More on Names and Predicates

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Everett Collin Carroll
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1 Lin115: Semantik I Maribel Romero 25. Nov / 2. Dez Lexicon: Predicates. More on Names and Predicates Predicates in NatLg denote functions (namely, schoenfinkelized characterisitc functions of sets): functions of type <e,t> (for intransitive verbs or, more generally, for 1-place predicates) (1) a. [[snore]] w = λx e. [x snores in w] b. [[student]] w = λx e. [x is a student in w] c. [[vegetarian]] w = λx e. [x is vegetarian in w] functions of type <e,<e,t>> (for transitive verbs or, more generally, for 2-place predicates) (2) a. [[kiss]] w = λx e. [λy e. y kisses x in w] DO SU Patient Agent b. [[advisor-of]] w = λx e. [λy e. y advises x in w] c. [[fond-of]] w = λx e. [λy e. y is fond of x in w] d. [[from]] w = λx e. [λy e. y is from x in w] functions of type <e,<e,<e,t>>> (for ditransitive verbs or, more generally, for 3- place predicates) (3) a. [[introduce]] w = λx e. [ λy e. [λz e. z introduces x to y in w] ] DO IO SU Patient Goal Agent 2. Semantic rules (so far). (4) Non-Branching Nodes: If α has the form α, then [[α]] w = [[β]] w. β (5) Functional Application: If α has the form α, then [[α]] w = [[β]] w ([[γ]] w ) or [[α]] w = [[γ]] w ([[β]] w ), whatever is defined. β γ 1

2 3. Sample derivations With one-place predicates. With a VERB: snore (6) S David V snores [[David]] w = d [[]] w = d [[snores]] w = λx e. [x snores in w] [[V]] w = λx e. [x snores in w] [[]] w = λx e. [x snores in w] [[ [David] SNsu [snores] SV ]] w = 1 if and only if [[[snores] SV ]] w ([[[David] SNsu ]] w ) = 1 iff λx e.[x snores in w] (d) = 1 iff d snores in w With an ADJECTIVE. Assumption: the verb to be is semantically vacuous. (7) S David (is) AdjP A vegetarian [[David]] w = [[]] w = d [[vegetarian]] w = [[A]] w = [[AP]] w = [[]] w = λx e. [x is vegetarian in w] [[ [David] SNsu [(is) vegetarian] SV ]] w = 1 iff [[[(is) vegetarian] SV ]] w ([[[David] SNsu ]] w ) = 1 iff λx e.[x is vegetarian in w] (d) = 1 iff d is vegetarian in w 2

3 With a NOUN. Assumption: for the time being, the Determiner a is semantically vacuous. (8) S David (is) NP (a) N student QUESTION 1: Do the semantic computation of (8) step by step. 3

4 3.2. With two-place predicates. With a VERB: kiss (9) S SV David V NPob kisses Frederick [[Frederick]] w = [[NPob]] w = f [[kisses]] w = [[V]] w = λx e. [λy e. y kisses x in w] [[ [kisses] V [Frederick] SNob ]] w = [[[kisses] V ]] w ([[[Frederick] SNob ]] w ) λx e. [λy e. y kisses x in w] (f) λy e. y kisses f in w [[David]] w = [[]] w = d [[[David] SNsu [kisses Frederick] SV ]] w = 1 iff [[[kisses Frederick] SV ]] w ([[[David] SNsu ]] w ) = 1 iff λy e. [y kisses f in w] (d) = 1 iff d kisses f in in With a PREPOSITION and with an ADJECTIVE. (10) S David (is) NP from N Croatia QUESTION 2: Do the semantic computation of (10) step by step. QUESTION 3: Replace from in (10) with fond-of. Do the semantic computation step by step. 4

5 3.3. With three-place predicates. With a VERB: introduce. Assumption: the IO preposition to is semantically vacuous. (11) S David V' PPio introduced NPdo (to) Gina Frederick [[Frederick]] w = [[NPdo]] w = f [[introduced]] w = λx e. [λy e. [λz e.z introduces x to y in w] ] [[introduced [Frederick] SNob ]] w = [[introduced]] w ([[[Frederick] SNob ]] w ) = λx e.[λy e.[λz e.z introduces x to y in w]] (f) = λy e.[λz e.z introduces f to y in w] [[Gina]] w = [[NPio]] w = g [[ [introduced Frederick] V' [(to) Gina] SNio ]] w = [[[introduced Frederick] V' ]] w ( [[[(to) Gina] SNio ]] w ) = λy e.[λz e. z introduces f to y in w] (g) = λz e. z introduces f to g in w [[David]] w = [[]] w = d [[[David] SNsu [introduced Frederick (to) Gina] SV ]] w = 1 iff [[[introduced Frederick (to) Gina] SV ]] w ([[[David] SNsu ]] w ) = 1 iff λz e. [z introduces f to g in w] (d) = 1 iff d introduces f to g in w 5

6 4. Lexicon: the functions and, or, not. (12) S David (is) ConjP AdjP Conj' nice and/or AdjP smart Conjunction and: If we treat predicates intuitively as sets, then and denotes the intersection operation, see (13). If we treat predicates as functions and assume the structure in (12), then and denotes a function that takes the two predicate functions in turn and uses the symbol from Propositional Logic to establish a link between the two, as in (14). (13) [[David]] w ( [[nice]] w [[smart]] w ) (14) [[and]] w = λp <e,t>. [λq <e,t>. [λu e. [P(u) Q(u)]] ] QUESTION 4: Why is the lexical entry (15) ill-formed? (15) [[and]] w = λp <e,t>. [λq <e,t>. P Q ] (16) Partial semantic computation of (12) with and: [[smart]] w = [[AdjP]] w = λx e. x is smart in w [[and [smart] AdjP ]] w = [[and]] w ([[[smart] AdjP ]] w ) = λp <e,t>. [λq <e,t>.[λu e.[p(u) Q(u)]]] (λx e.x is smart in w) = λq <e,t>.[λu e. [(λx e.x is smart in w) (u) Q(u)]] = λq <e,t>.[λu e. [ u is smart in w Q(u)]] [[nice]] w = [[AdjP]] w = λy e. y is nice in w [[nice [and smart] Conj' ]] w = [[[and smart] Conj' ]] w ([[[nice] AdjP ]] w ) = λq <e,t>. [λu e. [ u is smart in w Q(u)]] (λy e. y is nice in w) = λu e. [ u is smart in w (λy e. y is nice in w) (u)] = λu e. [ u is smart in w u is nice in w ] 6

7 Disjunction or If we treat predicates intuitively as sets, then or denotes the union operation, see (17). If we treat predicates as function and assume the structure in (12), then or denotes the function in (18), which uses the symbol from Propositional Logic to establish a link between the two smaller predicate functions, as in (18). (17) [[David]] w ( [[nice]] w [[smart]] w ) (18) [[or]] w = λp <e,t>. [λq <e,t>. [λu e. [P(u) Q(u)]] ] QUESTION 5: Do the semantic computation of (12) with or. Negation not Assumption: Ignore the auxiliary does. (19) S INFL' David (does) not V snore QUESTION 6: What operation does NatLg negation correspond to in set-theoretical terms? Write a schema similar to (13) and (17) with the appropriate set-theoretical operation. QUESTION 7: Now assume that predicates denote functions and define the lexical entry for not in (20). (20) [[not]] w = 7

Introduction to Semantics. Common Nouns and Adjectives in Predicate Position 1

Introduction to Semantics. Common Nouns and Adjectives in Predicate Position 1 Common Nouns and Adjectives in Predicate Position 1 (1) The Lexicon of Our System at Present a. Proper Names: [[ Barack ]] = Barack b. Intransitive Verbs: [[ smokes ]] = [ λx : x D e. IF x smokes THEN