The ambiguous "any": I was playing a game with some friends, and we came across the rule: " If you have more cards than any other player, then discard a card. " Does this mean "than all other players", or "than some other player"? Our group's opinion was divided (incl. across many native English speakers). In our class terms, it's not always clear whether "any" means for-all, or for-some (there-exists). Or maybe more accurately, in the phrase "for any x", does x necessarily mean an arbitrary player?
ASIDE: Linguistics students, or those who are so sure the rule clearly intended "than all other players": Switching "x>y " to"x<y " changes from an active voice to a passive voice but may also reverse your interpretation of the English quantifier "any": "If any player has fewer points than you, ..."
In your proof-writing (and your English writing, and your informal writing), think about replacing "any" with either "every" or with "some", to make your meaning clear.