Background on Generalized Quantifier Theory

Transcription

1 Background on Generalized Quantifier Theory More elaborate background material (online on the web): winter/courses/synsem.html 1 The modeltheoretic perspective on semantics Empirical motivations of modeltheoretic semantics: Entailment between sentences Truth/Falsity of sentences in situations Denotations: Expressions denote abstract mathematical objects. Sentences (in simple versions of MT semantics) denote either true or false. The truth-conditionality criterion: Let S and S be sentences. The theory should make sure that the following conditions hold in the same cases. (A) S intuitively entails S. (B) Whenever the theory takes S to denote true, it also takes S to denote true. Compositionality: The denotation of an expression is determined by the denotations of its parts and the ways they combine with each other. Architecture of Modeltheoretic Semantics: 1. Model : An abstract mathematical structure including (a) A frame : the set of domains of legitimate denotations. (b) An interpretation function : mapping lexical expressions to elements of domains in. (Or to sets of such elements in case of ambiguous lexemes). 2. Denotation function : a function that for each model maps (simple or complex) expressions to (sets of) elements of domains in. Thus extends the function of. 1

2 G G The big questions: 1. What is? (=ontology) 2. What is? (=lexical semantics) 3. What is? (=compositional semantics/syntax-semantics interface) 2 Types and model structure Definition 1 (extensional types) is the type of entities, is the type of truth values. The set of types is the smallest set TYPE that satisfies: If and then. Each type classifies a domain according to the following definition. Definition 2 (typed domains) is an arbitrary non-empty set.! #"%$'&. If and are types then (*) +-,/. ), the set of functions from ( to 0). Example: the domain 1, including all possible denotations of given a model where :<;>=@?A68B*C7ED4F 8 & : H 3I5J6K7 8ELM "N;>=@?A6 8ELM " CO7ED4F 8 LM " G#Q PH 3I5J6K7 8ELM "N;>=@?A6 8ELM " CO7ED4F 8 LM $ GIR H 3I5J6K7 8ELM "N;>=@?A6 8ELM $ CO7ED4F 8 LM " G#S H 3I5J6K7 8ELM "N;>=@?A6 8ELM $ CO7ED4F 8 LM $ G#V H 3I5J6K7T8 LM $U;>=@?A68 LM " CO7ED4F 8 LM " GXW H 3I5J6K7T8 LM $U;>=@?A68 LM " CO7ED4F 8 LM $ G#Y H 3I5J6K7 8ELM $U;>=@?A6 8ELM $ CO7ED4F 8 LM " H 3I5J6K7T8 LM $U;>=@?A68 LM $ CO7ED4F 8 LM $ predicates (e.g. tall) Function application: when glued together, expressions of type E and give a complex expression of type, with the meaning of applying one denotation to the other. Examples: JZ1 and [\1 give andz1 give JZ1 [\1 and don t allow function application: neither of these types is a prefix of the other. Example: analyze the types in Tina is tall. 2

3 3 Generalized quantifiers 3.1 Synopsis Three general problems: 1. The semantics of determiners: every, some, most, three etc. 2. Coordination of noun phrases: Mary or some other student, every student and every teacher, neither Mary nor Sue 3. The mapping from categories to types. Answers of Montague Grammar/GQ theory: The mapping from categories to types is a function. The type of all NPs is 1 Hence: a. The type of all determiners is 1 predicates) to GQs. = sets of sets of entities = generalized quantifiers. > 1 : functions from sets (one place b. Proper names like Mary and John denote sets of sets (=GQs). Coordination is uniformly boolean: and = intersection or = union not = complementation neither A nor B = not A and not B 3.2 Quantifiers examples (1) Every man ran. Let every man denote a set of sets: the set of subsets of men: (2) HZCO7 6 8 & In type-theoretical terms: every man denotes an J Application of this function to the VP denotation: function. (3) D K6 8 O HZCO7 6 8 & CO7 6 8 D K6 8 every member of the set CO7 6 8 is a member of the set D A6 8 that include the set of 3

4 (4) Some man ran. (5) D K6 8 O HZCO7 6 8 & CO7 6 8 D A6 8 there is an entity that is a member of both CO7E6 8 and D K6 8 (6) No man ran. (7) D K6 8 O HZCO7 6 8 CO7 6 8 D A6 8 (8) exactly five men: (9) most men: H CO7E6 8 2 & H CO7 6 8 ẌT& CO7 6 8 & Also proper names like Tina denote GQs from now on: (10) Tina ran. (11) D K6 8 O H 3I & the set of runners is in the set of subsets of that contain 3I I5J6K78@ D K Determiners examples Determiners as functions from sets to GQs (type 1 (12) ZD4F 8 H & Alternatively, we can view determiners as relations between subsets of - (13) ZD4F 8 0 iff (14) \=/C 8 0 iff (15) C= iff Terminology: We call determiner over. 3.4 Coordination examples (16) Mary or some other student a. $ $ (17) every student and every teacher "!# \[J ): (a relation between subsets of ) a HZC 8 % &'& $ H (\3 *)+96@3%8 #C 8 &X, & HZC 8 -/.-021 % 314\3 *)+96@ C 8 J& 4 :

5 a. $ $ $ H\3 )+T6/3 8 & $ H 3 Z7?+ZD 8 & H 3 )+T6/3 8 /5 3 Z7?+TD 8 & H 3 )+T6/348& 3 T7?+ZD#8 & (18) neither Mary nor Sue a. $ $ HZC 8 % & $ H C 8 -/5-8 % & H 8 - & 4 Generalized Quantifiers and Monotonicity 4.1 Quantifier monotonicity examples (19) a. Some man ran quickly Some man ran b. Some man ran quickly Some man ran (20) a. No man ran quickly No man ran b. No man ran quickly No man ran (21) a. Exactly five men ran quickly Exactly five men ran b. Exactly five men ran quickly Exactly five men ran Some man is called an upward monotone noun phrase. No man is called a downward monotone noun phrase. Exactly five men is called a non-monotone (neither upward nor downward monotone) noun phrase. 4.2 Determiner monotonicity examples (22) a. Some blue car arrived Some car arrived b. Some blue car arrived Some car arrived (23) a. Every blue car arrived Every car arrived b. Every blue car arrived Every car arrived (24) a. Exactly five blue cars arrived Exactly five cars arrived b. Exactly five blue cars arrived Exactly five cars arrived 5

6 G 4.3 Monotonicity in general Definition 3 (monotonicity) Let G be a function from!! to, where is a partially ordered set. We say that G is upward monotone on its -th argument iff is partially ordered and for all : if and then G G We say that G is downward monotone on its -th argument iff under the same conditions: G Corollary 1 (quantifier monotonicity) A generalized quantifier 1. upward monotone iff for all 2. downward monotone iff for all : if then. : if then. is: Universal 1 All lexical quantifiers (e.g. proper names, everyone, nobody, etc.) are monotone. Corollary 2 (determiner monotonicity) A determiner 1. upward left-monotone ( MON) iff for all 0 then %. 2. downward left-monotone ( MON) iff for all % 0 then. 3. upward right-monotone (MON ) iff for all then *. 4. downward right-monotone (MON ) iff for all then. Fact 3 A determiner is MON (MON ) iff for every (downward) monotone quantifier.! and and and and is: : if : if : if : if : is an upward Examples: some: every: not every: exactly five: most: "!$# "!$#!# %&!#'% %(!# Question: Are all lexical determiners right-monotone? The answer depends on the precise semantics/pragmatics of numerals (one, two, etc.). 6

7 G Q Q G Q Q G 4.4 Continuity (25) Exactly five students ran very quickly Exactly five students ran Exactly five students ran quickly (26) Exactly five big yellow taxis took away my old man Exactly five taxis took away my old man Exactly five big/yellow taxis took away my old man We say that exactly five is continuous on both arguments. Definition 4 (least upper bound) Let ' be a partially ordered set. We say that 4 is the greatest lower bound (glb) of 1 4 and denote 1 5 iff: 1. 1 ; 2. ; 3. for all 8 % : if 8 1 and 8 then 8. Symmetrically: least upper bound (. ). Definition 5 (continuity) Let G be a function from +!! to, where is a partially ordered set. We say that G is continuous on its -th argument iff ' is partially ordered and for all 1 1 : if and then is defined and satisfies ' Corollary 4 (quantifier continuity) A generalized quantifier [ is continuous iff for all : if 2 and then. Corollary 5 (determiner continuity) A determiner 1. left-continuous iff for all and 0 then %. 2. right-continuous iff for all then *. and Universal 2 All lexical determiners are right-continuous. Notes: Lexical determiners are not always left-continuous (cf. most).! is: : if 0 and : if and Complex determiners are not always right-continuous (cf. exactly three or exactly five). Hence: not all NPs are continuous (cf. exactly three or exactly five children). Fact 6 Any continuous determiner/quantifier (over a finite domain) is a finite conjunction of (upward or downward) monotone determiners/quantifiers respectively. 7

8 4.5 Conservativity (27) Every man ran Every man is a man who ran (28) Some man ran Some man is a man who ran (29) Exactly five men ran Exactly five men are men who ran... and so on for all determiners! Definition 6 (conservativity) A determiner for all : 0. 7! [ is conservative iff Universal: All natural language determiners (simple and complex) are conservative. 5 Semantic effects on grammaticality 5.1 Existential there sentences Question: Which NPs can appear in there sentences and why? (30) There is/are a cat/some cat/no cat/three cats/less/more than ten cats/between five and ten cats/many cats/few cats in the garden. (31) *There is/are every cat/most cats/all cats/the cat(s)/neither cat in the garden. Barwise and Cooper: The answer has to do with the semantics of there sentences. A determiner is allowed to appear in there sentences if and only if it does not make them tautological or contradictory. (32) A sentence there is/are NP is true iff is in the quantifier denoted by the NP. For instance: (33) There are three cats. 3I?D$ : is true whenever it is defined. A determiner is called negative strong iff for every : is false whenever it is defined. A determiner that is neither positive strong nor negative strong is called weak. (34) A determiner is called positive strong iff for every The determiners in (30) are weak. Those in (31) are strong. Generally: strong determiners are those that make there sentences semantically trivial (=tautological or contradictory whenever defined). Consequently they are ruled out. 8

9 5.2 Negative polarity items Question: What licenses negative polarity items? (35) a. John hasn t ever been to Moscow. b. *John has ever been to Moscow. (36) a. John didn t see any birds on the tree. b. *John saw any birds on the tree. (37) a. No student here has ever been to Moscow. b. *Some/every student here has ever been to Moscow. (38) a. Neither John nor Mary saw any birds on the tree. b. *Either John or Mary saw any birds on the tree. (39) a. None of John s students has ever been to Moscow. b. *One of John s students has ever been to Moscow. (40) a. Not a single student here has ever been to Moscow. b. *A single student here has ever been to Moscow. (41) a. Not more than five students here have ever been to Moscow. b. *More than five students here have ever been to Moscow. (42) a. Fewer than five students here have ever been to Moscow. b. *More than five students here have ever been to Moscow. (43) a. At most four students here have ever been to Moscow. b. *At least four students here have ever been to Moscow. (44) a. Less than half the students here have ever been to Moscow. b. *More than half the students here have ever been to Moscow. (45) a. Neither any students nor any teachers attended the meeting. b. *Either any students or any teachers attended the meeting. (46) a. John neither praised nor criticized any student. b. *John either praised or criticized any student. (47) a. Every/no/at most one student who has ever been to Moscow knows about the weather there. 9

10 b. *Some/at least one student who has ever been to Moscow knows about the weather there. (48) If John ever goes to Moscow he will know about the weather there. The Ladusaw-Fauconnier Generalization: Negative polarity items occur within arguments of monotonic decreasing functions but not within arguments of monotonic increasing functions. 6 Review exercises 1. Make sure to have carefully read either Keenan (1996) or Barwise and Cooper (1981), and, if time and mathematical interest permit, Keenan and Westerståhl (1996) as well. 2. Give types corresponding to the following descriptions: functions from truth-values to functions from truth-values to truth-values functions from functions from entities to truth-values to truth-values functions from functions from functions from entities to truth-values to entities to truth-values Describe in words the following types: ZJ J4I JZ1 >1 3. Consider the following sentences. (i) Tina ran quickly. (ii) Tina ran. (iii) Tina is very tall. (iv) Tina ran very quickly. Answer the following questions: a. Give a binary tree structure for sentence (i). b. Deduce from this structure a type for the adverb quickly. c. What restriction on the set of functions quickly can denote can you infer from the entailment (i) (ii)? d. Give a type for the word very in sentence (iii) below. 10

11 e. Give the same word a type also in (iv), in consideration of your previous answers. f. You have reached a challenge for the principle of one type per category. Can you think of a way to maintain the idea despite this fact? Alternatively, can you relax the restriction in a principled way? 4. Give denotations for the following NPs: more than four women, at most five children, between four and ten cows, all but five students, less than half the men, not every goat, more children than adults. 5. For each NP in 4 say whether it is MON, MON or non-monotone. 6. Give a relational denotation for the determiners in 4: more than four, at most five, between four and ten, all but five, less than half the, not every. 7. For each determiner in 6 specify its left and right monotonicity. 8. Show that the following artificial determiner is not conservative: 0 iff Hint: give and such that 0 and 0 do not have the same truth-value. Can you think of a syntactic element with the meaning of in a sentence of the form students ran? After you find such an element: give syntactic arguments for whether it is of the same category of the word all or not. Conflicting arguments are welcome you don t have to decide on this matter. 9. Show that most is not left-continuous: describe a situation showing that an entailment like (26) does not hold for this determiner. 10. Consider the sentence there are cats in the garden, as well as the questionable sentence *there is John in the garden. What problem do these cases pose to the Barwise & Cooper proposal? Can you think of a modification that would improve the situation? References Barwise, J. and Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 4: Keenan, E. (1996). The semantics of determiners. In Lappin, S., editor, The Handbook of Contemporary Semantic Theory. Blackwell. Keenan, E. and Westerståhl, D. (1996). Generalized quantifiers in linguistics and logic. In van Benthem, J. and ter Meulen, A., editors, Handbook of Logic and Language. Elsevier, Amsterdam. 11