Superlative Quantifiers as Modifiers of Meta-Speech Acts

The superlative quantifiers, at least and at most, are commonly assumed to have the same truth-conditions as the comparative quantifiers more than and fewer than. However, as Geurts & Nouwen (2007) have demonstrated, this is wrong, and several theories have been proposed to account for them. In this paper we propose that superlative quantifiers are illocutionary operators; specifically, they modify meta-speech acts. Meta speech-acts are operators that do not express a speech act, but a willingness to make or refrain from making a certain speech act. The classic example is speech act denegation, e.g. I don’t promise to come, where the speaker is explicitly refraining from performing the speech act of promising. What denegations do is to delimit the future development of conversation, that is, they delimit future admissible speech acts. Hence we call them meta-speech acts. They are not moves in a game, but rather commitments to behave in certain ways in the future. We formalize the notion of meta speech acts as commitment development spaces, which are rooted graphs: The root of the graph describes the commitment development up to the current point in conversation; the continuations from the root describe the admissible future directions. Superlative Quantifiers 2 We define and formalize the meta-speech act GRANT, which indicates that the speaker, while not necessarily subscribing to a proposition, refrains from asserting its negation. We propose that superlative quantifiers are quantifiers over GRANTs. Thus, Mary petted at least three rabbits means that the minimal number n such that the speaker GRANTs that Mary petted n rabbits is n = 3. In other words, the speaker denies that Mary petted two, one, or no rabbits, but GRANTs that she petted more. We formalize this interpretation of superlative quantifiers in terms of commitment development spaces, and show how the truth conditions that are derived from it are partly entailed and partly conversationally implicated. We demonstrates how the theory accounts for a wide variety of phenomena regarding the interpretation of superlative quantifiers, their distribution, and the contexts in which they can be embedded. 1. THE MEANING OF SUPERLATIVE QUANTIFIERS 1.1. Commonly held intuitions What do superlative quantifiers, like at least and at most, mean? Nonlinguists have a clear intuition—for example, note the following discussion of at least from a book about computer databases: One important rule to remember is that there should be at least (n−1) joins in an n-table query; thus, you need at least two joins for a three-table query, at least three joins for a query that involves four tables, and so on. The words “at least” are important: there could be more than (n− 1) joins. . . but if your multitable query has less than (n − 1) joins, the result will be [bad] (A. Kriegel and B. M. Trukhnov, SQL Bible, p. 319). According to this intuition, at least x means x or more, but not less; at most x means x or less, but not more. In a context in which only integers are relevant, things are even simpler: at least x means greater than x − 1, and at most x means fewer than x + 1. Thus, since it is impossible to pet a non-integer number of rabbits, (1-a) would mean (1-b). (1) a. John petted at least three rabbits. b. John petted more than two rabbits.1 Vol. 6: Formal Semantics and Pragmatics: Discourse, Context, and Models 3 Ariel Cohen & Manfred Krifka Similarly, (2-a) would be equivalent to (2-b). (2) a. John petted at most three rabbits. b. John petted fewer than four rabbits. 1.2. Keenan and Stavi (1986) Until recently, such intuitions were widely shared by linguists as well, and have been formalized by Keenan & Stavi (1986). According to their theory, both superlative (at least, at most) and comparative (more than, fewer than) quantifiers are treated simply as generalized quantifiers, i.e. relations between sets. Thus, the meaning of (1-a) is simply (3-a), where R is the set of rabbits, and the meaning of (2-a) is simply (3-b). (3) a. |R∩λx .pet(j, x)| ≥ 3 b. |R∩λx .pet(j, x)| ≤ 3 Besides being intuitive, this definition has two important advantages. One advantage is that it gets the truth conditions right: if John petted two or fewer rabbits, (1-a) is false, and if he petted four or more rabbits, (2-a) is false. The second advantage is that these truth conditions are extensional, which is as it should be. For example, suppose Mary likes rabbits but no other animal. Then, the extensions of the predicates rabbit and animal that Mary likes are the same. Note that the truth values of (1-a) and (4) are the same, indicating that superlative quantifiers are extensional. (4) John petted at least three animals that Mary likes. However, Keenan & Stavi’s theory, while getting the truth conditions right, fails to account for a number of phenomena. In particular, as Geurts & Nouwen (2007) point out, the meanings of superlative quantifiers differ from those of comparative quantifiers (more/fewer than) in subtle ways. One of the observations of Geurts & Nouwen is that the distribution of comparative quantifiers is more restricted than that of superlative quantifiers. For example, only the latter, but not the former, can take sentential scope: www.thebalticyearbook.org Superlative Quantifiers 4 (5) a. John petted three rabbits at most *fewer than . b. At least *More than , John petted three rabbits. Additionally, superlative quantifiers, but not comparative ones, may combine with quantifiers, proper names, and specific indefinites: (6) a. Mary petted at least *more than every young rabbit. b. Mary petted at most *fewer than Bugs Bunny. c. John petted at least *more than two rabbits, namely Bugs Bunny and Peter. There are, however, cases where comparative quantifiers are acceptable, and it is superlative quantifiers that are odd. Suppose John petted exactly three rabbits, and we know this. Based on this fact, we would be justified in uttering (1-b); however, it would be quite strange to utter (1-a) or (2-a).2 There are differences between superlative and comparative quantifiers not just in distribution, but also in interpretation: the former lack some readings that the latter have. For example, (7-a) is ambiguous: it can mean either that it is permissible for you to have fewer than three martinis (say, because you don’t like martinis), or that you may not have more than two martinis. In contrast, (7-b) is not ambiguous, and only receives the second reading. (7) a. You may have fewer than three martinis. b. You may have at most two martinis. 1.3. Geurts and Nouwen (2007) In order to account for these phenomena, Geurts & Nouwen (2007) argue against the commonly held intuition, and propose that comparative and superlative quantifiers have different interpretations. Following Krifka (1999a), they propose that comparative quantifiers are focus sensitive NP modifiers. Roughly, (1-b) means that there Vol. 6: Formal Semantics and Pragmatics: Discourse, Context, and Models 5 Ariel Cohen & Manfred Krifka is a property that is higher or equal on the relevant scale than the property of petting two rabbits, and this property applies to John.3 Formally, the meaning Geurts & Nouwen propose for more than α is: (8) λx .∃β(β > α∧ β(x)) The relevant scale is affected by focus, which explains the difference between (9-a) and (9-b). (9) a. John petted more than [two]F rabbits. b. John petted more than [two rabbits]F. Sentence (9-a), with focus on two, means that John petted a number of rabbits, and this number is greater than two; while (9-b), with focus on two rabbits, is compatible with John having petted exactly two rabbits, provided that he petted additional animals. Sentences (9-a) and (9-b) are evaluated with respect to different scales; and yet other scales account for examples such as the following: (10) a. I will be more than happy to send you the necessary forms. b. The telephone service here is less than satisfactory. Sentence (10-a) is presumably evaluated with respect to a scale involving properties such as being reluctant, indifferent, happy and ecstatic; (10-b) presumably involves properties such as being terrible, bad, satisfactory, good, and excellent. Importantly, Geurts & Nouwen restrict α and β in (8) to denote only first-order properties, i.e. expressions of type 〈e, t〉. The properties happy and satisfactory are clearly first-order. The property of being a group of two rabbits is also first-order in their system, since they treat groups as individuals. But propositions are not first-order properties, which is why comparative quantifiers cannot combine with them, and the unacceptability of the sentences in (5) is thereby explained. Similarly, quantifiers, names, and specific indefinites also do not denote first-order properties, which is why the sentences in (6) are bad. Regarding superlative quantifiers, Geurts & Nouwen propose that they are epistemic operators. Specifically, the meanings of (1-a) and (2-a) can be roughly paraphrased as (11-a) and (11-b), respectively. www.thebalticyearbook.org Superlative Quantifiers 6 (11) a. It is epistemically necessary that John petted three rabbits, and it is epistemically possible that he petted more. b. It is epistemically possible that John petted three rabbits but it is epistemically impossible that he petted more. Geurts & Nouwen demonstrate how their approach solves some of the problems with Keenan & Stavi’s theory. In particular, they can explain why (2-a) would be odd if it is known that John petted exactly three rabbits: according to Geurts & Nouwen’s theory, (2-a) entails that it is epistemically possible that John petted three rabbits. But in this case it is not only epistemically possible, but, in fact, epistemically necessary that John petted three rabbits, so the speaker makes a weaker statement than the one she can and ought to make. In other words, saying that it is epistemically possible that John petted three rabbits implicates that it i