Spring 2018 Ling 620 Introduction to Semantics of Questions: Questions as Sets of Propositions (Hamblin 1973, Karttunen 1977)

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1 Introduction to Semantics of Questions: Questions as Sets of Propositions (Hamblin 1973, Karttunen 1977) 1. Question Meanings and Sets of Propositions (1) The Semantics of Declarative Sentence Dave smokes a. Extension: Truth Value True b. Intension: Proposition [ λw : Dave smokes in w ] (2) Overarching Question: How should we treat the extension and intension of an interrogative sentence? We don t use questions to provide information, and so it seems that we should view their extensions/intensions differently from declarative sentences (3) The First Answer to Consider (Hamblin 1973) The extension of a question is the set of its possible answers (true or false) o (Where an answer to a question is a proposition, obtained by replacing the wh-word [interrogative pronoun] with a referring expression ) Thus, the extension of a question is a set of propositions Illustration: [[ Who smokes? ]] w = { [ λw : Barack smokes in w ], [ λw : Michelle smokes in w ], [ λw : Bernie smokes in w ], [ λw : Hillary smokes in w ], [ λw : Ted smokes in w ], } = { p : x. x D e & x is human & p = [ λw : x smokes in w ] } [[ What did Seth cook? ]] w = { [ λw : Seth cooked the pizza in w ], [ λw : Seth cooked the fish in w ], [ λw : Seth cooked the carrots in w ], [ λw : Seth cooked the pasta in w ], [ λw : Seth cooked the beans in w ], } = { p : x. x D e & x is not human & p = [ λw : Seth cooked x in w ] } 1

2 (4) Philosophical Motivation for this Approach (Hamblin 1973) To know the meaning of a declarative sentence is (in part) to know the conditions under which it s true (the truth-conditions) Similarly, to know the meaning of an interrogative sentence is (in part) to know the conditions under which it s been answered o So, knowing the meaning of a question is knowing what counts as a possible answer to that question o So, we could model the meaning of a question as the set of its possible answers (5) Empirical Motivation for this Approach (Karttunen 1977) This semantics will give us an elegant analysis of the compositional semantics of embedded questions (e.g., Dave knows what Seth cooked ). (6) A Key Issue Relating to Answers As noted in passing in (3), both Hamblin (1973) and Karttunen (1977) assume that the extension of a question only ever includes single positive propositions. Consequently, the following propositions would not be included in the denotation of the question Who smokes? a. Nobody smokes. b. Everybody smokes. c. Most of the first years smoke. d. Seth and Rajesh smoke. e. Seth. Informally speaking, though, all these sentences would be appropriate, pragmatically acceptable answers (responses) to the question Who smokes? Consequently, the set of objects that Hamblin (1973) and Karttunen (1977) assign as the meaning of question will not be identical to our everyday, pretheoretical concept of a possible answer to the question 2

3 (7) Some Terminology a. Semantic Answer for Question Q: Member of [[Q]] (i) Question: What time is it? (ii) Semantic Answer: It is 3:14 PM b. Pragmatic Answer for Question Q: A pragmatically acceptable response to Q, which helps the utterer of Q obtain the information that prompted the utterance of Q. (i) Question: What time is it? (ii) Semantic Answer: Your lunch break is in fifteen minutes, Jim. There is also a further controversy about the status of so-called negative answers to whquestions, like (6a) (8) Two Perspectives on Negative Answers a. Negative Answers aren t Actually Answers One extremely common view is that asking a Q presupposes that there is a (true) positive answer for it. o Thus, the question Who smokes? presupposes that there is someone that smokes. If this is right, then a negative answer like (6a) Nobody smokes contradicts a presupposition of the question. Consequently, such answers are not really direct, sincere answers to the question, but rather ways of informing someone that the presupposition of their question fails to hold. An Analogy: P1: The second season of Firefly was awesome! P2: There wasn t any second season! What are you talking about? P1: Which of the first years smoke? P2: Uhm, none of them. What decade do you think this is? b. Negative Answers are Indeed Actually Answers Some folks, however, are not convinced that a question Q really does presuppose that there is a true positive answer for Q (Groenendijk & Stokhoff 1982). o Consequently, (6a) is indeed a direct, sincere answer to the question Who smokes? (Indeed, it could be a semantic answer ) 3

4 (9) Major Question: Assuming that the general view in (3) is correct, how do we compositionally derive those sets as meanings? (10) Two Approaches to Deriving Sets as Question Meanings a. Hamblin 1973: Alternative semantics (a.k.a., Hamblin semantics, Hamblin alternatives ) b. Karttunen 1977: Special question operators and interpretation rules 2. Alternative Semantics for Questions (Hamblin 1973) (11) Key Question: How do we derive a set of propositions as the meaning of a question? (12) Hamblin s Key Idea, Step One Let s first reconsider how the denotation of a question relates to the denotation of a declarative sentence. a. Denotation of a Question: A non-singleton set of propositions [[ Who smokes? ]] = { [ λw : Barack smokes in w ], [ λw : Michelle smokes in w ], [ λw : Bernie smokes in w ], [ λw : Hillary smokes in w ], [ λw : Ted smokes in w ], } b. Denotation of a Declarative Sentence: A singleton set of propositions [[ Barack smokes ]] = { [ λw : Barack smokes in w ] } If we adopt this view, then declaratives and interrogatives have the same semantic type (sets of propositions). Side-Note: Hamblin s (1973) system maps expressions directly to denotations The denotation of a sentence is (akin to) its intension The denotation of a question, though, is more akin to its extension Okay but how do we design a system to derive these sets of propositions as meanings?... 4

5 (13) The Key Ingredients for Declarative Clauses a. The Lexicon: In Hamblin s (1973) system, the denotation of every (non-interrogative) lexical item is a (singleton) set. (i) Names denote singletons of entities [[ Barack ]] = { Barack } [[ Joe ]] = { Joe } [[ Hillary ]] = { Hillary }, (ii) Intransitive verbs denote singletons of <e,<s,t>> functions [[ smokes ]] = { [λx : [λw : x smokes in w ]] } (iii) Transitive verbs denote singletons of <e, <e, <s,t>>> functions [[ cooked ]] = { [λy : [λx : [λw : x cooked y in w ]]] } b. The Key Rule: Point-wise Function Application (PWFA) If X has two daughters Y and Z, and [[Y]] is a set of objects of type α, while [[Z]] is a set of objects of type <α, β>, then [[ Z ]] = { f(x) : f [[Z]] and x [[Y]] } (14) Illustration of Compositional Semantics for Declaratives a. [[ Barack smokes ]] = (by PWFA) b. { f(x) : f [[smokes]] and x [[Barack]] } = (by Lexicon) c. { f(x) : f { [λx : [λw : x smokes in w ]] } and x {Barack} } = d. { [λx : [λw : x smokes in w ]](Barack) } = (by LC) e. { [λw : Barack smokes in w ] } 5

6 (15) The Key Ingredients for Interrogative Clauses, Part 1 Interrogative pronouns denote non-singleton sets of entities. a. [[ who ]] = { x : x D e & x is human } = { Barack, Joe, Hillary, Ted, Bernie, Seth, } b. [[ what ]] = { y : y D e & y is not human } = { the fish, the carrots, the pasta, the pizza, the beans, } With just these ingredients in (13b) and (15) we can get the desired meaning for wh-questions! (16) Illustration of Hamblin Semantics for Questions a. [[ who smokes ]] = (by PWFA) b. { f(x) : f [[smokes]] and x [[who]] } = (by Lexicon) c. { f(x) : f { [λx : [λw : x smokes in w ]] } and x { x : x D e & x is human } } = d. { [λx : [λw : x smokes in w ]](x) : x D e & x is human } = (by LC) e. { [λw : x smokes in w ] : x D e & x is human } = { [ λw : Barack smokes in w ], [ λw : Michelle smokes in w ], } With just these ingredients, we also get an appropriate semantics for multiple wh-questions! (17) Additional Result: Multiple Wh-Questions a. [[ who cooked what ]] = (by PWFA) b. { f(x) : x [[who]] and f [[cooked what]] } = (by PWFA) c. { f(x) : x [[who]] and f {g(y) : y [[what]] and g [[cooked]] } } = (by Lexicon) 6

7 d. { f(x) : x [[who]] and f {g(y) : g { [λy : [λx : [λw : x cooked y in w ]]] } and y { y : y D e & y is not human } } } = (by set theory) e. { f(x) : x [[who]] and f {[λy : [λx : [λw : x cooked y in w ]]](y) : y D e & y is not human}} = f. { f(x) : x [[who]] and f { [λx : [λw : x cooked y in w ]] : y D e & y is not human}} = g. { f(x) : x { x : x D e & x is human } and f { [λx : [λw : x cooked y in w ]] : y D e & y is not human}} = (by LC) (by Lexicon) (by set theory) h. { [λx : [λw : x cooked y in w ]](x) : x { x : x D e & x is human } and y D e & y is not human } = (by LC) i. { [λw : x cooked y in w ] : x { x : x D e & x is human } and y D e & y is not human } = (by set theory) i. { [λw : x cooked y in w ] : x, y D e & x is human and y is not human } = { [ λw : Barack cooked the fish in w ], [ λw : Barack cooked the carrots in w ], [ λw : Michelle cooked the fish in w ], [ λw : Michelle cooked the carrots in w ], [ λw : Joe cooked the fish in w ], [ λw : Joe cooked the carrots in w ], } (18) Fundamental Technical Problem: Hamblin Semantics and Predicate Abstraction This system of alternative semantics does not integrate well with our rule of Predicate Abstraction (Kratzer & Shimoyama 2002, Shan 2004). a. What We Want: [[ [1 [t 1 cooked what]] ]] = { [λx : [λw : x cooked y in w ]] : y is non-human } b. The Problem: The complex way variable assignments interact with alternative semantics prevents us from stating a rule that will deliver exactly the result in (18a). 7

8 (19) A Possible Work-Around Following Hamblin (1973), we could assume that wh-movement in questions is semantically vacuous, and so is undone by LF (i.e., moved wh-words reconstruct). a. Sentence: What did Seth cook? b. LF: [ Seth [ cooked what ] ] c. Predicted Meaning: { [ λw : Seth cooked x in w ] : x D e & x is not human } (20) The Key Ingredients for Interrogative Clauses, Part 2 For polar questions ( yes/no-questions ), we ll assume that there is an unpronounced question operator Q in the left periphery. This operator will predict that a polar question like Does Seth smoke will denote the set of propositions Seth smokes (Yes) and Seth doesn t smoke (No) a. Sentence: Does Seth smoke? b. LF: [ Q [ Seth smokes ] ] c. Meaning of Q: [[Q]] = { [λp <s,t> : p], [λp <s,t> : [λw : p(w ) = F ]] } d. Predicted Meaning of LF (20b): { [ λw : Seth smokes in w ], [ λw : Seth does not smoke in w ] } yes no (21) Co-Occurrence of Q and Interrogative Pronouns Nothing in our system would prevent the Q-operator in (20c) from co-occurring with a wh-word. a. Interpretable LF: [ Q [ Who smokes ] ] (~ Does who smoke?) The meaning that we predict for such structures, however, is quite bizarre. b. Predicted Answer Set for (21a): { [ λw : Seth smokes in w ], [ λw : Seth does not smoke in w ], [ λw : Barack smokes in w ], [ λw : Barack does not smoke in w ], [ λw : Joe smokes in w ], [ λw : Joe does not smoke in w ], } To avoid this issue, we might just assume that the syntax rules out LFs like (21a)... 8

9 (22) Sketch of Semantics for Other Interrogative Pronouns a. [[ where ]] = Set of locations b. [[ when ]] = Set of times c. [[ why ]] = Set of reasons d. [[ how ]] = Set of manners / adverbial meanings e. [[ how many ]] = Set of numbers / quantities (23) Some Issues Related to Which a. One Perspective on Which-Questions (Karttunen s, not Hamblin s) [[ which student smokes ]] w = { [λw : x smokes in w ] : x is a student in w } b. Key Issue: Due to the technicalities of Hamblin s (1973) system, the perspective in (23a) cannot be captured in his alternative semantics. c. Hamblin s (1973) Proposal for Which-Questions [[ which student smokes ]] = { [λw : x is a student in w and x smokes in w ] : x D e & x is human } o This result can be obtained in his general system (but we ll skip the details) d. Problem, Acknowledged by Hamblin o Intuitively, it feels like if we answer the question Which student smokes with Seth, then we are presupposing that Seth is a student. o In Hamblin s proposal in (23c), however, it comes out that we are asserting that Seth smokes. o In Hamblin s (1973) view, this was OK, because there wasn t yet a formal way of modeling the difference between presupposition and assertion (and so this was good enough for what they could do at the time!) 9

10 3. Deriving Answer Sets Without Alternatives (Karttunen 1977) As mentioned in (5) above, one of the key contributions made by Karttunen (1977) is the following methodological point: (24) Focus on Embedded Interrogatives We don t actually have any linguistic judgments about what kind of semantic object interrogative clauses should denote But we do have linguistic judgments about the truth-conditions of sentences with embedded interrogatives. Therefore, a good way of progressing in our theorizing about interrogative semantics (generally) would be to focus on getting a good theory of embedded interrogatives o Ceteris paribus, we should assume that embedded interrogatives and matrix interrogatives have the same denotation o And so the best theory for embedded interrogatives should determine how we think about matrix interrogatives (25) An Illustration: Hamblin (1973) vs. Wachowicz (1974) a. Hamblin (1973): All questions denote sets of propositions. b. Wachowicz (1974): Questions denote properties. [[ who smokes ]] = [ λx : x smokes ] [[ who ate what ]] = [ λy : [ λx : x ate y ] ] When we ask a question, we are requesting that the addressee provide us arguments (or pairs of arguments) for the predicate that will yield True c. Embedded Questions Help Decide: Any verb that can embed a single wh-question can embed a multiple wh-question In Hamblin s (1973) system, this follows, since all questions are of the same type (sets of propositions) This doesn t directly follow in Wachowicz s (1974) system, since single whquestions and multiple wh-questions are of different types (<et>, and <eet>) 10

11 Consequently, Karttunen (1977) follows Hamblin (1973) in viewing questions as denoting sets of propositions. However (26) Karttunen s (1977) Addition to the Meaning of Questions A question doesn t denote the set of all it s possible (semantic) answers Rather, at a world w, a question denotes the subset of (semantic) answers that are true at world w. a. Hamblin s Semantics for Who smokes? : { p : x. x D e & x is human & p = [ λw : x smokes in w ] } b. Karttunen s Semantics for Who smokes? (at world w): { p : x. x D e & x is human & p = [ λw : x smokes in w ] & p(w) = T } (27) Empirical Motivation for the Addition I do not have a knock-down argument against Hamblin s original proposal; as far as I can see, it could be made to work just as well as my own. However, under my analysis, the meaning of [certain] verbs can be explicated in a more straightforward way (Karttunen 1977: 10) a. Observation: When tell takes a declarative clause as complement, the proposition that was told can be false (i) When tell takes an interrogative clause as complement, the proposition that was told cannot be false (ii) (i) Dave told Mary Michelle smokes. (Dave could have lied) (ii) Dave told Mary who smokes. (Entails that Dave told the truth) b. Analysis: (i) Special Rule for Tell Taking Interrogative Complement: [[ tell CP interrogative ]] w = [ λy : [ λx : p [[ CP interrogative ]] w. [[tell]] w (p)(y)(x) = T ] (ii) Discussion: If [[ CP interrogative ]] w denotes all and only the true answers to the question, then the rule in (27b) would capture observation (27a). We could put the constraint that p(w) = T into the special rule, but Karttunen (1977) says this would miss an important generalization (consider recall, indicate, specify, depend on, matter to, ) 11

12 One other, major difference between Karttunen s (1977) system and Hamblin s (1973) is how the compositional semantics works Important Note: Since Karttunen s (1977) system is written in the framework of classic Montague Grammar, I won t be presenting it to the letter, but instead will be adapting it to our Heim and Kratzerbased formalism Karttunen s analysis for wh-questions builds upon his analysis of alternative questions, and so we ll start with the latter 3.1 Alternative Questions and Polar Questions (28) Illustrative Examples of Alternative Questions a. Did Bill walk to work, or did he drive? b. Does Bill like Sue, or does Mary [like Sue]? c. Does Bill, or Mary, like Sue? (Ambiguous; one reading is an Alt-Q) Syntactically, these questions either look like disjunctions of polar questions, or polar questions formed from disjunctions Semantically, they are not answered by yes or no (unlike polar questions), but instead by asserting one of the disjuncts in the question (29) Polar Questions as Alternative Questions We could view polar questions as a limiting case of alternative question. That is, perhaps the polar question in (29a) is simply elliptical for the alternative question in (29b). a. Does Barack smoke? b. Does Barack smoke, or does he not smoke? (30) Analytic Desideratum As stated above (26), Karttunen assumes that the extension of Q at w is the true answer(s) to Q. Therefore, if Bill walked to work in w, then [[(28a)]] w should be (the characteristic function of) the set { [ λw : Bill walked to work in w ] } And, if Bill drove to work in w, then [[(28a)]] w should be (the characteristic function of) the set { [ λw : Bill drove to work in w ] } 12

13 (31) Key Ingredient: The? Operator There is a phonologically null operator,?, which sits in the left periphery of a question. This operator does not on its own form a question, but is a key ingredient of questions. a. Semantics for? : [[? ]] w = [ λp <st> : [ λq <st> : q(w) = T & q = p ] ] b. Illustration: (i) [[? Barack smokes ]] w = (by IFA, FA) (ii) [[? ]] w ([λw : Barack smokes in w ]) = (by Lexicon) (iii) [ λp <st> : [ λq <st> : q(w) = T & q = p ] ] ([λw : Barack smokes in w ]) = (by LC) (iv) [ λq <st> : q(w) = T & q = [λw : Barack smokes in w ] ] c. Note: If Barack indeed smokes in w, then [[? Barack smokes ]] w will be (the characteristic function of) the set {[λw : Barack smokes in w ]} If Barack doesn t smoke in w, then [[? Barack smokes ]] w will be (the characteristic function of) the empty set. With this?-operator, Karttunen introduces a syntax and semantics for alternative questions (32) The Analysis of Alternative Questions a. Sentence: Did Bill walk to work, or did he drive? b. Proposed LF: [ whether [ [? Bill walked to work ] or [? Bill drove to work ] ] ] c. Proposed Semantic Rule: [[ whether [ [? XP] or [? YP] ] ]] w = d. Illustration: [[(32b)]] w = [ λq <st> : [[? XP ]] w (q) = T or [[? YP ]] w (q) = T ] [ λq <st> : [[? Bill walked to work]] w (q) = T or [[? Bill drove to work]] w (q) = T ] = [ λq <st> : q(w) = T and q = [λw : Bill walked to work in w ] or q(w) = T and q = [λw : Bill drove to work in w ]] 13

14 (33) Discussion a If Bill walked to work (in w), but didn t drive, then [[(32b)]] w is (the characteristic function of) the set { [λw : Bill walked to work in w ] } b. If Bill drove to work (in w), but didn t walk, then [[(32b)]] w is (the characteristic function of) the set { [λw : Bill drove to work in w ] } Thus, as desired, the extension of question (32a) will be the set of its true answers! c. A Potential Problem (Acknowledged by Karttunen): An alternative question like (32a) seems to presuppose that exactly one of the two propositions is true This isn t captured by Karttunen s semantics above o Nothing in the meaning in (32d) would rule out w being a world where both Bill walked to work and he drove to work (somehow) (34) Polar Questions as Alternative Questions a. Sentence: Did Bill walk to work? b. LF: [ whether [ [? Bill walked to work ] or [? Bill didn t walk to work ] ] ] c. Predicted Meaning: [[ (34) ]] w = (by (32c)) [ λq <st> : [[? Bill walked to work]] w (q) = T or [[? Bill didn t drive to work]] w (q) = T ] = (by rules) [ λq <st> : q(w) = T and q = [λw : Bill walked to work in w ] or q(w) = T and q = [λw : Bill didn t walk to work in w ]] d. Discussion: If Bill walked to work (in w), then [[(34c)]] w is (the characteristic function of) {[λw : Bill walked to work in w ]} (which we may consider the yes answer) If Bill didn t walked to work (in w), then [[(34c)]] w is (the characteristic function of) {[λw : Bill didn t walk to work in w ]} (which we may consider the no answer) 14

15 (35) Special Rule for Know With Interrogative Complements [[ know CP interrogative ]] w = [ λx : p. [[ CP interrogative ]] w (p) = T à [[know]] w (p)(x) = T ] Note the similarity to our rule for interrogative-embedding tell o (Nothing in Karttunen s system captures this similarity directly) (36) Illustration: Know Embedding a Polar Question a. Sentence: Dave knows whether Barack smokes. b. Assumed LF: [Dave knows [whether [[? Barack smokes] or [? Barack doesn t smoke] ] c. Predicted Meaning: (i) [[(36b)]] w = T iff (ii) (iii) p. [[ whether [ [? Barack smokes ] or [? Barack doesn t smoke ] ] ]] w (p) = T à [[know]] w (p)(dave) = T p. [ λq <st> : q(w) = T and q = [λw : Barack smokes in w ] or q = [λw : Barack doesn t smoke in w ]](p) = T à [[know]] w (p)(dave) iff iff (iv) p. p(w) = T and p = [λw : Barack smokes in w ] or p = [λw : Barack doesn t smoke in w ] à [[know]] w (p)(dave) If proposition p is true, and is either Barack smokes or Barack doesn t smoke, then Dave knows p Dave knows the true answer to Does Barack smoke? 15

16 3.2 Wh-Questions, Single and Multiple With the semantics for the? -operator in (31) and for know in (35), Karttunen (1977) is able to build a semantics for wh-questions, using the following two ingredients. (37) Semantics for Wh-Words Interrogative pronouns are assumed to have exactly the semantics of indefinites. a. [[which]] w = [[some]] w = [ λp: [ λq: x. P(x) = T & Q(x) = T ] ] b. [[who]] w = [[which person]] w = [ λq: x. x is a person in w & Q(x) = T ] c. [[who]] w = [[which thing]] w = [ λq: x. x is a not a person in w & Q(x) = T ] d. [[when]] w = [[which time]] w = (38) Special Compositional Rule for Wh-Questions [[ wh-phrase [ n [ XP ] ]] w,g = [ λq : [[wh-phrase]] w ([λx : [[XP]] w,g(n/x) (q) = T]) = T ] (39) Illustration: What did Seth eat? a. Assumed LF: [ what [ 1 [? [ Seth eat t 1 ] ] b. Predicted Meaning: [[(39a)]] w = (by (38)) (i) [ λq : [[what]] w ([λx : [[?[Seth eat t 1 ] ]] w,g(1/x) (q) = T]) = T ] = (by (37)) (ii) [ λq : [ λq: x. x is a not a person in w & Q(x) = T ] ([λx : [[?[Seth eat t 1 ] ]] w,g(1/x) (q) = T]) = T ] = (by LC) (iii) [ λq : x. x is a not a person in w & [[?[Seth eat t 1 ] ]] w,g(1/x) (q) = T ] = (by (31a), IFA) (iv) (v) [ λq : x. x is a not a person in w & [ λr <st> : [ λp <st> : p(w) = T & p = r ] ] ([λw : [[Seth eat t 1 ]] w, g(1/x) ])(q) = T ] = (by LC) [ λq : x. x is a not a person in w & q(w) = T & q = [λw : [[Seth eat t 1 ]] w, g(1/x) ] = (by FA, PR) (vi) [ λq : x. x is a not a person in w & q(w) = T & q = [λw : Seth ate x in w ] ] q is a true proposition of the form Seth ate x, where x is non-human 16

17 (40) Some Discussion The rule in (38) is rather complex, and not at all intuitive. The rule stipulates that the meaning of a wh-question will be a set of propositions. The rule also redundantly stipulates the binding of the wh-trace, which ideally should somehow follow from our independent rule of PA (41) Consequences for Embedded Wh-Questions a. Sentence: Dave knows what Seth ate. b. Assumed LF: [ Dave [ knows [ what [ 1 [? [ Seth eat t 1 ] ] c. Predicted Meaning: (Given the ingredients in (31), (35), (37), (38)) p. x. x is a not a person in w & p(w) = T & p = [λw : Seth ate x in w ] à [[know]] w (p)(dave) If p is a true proposition of the form Seth ate x, where x is non-human, then Dave knows p (42) A Crucial Problem for This Semantics a. The Scenario: Seth ate the cake and the pasta, but nothing else. Dave knows that Seth ate the cake, and that he ate the pasta. However, he also wrongly believes that Seth ate the chicken. b. The Problematic Intuition: In scenario (42a), sentence (41a) is false. c. The Problematic Prediction: In scenario (42a), sentence (41a) is true. After all, the truth-conditions predicted in (41c) will hold in scenario (42a) d. Karttunen s Response Perhaps the problematic prediction in (42c) isn t really so bad After all, to capture the intuition in (42b), it seems we d have to strengthen the meaning of what Seth ate, so that it includes all the true propositions of the form Seth didn t eat x But then what Seth ate would end up semantically equivalent to what Seth didn t eat, and that seems wrong too 17

18 (43) Illustration: Which students smoke a. Assumed LF: [ which students [ 1 [? [ t 1 smoke ] ] b. Predicted Meaning: [[(43a)]] w = (by (38)) (i) [ λq : [[which students]] w ([λx : [[?[ t 1 smoke ] ]] w,g(1/x) (q) = T]) = T ] = (by (37)) (ii) [ λq : [ λq: x. x is a student in w & Q(x) = T ] ([λx : [[?[ t 1 smoke ] ]] w,g(1/x) (q) = T]) = T ] = (by LC) (iii) (iv) (v) [ λq : x. x is a student in w & [[?[ t 1 smoke ] ]] w,g(1/x) (q) = T ] = (by (31a), IFA) [ λq : x. x is a student in w & [ λr <st> : [ λq <st> : q(w) = T & q = r ] ] ([λw : [[t 1 smoke]] w, g(1/x) ])(q) = T ] = (by LC) [ λq : x. x is a student in w & q(w) = T & q = [λw : [[t 1 smoke]] w, g(1/x) ] = (by FA, PR) (vi) [ λq : x. x is a student in w & q(w) = T & q = [λw : x smokes in w ] ] q is a true proposition of the form x smokes, where x is a student (44) Consequence for Embedded Which-Questions a. Sentence: Dave knows which students smoke. b. Assumed LF: [ Dave [ knows [ which students [ 1 [? [ t 1 smoke ] ] c. Predicted Meaning: (Given the ingredients in (31), (35), (37), (38)) p. x. x is a student in w & p(w) = T & p = [λw : x smokes in w ] à [[know]] w (p)(dave) If p is a true proposition of the form x smokes, where x is a student, then Dave knows p 18

19 (45) Crucial Problem for This Semantics a. The Scenario: Bill and Jim are the only students who smoke. Seth knows that Bill smokes and that Jim smokes, but he doesn t know that they are students. b. The Problematic Intuition: (i) (ii) There s a way of taking (44a) to be true in scenario (45a), but it s not the most salient reading. The most salient reading of (44a) actually entails that it s false in scenario (45a), precisely because Dave doesn t know who the students are. c. The Problematic Prediction: In scenario (45a), sentence (44a) is univocally true. After all, the truth-conditions predicted in (44c) will hold in scenario (45a) Karttunen offers no solution to this problem It remains a problem until the work of Groenendijk & Stokhoff (1982) One last important observation is that the ingredients above are all sufficient for a compositional analysis of multiple wh-questions (46) Illustration: Who ate what? a. Assumed LF: [ who [ 1 [ what [ 2 [? [ t 1 ate t 2 ] ] b. Predicted Meaning: [[(46a)]] w = (by (38)) (i) [ λq : [[who]] w ([λx : [[what [ 2 [? [ t 1 ate t 2 ] ]] w,g(1/x) (q) = T]) = T ] = (by (38)) (ii) [ λq : [[who]] w ([λx : [ λp : [[what]] w ([λy : [[? [ t 1 ate t 2 ] ]] w, g(1/x)(2/y) (p) = T]) = T] (q) = T]) = T ] = (by Lex.) (iii) [ λq : [ λq: x. x is a person in w & Q(x) = T ] ([λx : [ λp : [ λq: y. y is not a person in w & Q(y) = T ] ([λy : [[? [ t 1 ate t 2 ] ]] w, g(1/x)(2/y) (p) = T]) = T] (q) = T]) = T ] = (by LC) Continued on next page 19

20 (iv) [ λq : [ λq: x. x is a person in w & Q(x) = T ] ([λx : [ λp : y. y is not a person in w & [[? [ t 1 ate t 2 ] ]] w, g(1/x)(2/y) (p) = T] (q) = T]) = T ] = (by LC) (v) [ λq : [ λq: x. x is a person in w & Q(x) = T ] ([λx : y. y is not a person in w & [[? [ t 1 ate t 2 ] ]] w, g(1/x)(2/y) (q) = T] = T ] = (by LC) (vi) (vii) (viii) [ λq : x. x is a person in w & y. y is not a person in w & [[? [ t 1 ate t 2 ] ]] w, g(1/x)(2/y) (q) = T ] = (by Lex., rules) [ λq : x. x is a person in w & y. y is not a person in w & [ λp <st> : [ λq <st> : q(w) = T & q = p ] ] ([λw : x ate y in w ])(q) = T ] = (by LC) [ λq : x. x is a person in w & y. y is not a person in w & q(w) = T & q = [λw : x ate y in w ] ] q is a true proposition of the form x ate y, where x is human and y is non-human (compare to (17)) (47) Karttunen (1977): Some Summary Points a. Karttunen s system provides a semantics for questions that builds on Hamblin s (1973) key idea that questions denote sets of propositions. b. Karttunen s compositional system straightforwardly extends to the case of embedded questions (which is its empirical focus) c. Like Hamblin (1973), Karttunen (1977) easily captures both single and multiple wh-questions. d. Like Hamblin (1973), Karttunen (1977) easily captures polar questions. e. Unlike Hamblin (1973), Karttunen (1977) also captures alternative questions and which-questions f. Outstanding Problems for Karttunen s (1977) System: (i) (ii) Wrongly predicts that Dave knows Q can be true when Dave believes some false answers to Q. Fails to capture the most salient (de dicto) reading of an embedded which question 20