Consider this chess puzzle. White to checkmate in two. It appeared recently in the Boston Globe, and what startled me about it was that I had thought it had been proven that you can’t checkmate with a lone knight (and a king, of course). This is a counterexample, a strange circumstance that can arise in a legal game of chess. This fact is a higher-order truth of chess, namely that the “proof” that you can never checkmate with a lone knight and king is unsound. Now let’s consider chmess (I made up the term); it’s the game that you get by allowing the king to move two spaces, not one, in any direction. I made it up and I don’t know if anybody has ever played it. I have no idea whether it’s worth playing. Probably it isn’t, but it doesn’t matter; it’s not even worth our attention long enough to figure out. But a moment’s reflection reveals that there are exactly as many higher-order a priori truths of chmess as there are of chess, namely, an infinity of them. And no doubt they would be roughly as difficult to discover and to prove as the higherorder truths of chess. There are people who make a living working out the truths of chess and certainly it’s been a big avocation for many other people. But I doubt if anybody yet has spent more than 5 minutes trying to work out the a priori truths, and the higher-order truths, of chmess.