Question I'm still puzzled by the answers to question 5792, on whether it is true that Mary won all the games of chess she played, when Mary never played any game of chess. Both respondents said that it is true. But is it meaningful to say "I won all the games I played, and I never played any game."? It seems to me that someone saying this would be contradicting himself.

Accepted:
March 19, 2015

Comments

I think you're right to at least this extent. If I say to someone "I won all the games of chess I played," the normal rules of conversation (in particular, the "pragmatics" of speech) make it reasonable for the other person to infer that I have actually played at least one game. Whether my statement literally implies this, however, is trickier.

Think about statements of the form "All P are Q." Although it may take a bit of reflection to see it, this seems to be equivalent to saying that nothing is simultaneously a P and a non-Q. We can labor the point a bit further by turning to something closer to the lingo of logic: there does not exist an x such that x is a P and also a non-Q. For example: all dogs are mammals. That is, there does not exist a dog that is a non-mammal.

Now go back go the games. If Mary says "All games I played are games I won," then by the little exercise we just went through, this becomes "There does not exist a game that I played and lost." But if Mary played no games at all, then that's true. No game is a game she played and lost because no game is a game she played.

It turns out that avoiding this conclusion isn't as easy as it might seem. We usually agree that "No X are Y" and "No Y are X" amount to the same thing. We can also agree that no animals are unicorns, because there aren't any unicorns at all. But if no animals are unicorns, then the principle we just noted entails that no unicorns are animals. which is already starting to sound awkward. Worse, we also usually agree that "No X are Y" amounts to "All X are non-Y," and so we get "All unicorns are non-animals."

There are approaches to logic that find ways around this sort of thing. But the carpet will have to bulge somewhere. Either the rules of inference will be a bit more complicated or we'll have to give up principles that seem appealing or we'll end up with some cases of "correct" inferences that seem peculiar. Different people will see the costs and benefits differently. My own view, which would not win me friends in certain circles, is that there's nothing deeply deep here. But not everyone agrees.