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CSSP 2003 The Fifth Syntax and Semantics Conference in Paris October 2-4, 2003 Bare Plurals: Kind-referring, Indefinites, Both, or Neither? Manfred Krifka Humboldt-Universität zu Berlin Zentrum für Allgemeine Sprachwissenschaft (ZAS), Berlin

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Two Interpretations of Bare Noun Phrases Two interpretations of bare NPs (bare plurals, bare mass nouns): »Existential:[[Dogs are barking.]] = w x[DOGS(w)(x) BARKING(w)(x)] [[Gold was found in the river.]] = w x[GOLD(w)(x) FOUND_IN_RIVER(w)(x)] Bare NPs appear to denote indefinite quantifiers based on properties like DOGS, = w x[x are dogs in w] e.g. [[dogs]] = w P x[DOGS(w)(x) P(w)(x)] »Generic:[[Dogs evolved 100,000 years ago.]] = w[EVOLVED_100000_YEARS_AGO(w)(CANIS)] [[Gold is a metal.]] = w[METAL(w)(AUREUM)] Bare NPs appear to be names of kind individuals, e.g. [[dogs]] = CANIS, the kind of dogs. Question: Are bare NPs basically indefinites, kind-referring, or ambiguous?

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Uniform Interpretation as Kinds: Carlson (1977) Interpretation of bare NPs as kind individuals, even in the indefinite interpretation: [[Dogs are barking.]] = w[[[are barking]](w)([[dogs]])] = w [ y x[R(x, y) BARKING(w)(x)](CANIS)] = w x[R(x, CANIS) BARKING(w)(x)] there is an x that is a realization (a specimen) of the kind Canis, and x is barking

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Arguments for Kind-Referring Interpretation First family of arguments (Carlson 1977): Only narrow scope interpretation, in contrast to true indefinites De re / de dicto interpretations: Minnie wants to talk to a psychiatrist. (a particular psychiatrist [de re], or any psychiatrist [de dicto]) Minnie wants to talk to some psychiatrists. (de re, or de dicto) Minnie wants to talk to psychiatrists. (only de dicto). Scope with respect to negation: [[A dog is barking and a dog is not barking. ]] (No contradiction) = w x[DOG(w)(x) BARKING(w)(y)] w x[DOG(w)(x) BARKING(w)(y)] [[Dogs are barking and dogs are not barking.]] (Contradiction) = w [ y x[R(x, y) BARKING(w)(y)](CANIS)] w [ y x[R(x, y) BARKING(w)(y)] (CANIS)] = w x[R(x, CANIS) BARKING(w)(y)] w x[R(x, CANIS) BARKING(w)(y)]

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Arguments for Kind-Referring Interpretation Second family of arguments (Carlson1977, Rooth 1985, Schubert & Pelletier 1987) : Anaphoric reference across kind and object interpretation # At the meeting, some Martians presented themselves as almost extinct. (sortal conflict) At the meeting, Martians presented themselves as almost extinct. (o.k.) # At the meeting, some Martians claimed [PRO to be almost extinct] (sortal conflict) At the meeting, Martians claimed [PRO to be almost extinct] (o.k.) = w[[[ __ claimed [PRO to be almost extinct]](w)([[Martians]])] = w [ y x[R(x, y) CLAIM(w)( w[ALMOST_EXTINCT(w)(y)])(x)](MART.)] = w x[R(x, MART.) CLAIM(w)( w[ALMOST_EXTINCT(w)(MART.)])(x)] There are some specimens of Martians x, and x claimed that Martians (= the kind) are almost extinct.

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Arguments for Indefinite Interpretation Some arguments for Ambiguity Hypothesis (Wilkinson 1991, Gerstner-Link & Krifka 1993) » No definite kind referring NP in episodic sentences The dog / Dogs evolved 100,000 years ago. The dog is barking. Dogs are barking. » Parallel distribution with indefinites in rules-and-regulation statements (Carlson 1995) A gentleman opens doors for ladies. Gentleman open doors for ladies. ?? The gentleman opens doors for ladies. » Parallel distribution with indefinites with respect to non-established kinds (Carlson 1977): The coke bottle / *The green bottle has a narrow neck. (* on kind-referring interpretation) Coke bottles / Green bottles have a narrow neck. A coke bottle / A green bottle has a narrow neck.

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Arguments for Indefinite Interpretation » Non-equivalence e.g. in Italian (Longobardi 2001): Elefanti di colore bianco possono creare grade curiosità. White-colored elephants can create great curiosity.(Indefinite o.k.) *Elefanti di colore bianco sono estinti. White-colored elephants are extinct.(*Kind reference)

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The Theory of Chierchia (1998), Reference to Kinds across Languages Goals: »Account for interpretations of bare NPs by general principles of type shift »Account for differences between languages (Germanic, Romance, Slavic, Chinese) by a system of linguistic types and the presence or absence of overt operators. Claims: »Bare mass nouns always refer basically to kinds. »Bare plurals are basically properties, but they are always shifted to kinds. »Apparently non-kind-referring uses are due to various type shifts.

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Chierchia (1998): Ontological Requirements Domain forms a join semi-lattice, abcd

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Chierchia (1998): Ontological Requirements Domain forms a join semi-lattice, with join operation, abcd b c

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Chierchia (1998): Ontological Requirements Domain forms a join semi-lattice, with join operation, part relation, abcd b c

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Chierchia (1998): Ontological Requirements Domain forms a join semi-lattice, with join operation, part relation, abcd b c

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Chierchia (1998): Ontological Requirements Domain forms a join semi-lattice, with join operation, part relation, abcd b c a b c a b c d

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Chierchia (1998): Ontological Requirements Domain forms a join semi-lattice, with join operation, part relation, set of atoms AT. abcd b c a b c a b c d atoms

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Chierchia (1998): Nominal meanings [[ Extension of singular count noun, in world w: [[dog]](w) = DOG(w), a set of atoms.

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Chierchia (1998): Nominal meanings [[ Extension of plural count noun, in world w: [[dogs]](w) = DOGS(w) = x[ DOG(w)(x) y x[AT(y) DOG(w)(y)]] DOGS is a cumulative property: If DOGS(w)(x) and DOGS(w)(y) then DOGS(w)(x y)

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Chierchia (1998): Nominal meanings Meaning of definite article : DOGS(w) = the maximal individual that falls under DOGS(w) DOGS(w) exists because DOGS(w) is a cumulative predicate.

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Chierchia (1998): Kinds Kinds have a hybrid nature: » They are individual concepts (functions from worlds to individuals) » They are systematically related to properties (applying to the specimens) Mapping of properties to kinds by Down Operator If P is a property, then P = w[ [ P(w)]] Cf. ter Meulen (1980), hybrid nature of mass nouns: » Predicate use, This ring is gold. » Referring use, Gold is a metal.

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Chierchia (1998): Kinds Not every property is related to a kind: » For every world w, P(w) must be defined; this is the case with cumulative properties like DOGS but not with non-cumulative properties like DOG. » Chierchia restricts the down operator further: If P is a property, then P = w[ P(w)], provided this is an element of the set K AT of kinds. (dogs in this building does not correspond to a kind) Note: We must allow for partial properties and individual concepts, otherwise we cannot handle extinct kinds or imaginary kinds, like the dodo or the unicorn. [[dodos]] = w[DODOS(w)], defined only in worlds DODOS = w[ DODOS(w)] in which dodos exist Up operator maps kinds to the property that applies to their specimens: If k K, then k = w x[x k(w)]

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Chierchia (1998): Type Shifting Noun phrase interpretation by type shifting Partees type shifting operators: »Individual type shift :P ==> w P(w) »Existential type shift : P ==> w P x[P(w)(x) P (w)(x)] » Predicational shift BE : w P x[P(w)(x) P (w)(x)] => P Type shift may be indicated overtly, by articles: » Individual type shift: the dog » Existential type shift: a dog, as in a dog barked. Type shift may happen covertly, by coercion: » Predicational type shift: a dog, as in Fido is a dog. » Definite and indefinite interpretation of bare NPs in Slavic. Blocking principle: If a language has an overt operator to express a type shift, it has to be used, i.e. covert type shift is blocked. Chierchias operators as type shifters: »Down shift : P ==> w[ P(w)], if w[ P(w)] K, else undefined. » Up shift : k ==> w x[x k(w)], if k K, else undefined. No generic determiners, hence these shifts are always covert, never blocked.

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Chierchia (1998): Predication Types Regular Kind Predicatios:Dodos are extinct. Characterizing Statements:Lions have a mane. Derived Kind Predications:Dogs are barking.

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Chierchia (1998): Regular Kind Predications - With bare mass terms:Gold is a metal. w[METAL(w)(AUREUM)] - With bare pluralsDodos are extinct. w[EXTINCT(w)( DODOS) Mass terms are names of kinds, bare plurals are basically properties that are shifted to kinds by bare singulars cannot be shifted, hence *Dodo is extinct.

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Chierchia (1998): Characterizing Statements Characteristic Statements Kind predication Dogs have a tail.Dogs evolved 100,000 years ago. A dog has a tail.*A dog evolved 100,000 years ago. (taxonomic reading o.k.) Treatment of characterizing statements by dyadic generic operator (Krifka e.a. 1995): [[A dog has a tail]] = w[GEN(w) ( w x[DOG(w)(x)]) ( w x y[TAIL(w)(y) HAS(w)(y)(x)])] Characterizing statements with bare NPs: - With bare mass termsGold is shiny. w[GEN(w)( AUREUM)(SHINY)] - With bare pluralsLions have a mane. w[GEN(w)( LION)(HAVE_A_MANE)] (not w[GEN(w)(LIONS)(HAVE_A_MANE), as this would also allow for *Lion has a mane w[GEN(w)(LION)(HAVE_A_MANE) - but how could this derivation be prevented?)

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Chierchia (1998): Derived Kind Predications Example: Dogs are barking. DKP rule: If the verbal predicate P basically applies to objects, and k denotes a kind, then interpret w[P(w)(k)] as w x[ k(w)(x) P(w)(x)] Dogs are barking. * w[BARKING(w)( DOGS)], not interpretable due to sort mismatch = w x[ DOGS(w)(x) BARKING(w)(x)], by DKP rule Narrow scope interpretation, if DKP rule is triggered locally: John didnt see dogs. LF: [dogs 1[John didnt see t 1 ]] (style of Heim & Kratzer 1998) interpretation (after type shift DOGS ==> DOGS): w [ x[ [SEE(w)(x)(JOHN)]]( DOGS)] after application: w [ [SEE(w)( DOGS)(JOHN)]] local application of DKP: w x[ DOGS(w)(x) SEE(w)(x)(JOHN)] Notice: x has arrow scope over due to local triggering of DKP rule

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Chierchia (1998): Problems with the DKP rule Problems of the DKP rule: » DKP rule not couched in type shift format Remedy: Assume a sequence of type shifts, DOG ==> DOGS ==> DOGS ==> DOGS ==> DOGS pluralization type requirement DKP-rule DKP-rule » But now some type shifts are unmotivated: - Shift DOGS ==> DOGS unmotivated, as the resulting structure is not interpretable » There is a simpler derivation in which every step is motivated: DOG ==> DOGS ==> DOGS pluralization type requirement Dogs are barking. * w[BARKING(w)(DOGS)] -- type clash! after existential shift: w x[DOGS(w)(x) BARKING(w)(x)] Chierchia argues that existential shift is dispreferred because it has existential impact (i.e. a more specific meaning). But even Chierchias DKP type shift sequence has existential impact!

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A Revised Type Shift Theory for Bare NPs Goals: » Assume locally coerced type-shifting and blocking principle. » Replace DKP rule type shifting in accordance with general principles. » Give semantics for regular kind predications, characterizing statements and non-generic statements. » Account for differences between languages.

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Type Shifts and Interpretation Shakenbake Semantics (Emmon Bach). Convention: {A, B} = A(B) or B(A), whatever is well-formed. Interpretation of binary branching constituents [ ]: [[[ ]]] = w[{ [[ ]](w), [[ ]](w)]}] or w[{ [[ ]], [[ ]](w)]}], w[{ [[ ]](w), [[ ]]]}], w[{ [[ ]], [[ ]]]}], whatever is well-formed If this fails: [[[[ ]]] = w[{ TS[[ ]](w), [[ ]](w)]}] or w[{ [[ ]](w), TS[[ ]](w)]}], where TS is a possible type shift operation not blocked by overt operators If this fails: Iterate the last step (i.e. apply more type shifts) Important type shifts: Max Individual : Predicate P ==> P Existence : Predicate P ==> P x[P(x) P(x)] Property BE:Existential quantifier P x[P(x) P(x)] ==> P Kind :Property P ==> P, = w[ P(w)]

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Semantics of Count Nouns Krifka (1995), comparative study of English / Chinese Mass nouns are properties of individuals [[gold]] = GOLD, = w x[GOLD(w)(x)] Count nouns are relations between numbers and individuals [[dog]] = DOG = w n x[DOG(w)(n)(x)] The number argument can be filled by a number word: [[one dog]]= w[[[dog]](w)([[one]](w))] = w[ n x[DOG(w)(n)(x)](1)] = w x[DOG(w)(1)(x)] Count noun relations are extensive measure functions: - If DOG(w)(n)(x) and DOG(w)(m)(x), then n = m - If DOG(w)(n)(x) and DOG(w)(m)(y) and x, y do not overlap, i.e. z[z x z y] then DOG(w)(n+m)(x y) With this, DOG(w)(n) is a quantized predicate, i.e. if DOG(w)(n)(x) and y

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Number Agreement within NP Potential problem of this theory of count nouns: one dog, but two dogs. But this may be just syntactic/morphological agreement: one, a, every: singular agreement two, three, many, few, all: plural agreement This agreement is semantically irrelevant: Decimal fractions induce plural agreement, even with one point zero: American households have, on average, zero point seven cat-s and one point zero dog-s. Many languages with nominal plural lack agreement, e.g. Hungarian egy kutya két kutya kutyák a kutya a kutyák one dog two dog dog-s the dog the dog-s

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Semantically relevant number In bare plurals and definite plurals in English (or Hungarian), number is relevant, it existentially quantifies over the number argument. [[dog-s]] = DOGS = w x n[DOG(w)(n)(x)] The number n is unrestricted: Did you eat apples? Yes, one. / *No, one. Scalar implicature forces number choice in cases like This is an apple. (vs. These are apples). Semantically relevant singular in bare singulars, e.g. Slavic languages like Czech: [[pes]] = w x[DOG(w)(1)(x)]

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Treatment of Articles Indefinite article:[[a]] = w R P x[R(1)(x) P(x)] [[a dog]] = w[{[[a]](w), [[dog]](w)}] = w[[[a]](w)([[dog]](w))] = w P x[DOG(w)(1)(x) P(x)] Combination with VP:[[[[a dog] [is barking]]]] = w[[[a dog]](w)([[is barking]](w))] = w[ P x[DOG(w)(1)(x) P(x)](BARKING(w))] = w x[DOG(w)(1)(x) BARKING(w)(x)]

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Existential type shift with bare NPs Bare NPs in episodic sentences: [[[dogs [are barking]]]] = w[{[[dogs]](w), [[are barking]](w)}], functional application impossible existential type shift: w[{[ [[dogs]]]](w), [[are barking]](w)}] = w[ P x n[DOG(w)(n)(x) P(x)](BARKING(w))] = w x n[DOG(w)(n)(x) BARKING(w)(x)]

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Existential type shift leads to narrow scope Dogs arent barking LF: [dogs 1[arentt [t 1 barking]]] Interpretation: [[[dogs 1[arentt [t 1 barking]]]]] = w[{[[dogs]](w), [[ 1[arentt [t 1 barking]]]](w)}] = w[{[[dogs]](w), 1 [[[arent [t 1 barking]]]]}] = w[{[[dogs]](w), 1 [[[arent [t 1 barking]]] t 1 1 (w)] = w[{[[dogs]](w), 1 [ [{BARKING(w), 1 }]]}] = w[ 1 [ [{BARKING(w), 1 }]([[dogs]](w))] = w[ [{BARKING(w), [[dogs]](w)}]] type shift necessary at this point; existential shift only option: = w[ [{BARKING(w), [[[dogs]]](w)}]] = w [ [[[dogs]]](w)(BARKING(w))] = w [ P n[DOG(w)(n)(x) P(w)(x)](BARKING(w))] = w x n[DOG(w)(n)(x) BARKING(w)(x)] Local triggering of type shift leads to narrow scope.

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Existential type shift leads to narrow scope Why not *Dog is barking? 1. Because [[dog]] is not a property, but a relation between numbers and individuals. No type shift defined for such relations. 2. Even if [[dog]] were a property, or a type shift by specifying the number as 1 were defined, Existential type shift is blocked by indefinite article, a. Why no type shift to a definite interpretation? This is blocked by the overt definite article, the

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Some vs. bare NPs Why is existential type shift of dogs not blocked by some, e.g. Some dogs are barking? Because some does not just express existence, it triggers specific interpretations, they can be captured by choice functions. Example: Some dogs arent barking. w ƒ[ [BARKING(ƒ( x n[DOG(w)(n)(x)])] equivalent to: w x[ n[DOG(w)(n)(x) BARKING(w)(x)]] Derivation of reading: [[some dogs]] = ƒ( x n[DOG(w)(n)(x)]) Choice function variable ƒ is existentially bound at certain positions. Specific reading does not necessarily mean wide scope, cf. Every student read some book Choice function is bound under the scope of every, cf. Abusch (1993). Specific reading / choice function interpretation excludes characterizing interpretation: Some dogs bark. Characterizing only under taxonomic interpretation.

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Wide-scope bare NPs Wide-scope reading of certain bare NPs observed Carlson (1977). Parts of that machine arent working. The police is looking for persons in this building. Chierchia (1998): These NPs do not correspond to kinds, hence existential type shift with the option for wide-scope interpretation: If kind type shift is ruled out, existential type shift becomes an option. Explanation within current theory: » Assume an existential type shift CF with choice function interpretation: P ==> ƒ(P), with ƒ a choice function, to be bound existentially. » This type shift is blocked by overt some. » However, some can have a partive interpretation (roughly, when the head N refers to a finite or given set): some parts of that machine means: some (but not all) parts some persons in this building means: some (but not all) persons i.th.b. » In these cases, some does not block choice function type shift, hence wide-scope interpretation of bare NPs is possible: The police is looking for persons in this building.

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Regular Kind Predications Genuine kind predications: Dodos are extinct. Assume type shift by, following Chierchia: [[Dodos are extinct]] = w[{EXTINCT(w), w x n[DODO(w)(n)(x)]}] type shift required: w[EXTINCT(w)( w x n[DODO(w)(n)(x)])] Why not *Dodo became extinct? Because is not defined only for properties, not for relations between numbers and entities, like DODO. Why not existential type shift? Because the result would violate sortal restrictions: EXTINCT is defined for kind individuals, not for objects.

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Characterizing statements Characterizing statements with bare NPs: Dogs bark. No type shift required, as we need a predicate in the restrictor: w[GEN(w)( w x n[DOG(w)(n)(x)])( w x[BARK(w)(x)])] Why not *Dog barks? Again, because DOG is a relation, not a property; we need properties for specifying the restrictor. How to derive characterizing statements with singular indefinites, like A dog barks or A lion has a mane? Recall: Indefinite article leads to quantifier interpretation, [[a dog]] = w P x[DOG(w)(1)(x) P(x)]. Shifting to a property interpretation by type shift BE: w[GEN(w) ( w[BE[ P x[DOG(w)(1)(x) P(x)]]])( w x[BARK(w)(x)])] = w[GEN(w) ( w x[DOG(w)(1)(x)])( w x[BARK(w)(x)])]

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Reflexive and control anaphora At the meeting, Martians claimed [PRO to be almost extinct] *At the meeting, some Martians claimed [PRO to be almost extinct] Martians 1 claimed [PRO 1 to be almost extinct]. w[{[[Martians]](w), [[claimed]](w)({[[PRO 1 ]] PRO 1 [[Martians]] (w), [[almost extinct]](w)}])}] = w[{[[Martians]](w), [[claimed]](w)([{[[Martians]](w), [[ almost extinct]](w)}])}] type mismatch (twice) with [[Martians]], requiring type shifts by and : = w[{ [[Martians]](w), [[claimed]](w)([{ [[Martians]](w), [[almost extinct]](w)}])}] = w[ [[Martians]](w)(CLAIMED(w)(ALMOST_EXTINCT(w)( [[Martians]]))] = w[ P x[ n[MARTIAN(w)(n)(x) P(x)] (CLAIMED(w)(ALMOST_EXTINCT(w)( w y n[MARTIAN(w )(n)(y)])))] = w x[ n[MARTIAN(w)(n)(x) CLAIMED(w)(ALMOST_EXTINCT(w)( w y n[MARTIAN(w )(n)(y)]))(x)]

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The role of information structure: Characterizing Statements Information structure in characterizing statements (Rooth 1995, Krifka 1995, 2001) Frenchmen wear a BERET. w[GEN(w)( w x[FRENCHMEN(w)(x)]) ( w x y[BERET(w)(y) WEAR(w)(y)(x)] FRENCHmen wear a beret. w[GEN(w)( w y[BERET(w)(y)]) ( w y x[FRENCHMEN(w)(x) WEAR(w)(y)(x)] Analysis by Krifka (2001): Restrictor must be deaccented, topical. Possible explanation of complexity requirement in Romance languages: NPs must be heavy enough to realize topic accent: Elefanti di colore bianco possono creare grande curiosità. *Elefanti possono creare grande curiosità.(Longobardi 2001)

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The role of information structure: Stage level and Individual level predicates Influence of episodic / stative contrast (cf. Carlson 1977, stage level and individual level predicates). Dodos walked towards the sailors. (episodic => non-generic) Dodos liked to eat grass. (stative => generic) Analysis by Erteshik-Shir & Cohen (2001): » Every sentence must have a topic. » In episodic sentences, a possible topic is the situation talked about. » Stative situations dont refer to a situation talked about, so something else must be the topic, this can be interpreted as the restrictor of a generic statement.

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The role of information structure in kind reference Shift by down operator only if NP can count as a topic, by its position and accent. Transistors were invented by Shockley. (kind-referring o.k,) Shockley invented transistors. (only taxonomic reading) Shockley invented the transistor.(kind-referring o.k.) Kind-referring interpretation of bare singulars only in topic position: Hindi (Dayal 1992), Brazilian Portuguese (Schmitt & Munn 1999), Hebrew (Doron 2003) namer / ha-namer hitara kan. tiger / DEF-tiger struck-roots here. The tiger became indigenous here profesor li xoker et ha-namer. professor Li investigates OBJ DEF-tiger Professor Li investigates the tiger (specific animal, or species). professor li xoker namer. Professor Li investigates a tiger. (only non-kind-referring, or taxonomic)

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Conclusion Bare NPs: Indefinites, Kind-referring, Both, or Neither? Answer: All of the above. They are basically neither indefinites nor kind-referring, but properties: [[dogs]] = w x n[DOG(w)(n)(x)] [[gold]] = w x[GOLD(w)(x)] But they can be shifted to indefinites: [[dogs]] = w P x[[[dogs]](w) P(x)] And they can be shifted to kind-referring NPs: [[dogs]] = [[dogs]] Hence, they are both kind-referring and indefinites. With nouns referring to finite set, they also can be shifted to choice function interpretation: CF[[persons in this building]] = w[ƒ([[persons in this building]](w))], where ƒ is bound existentially. Additional type shifts, e.g. bridging (Condoravdi 1992): A serial killer was haunting the campus. Students were aware of the danger.

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Appendix: Definite Generic NPs Examples: The dog evolved 100,000 years ago. The dodo is extinct. The lion has a mane. Such NPs do not refer to the same kind as bare plurals (Chierchia 1998, following Kleiber 1989) Lions are numerous. *The lion is numerous. Assume that definite generic NPs refer to atomic individuals that are related to plural kinds via operator Kind interpretation of lions: [[lions]]= w x n[LION(w)(n)(x)], = w x n[LION(w)(n)(x)], an individual concept, the function from worlds w that picks out the maximal individual that falls under the predicate lions in w. Kind interpretation of the lion: [[the lion]] = LIONS, = LEO LEONIS, an atomic individual of the sort kind.

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Appendix: Definite Generic NPs Type shift to specimens: If k is a kind, then Sk = w x[SPECIM(w)(k)(x)] Characterizing predications via membership relation: The lion (usually) has a mane. wGEN(w) (S[[the lion]])([[has a mane]])] No simple episodic sentences because this would require two type shifts: S and, and this reading can be achieved in simpler ways. [[The lion approached us.]] = w[{ S[[[[the lion]](w), [[approached us]](w)}] more complex than [[Lions approached us.]] = w[{ [[[[lions]](w), [[approached us]](w)}] Treatment of predicates like be rare as event-related: A tiger is rare. To encounter a tiger is rare. Type shift by BE: w[RARE(w)(BE[[a tiger]])] The tiger is rare. To encounter a specimen of the tiger is rare. Type shift by S: w[RARE(w)(S[[the tiger]])]

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The paper to this talk is to be published in the proceedings of SALT XIII and can be downloaded at: www.amor.rz.hu-berlin.de/~h2816i3x

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CS 285- Discrete Mathematics Lecture 4. Section 1.3 Predicate logic Predicate logic is an extension of propositional logic that permits concisely reasoning.

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