Two Puzzles about Requirements

I discuss the semantics of statements of minimum and maximum requirement. I show that, on standard assumptions, such statements receive a non-sensical interpretation. I solve this puzzle by pointing out certain semantic features shared by all end-point markers akin to minimum and maximum. 1 First puzzle: Minimum requirements My friend and me are having a bet in which I claim to be able to score at least 300 points in the game of scrabble we are about to start. The following would be an accurate paraphrase for the bet in question. (1) The minimum number of points I need to score to win the bet is 300. That is, the bet involves a minimum requirement: If I score 300, I win. If I score more, I also win. But if I score less, then I lose. To be sure, at first sight it seems obvious why (1) is interpreted as such, for I do need to score 300, and 300 is the minimum score that makes me win the bet. Yet, when we make things precise, and given common assumptions on the semantics of modal auxiliaries, it turns out that it is rather mysterious why (1) means what it means. The common assumptions I am alluding to are, first of all, that need is a universal quantifier over possible worlds and, second, that the to-phrase in (1) restricts quantification over possible worlds (as in von Fintel and Iatridou’s [2]). In other words, “to p, need to q” is true if and only if all the p-worlds are q-worlds; i.e. if p entails q. At first sight, this view appears to make good predictions. For example, a case like (2) is now interpreted as saying that you went to the Twijnstraat in all the worlds where you got good cheese. (2) To get good cheese, you need to go to the Twijnstraat. However, when we apply the above assumptions to (1), the outcome is very puzzling. Note first the following: in the scenario I sketched about the scrabble bet, there are no worlds in which I win the bet while scoring fewer than 300 points. Furthermore, the worlds where I do win come in many variations: in some (but not all) of them I score 300 points, in some (but not all) my score is 301, in some (but not all) it is 302, 310, or even 550. The problem is that the most obvious referent for the minimum number of points I need to score is the smallest number x such that I scored x points is true in all relevant worlds. However, for no value for x is this open sentence true in all bet-winning worlds.