has no husband" does not straightforwardly imply "Chris's husband is unmarried"; it can only have that implication under some fairly gnarly rule of interpretation, which I challenge anybody to verbalize. On Wed, Jan 20, 2016 at 2:01 PM, James Propp <jamespropp@gmail.com> wrote:
Actually, I just noticed the heterosexism in my original email. "If a woman is married then her husband is married" is false (though if the woman's husband isn't married, her wife is!).
Jim
On Wednesday, January 20, 2016, James Propp <jamespropp@gmail.com> wrote:
Thanks, Dan; I tried that address and got a response from Smullyan a few hours later!
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. :-)
Jim Propp
On Tue, Jan 19, 2016 at 3:01 PM, Dan Asimov <asimov@msri.org <javascript:_e(%7B%7D,'cvml','asimov@msri.org');>> wrote:
The Lehman College math / comp. sci. webpage lists this e-address:
rsmullyan@verizon.net <javascript:_e(%7B%7D,'cvml','rsmullyan@verizon.net');> <mailto: rsmullyan@verizon.net <javascript:_e(%7B%7D,'cvml','rsmullyan@verizon.net');>>.
(But this person is listed as "Raymond", not "Ray".)
—Dan
On Jan 19, 2016, at 7:31 AM, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
Does anyone on the list have contact info for Smullyan?
I want to send him my paradoxical pair of sentences "If the premise of this implication does not contain the letter p, then two is not even" and its formal contrapositive "If two is even, then the premise of this implication contains the letter p" (the first is true whereas the second is false); more importantly, I want to find out if he knows of other, better sentences of this kind. (I haven't read all his books; it's possible that he had this idea years before I did, and implemented it in a better way.)
I should probably send my question to Hofstadter too, but that seems less urgent. (The last time I saw Smullyan, he did not look that well.)
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