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Seth Cable Formal Semantics Spring 2017 Ling 620 1 Introduction To Seth Cable Formal Semantics Spring 2017 Ling 620 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 ‘Barack smokes’ a. Extension: Truth Value True b. Intension: Proposition [ λw’ : Barack 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 ∈ De & 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 ∈ De & x is not human & p = [ λw’ : Seth cooked x in w’ ] } 1 Seth Cable Formal Semantics Spring 2017 Ling 620 (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 (appropriately) 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, pre- theoretical concept of ‘a possible answer to the question’ 2 Seth Cable Formal Semantics Spring 2017 Ling 620 (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 wh- questions, 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 Seth Cable Formal Semantics Spring 2017 Ling 620 (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 Seth Cable Formal Semantics Spring 2017 Ling 620 (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 Seth Cable Formal Semantics Spring 2017 Ling 620 (15) The Key Ingredients for Interrogative Clauses, Part 1 Interrogative pronouns denote non-singleton sets of entities. a. [[ who ]] = { x : x ∈ De & x is human } = { Barack, Joe, Hillary, Ted, Bernie, Seth, … } b. [[ what ]] = { y : y ∈ De & 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 ∈ De & x is human } } = d. { [λx : [λw’ : x smokes in w’ ]](x) : x ∈ De & x is human } = (by LC) e. { [λw’ : x smokes in w’ ] : x ∈ De & 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 Seth Cable Formal Semantics Spring 2017 Ling 620 d. { f(x) : x ∈ [[who]] and f ∈ {g(y) : g ∈ { [λy : [λx : [λw’ : x cooked y in w’ ]]] } and y ∈ { y : y ∈ De & y is not human } } } = (by set theory) e. { f(x) : x ∈ [[who]] and f ∈ {[λy : [λx : [λw’ : x cooked y in w’ ]]](y) : y ∈ De & y is not human}} = (by LC) f. { f(x) : x ∈ [[who]] and f ∈ { [λx : [λw’ : x cooked y in w’ ]] : y ∈ De & y is not human}} = (by Lexicon) g. { f(x) : x ∈ { x : x ∈ De & x is human } and f ∈ { [λx : [λw’ : x cooked y in w’ ]] : y ∈ De & y is not human}} = (by set theory) h. { [λx : [λw’ : x cooked y in w’ ]](x) : x ∈ { x : x ∈ De & x is human } and y ∈ De & y is not human } = (by LC) i. { [λw’ : x cooked y in w’ ] : x ∈ { x : x ∈ De & x is human } and y ∈ De & y is not human } = (by set theory) i. { [λw’ : x cooked y in w’ ] : x, y ∈ De & 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).
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