1 Semantics in Generative Grammar Angelika Kratzer November 26, 2007 Constraints on quantifier denotations 1 Recommendation: Jan van Eijck (1991): Quantification. In Arnim von Stechow & Dieter Wunderlich (eds.): Semantik/Semantics. Berlin (de Gruyter). Chapter 21, 459 - 487. The following notes are very much indebted to the pedagogy in van Eijck’s chapter. The library has the handbook in the reference section. 1. Where we are We have been translating the English determiners every, some, and no as follows: a. T(every) = λPλQ ∀x(P(x) → Q(x)) b. T(some) = λPλQ ∃x(P(x) & Q(x)) c. T(no) = λPλQ ¬∃x(P(x) & Q(x)) 2. Quantifiers as relations between sets When thinking about universals of quantifier denotations, it is convenient to think of quantifiers as relations between sets of individuals. If A and B are sets of individuals, we have: Fill in the appropriate conditions: a. Revery (A, B) iff b. Rsome (A, B) iff c. Rno (A, B) iff d. Rexactly two (A, B) iff e. Rat least two (A, B) iff2 f. Rat most two (A, B) iff g. Rmore than half of (A, B) iff h. Rall but 5 of (A, B) iff i. Rthe 5 (A, B) iff The most important question Which binary relations between sets of individuals are possible quantifier denotations in natural languages? 3. Constraints on quantifier denotations in natural languages: Extension Extension If A, B ⊆ E ⊆ E’, then RE (A, B) iff RE’ (A, B) Quantifier relations that satisfy Extension remain stable under extension of the universe of individuals. This means that whether a quantifier relation holds between two sets A and B only depends on the minimal universe A ∪ B. Illustration: the denotation of every satisfies Extension If Revery (A, B) holds under the assumption that the universe is E, then Revery (A, B) also holds when the universe is a superset of E. Revery (A, B) holds just in case A is a subset of B, and whether or not the subset relation holds between A and B only depends on A and B. The same is true for all quantifier relations listed in section 2, and arguably for all relations expressed by quantifiers in natural languages.3 Here is an artificial quantifier relation that does not satisfy Extension: If A, B ⊆ E, then Rnonex (A, B) iff A = B’. That is, the relation holds between A and B just in case A is the complement of B with respect to whatever the universe is. A B E E’ B A B A E’ Rnonex (A, B) holds, since A is the complement of B with respect to the universe E, which is A ∪ B in this case. When the universe expands to E’, A is no longer the complement of B, hence Rnonex (A, B) no longer holds.4 What to watch out for Extension is a constraint on relations between sets, and it is stated as such. To check whether a quantifier does or does not satisfy Extension, you have to find out whether it could happen that the relation expressed by the quantifier may hold between two sets A and B when the universe is E, but may no longer hold between those very same sets A and B when the universe is a superset of E. Crucially, Extension is NOT violated by the fact that sentence (1), for example, may be true when evaluated with respect to a universe E, but may very well become false when evaluated with respect to a universe E’ that is a superset of E. (1) Every child is quiet. Consider the following sets, where E ⊆ E’: A = {a: x is a child and x ∈ E} B = {a: x is quiet and x ∈ E} C = {a: x is a child and x ∈ E’} D = {a: x is quiet and x ∈ E’} (1) is true with respect to the universe E iff A ⊆ B. Suppose now that (1) is true if evaluated with respect to E. What happens if (1) is evaluated with respect to E’? In that case, the extension of child is C, rather than A, and the extension of be quiet is D, rather than B. If E’ contains noisy children, Revery (C, D) does not hold, and (1) winds up false. The reason why (1) may become false if the universe expands, then, is that the extensions of child and be quiet may change. Revery (A, B) continues to hold, of course, and Extension is still satisfied. A quantifier that might not satisfy Extension: (relatively) many (2) Many of our graduate students speak Portuguese. (3) The proportion of Portuguese speakers among our graduate students is higher than the proportion of Portuguese speakers in the general population.5 • Does many truly have such a reading? If it does, it would be defined as follows: Rmany (A, B) iff (fill in the condition) Recommendations: Barbara Partee (1989): Many quantifiers. Reprinted (2004) in Compositionality in Formal Semantics. Selected Papers by Barbara H. Partee, chapter 12, 241-258. Ariel Cohen (2001): Relative readings of many, often, and generics. Natural Language Semantics 9,