Semantics 1. Gerhard Jäger. May 15, (May 15, 2012) Semantics 1 Gerhard Jäger 1 / 19

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1 emntics 1 My 15, 2012 Gerhrd Jäger (My 15, 2012) emntics 1 Gerhrd Jäger 1 / 19

2 eterminiers Mening of determiner is 3-plce reltion between sitution, two reltions between situtions nd individuls, i.e., the menings of the P nd the respectively logicl determiners:, some: λpλqλs x(p(s,x) Q(s,x)) every, ll: λpλqλs x(p(s,x) Q(s,x)) no: λpλqλs x(p(s,x) Q(s,x)) (My 15, 2012) emntics 1 Gerhrd Jäger 2 / 19

3 eterminer λs. x(student (s, x) sleep (s, x)) P λqλs. x(student (s, x) Q(s, x)) λxλs.sleep (s, x) λpλqλs. x(p(s,x) Q(s,x)) every V λxλsstudent (s, x) λxλs.sleep (s, x) student sleeps (My 15, 2012) emntics 1 Gerhrd Jäger 3 / 19

4 eterminer λs. x(student (s, x) sleep (s, x)) P λqλs. x(student (s, x) Q(s, x)) λxλs.sleep (s, x) λpλqλs. x(p(s,x) Q(s,x)) V λxstudent (s, x) λxλs.sleep (s, x) student sleeps (My 15, 2012) emntics 1 Gerhrd Jäger 4 / 19

5 eterminer λs. x(student (s, x) sleep (s, x)) P λqλs. x(student (s, x) Q(s, x)) λxλs.sleep (s, x) λpλqλs. x(p(s,x) Q(s,x)) no V λxstudent (s, x) λxλs.sleep (s, x) student sleeps (My 15, 2012) emntics 1 Gerhrd Jäger 5 / 19

6 eterminers beyond predicte logic equivlent nottion of the determiners used so fr: 1 every: λp λqλs.({x P(s, x)} {x Q(s, x)}) : λpλqλs.({x P(s,x)} {x Q(s,x)} ) no: λpλqλs.({x P(s,x)} {x Q(s,x)} = ) bsiclly, determiner expresses 2-plce reltion between two sets ({x P(s,x)} nd {x Q(s,x)}) similr ptterns holds for ll determiners: 1 ote tht our met-lnguge is mixture of predicte logic nd set theory. (My 15, 2012) emntics 1 Gerhrd Jäger 6 / 19

7 eterminers beyond predicte logic two: λpλqλs. {x P(s,x)} {x Q(s,x)} 2 t most two: λpλqλs. {x P(s,x)} {x Q(s,x)} 2 exctly two: λpλqλs. {x P(s,x)} {x Q(s,x)} = 2 most: λpλqλs. {x P(s,x)} λx.p(s,x) > {x P(s,x)} {x Q(s,x)} 1 A is the crdinlity of the set A, i.e., the number of its elements. (My 15, 2012) emntics 1 Gerhrd Jäger 7 / 19

8 Quntifier Rising quntifiers in object position re not interpretble with our current mchinery??? P λqλs. x(book (s, x) Q(s, x)) V λyλxλs.red (s, x, y) λpλqλs. x(p(s,x) Q(s,x)) red λxλs.book (s, x) book both P nd denote functions domin of book : two-plce reltion red is three-plce reltion domin of red : individuls book is not n individul, but reltion (My 15, 2012) emntics 1 Gerhrd Jäger 8 / 19

9 Quntifier Rising solution: (one of severl possible solutions): syntx tree is modified before compisitionl interprettion strts originl syntctic structure: -structure derived syntctic structure for semntic interprettion: Logicl Form (LF) trnsition from -structure to LF is governed by trnsformtion rules (My 15, 2012) emntics 1 Gerhrd Jäger 9 / 19

10 Excursus: pronouns nd vribles so fr, interprettion is lwys uniquely determined: α hs unique vlue some expressions, such s pronouns, re context dependent He sleeps. comprble to vribles in predicte logic different occurrences of pronoun need not be co-referent desmbigution vi indices He sees him. He i sees him j. indices re nturl numbers; equl letters represent equl numbers nd different letter for different numbers (My 15, 2012) emntics 1 Gerhrd Jäger 10 / 19

11 Excursus: Pronomen und Vrible interprettion rule for pronouns he i = x i he i sees him j = λs.see (s,x i,x j ) (My 15, 2012) emntics 1 Gerhrd Jäger 11 / 19

12 Quntifier Rising trnsformtion rule Quntifier Rising : 1 replce the P-node α of generlized quntifier by P i 2 replce some -node β tht domintes α in -structure by the configurtion [ α i β] the lower P-node is informlly clled trce nd the trnsformtion itself movement (should be fmilir from yntx 0/yntx 1) sometimes trces re mrked by t (My 15, 2012) emntics 1 Gerhrd Jäger 12 / 19

13 Quntifier Rising interprettion of LF If node P i is lef (i.e., it is trce): P i = x i If [ 1 P i 2 ] is configurtion tht results from Quntifier Rising: 1 = P (λx i. 2 ) ote: This rule is n exception to the principle of type-driven interprettion. (My 15, 2012) emntics 1 Gerhrd Jäger 13 / 19

14 Quntifier Rising λs. x(book (s, x) red (s, p, x)) P λqλs. x(book (s, x) Q(s, x)) λs.red (s,p,x i ) λpλqλs. x(p(s, x) Q(s, x))) λxλsbook (s,x)) book P p V p λyλxλs.red (s, x, y) Peter reds λxλs.red (s,x,x i ) P i x i (My 15, 2012) emntics 1 Gerhrd Jäger 14 / 19

15 Multiple quntifiction A single sentence my contin more thn one quntifier: Every child bought cookie. Every referee shows some tem two red crds. for n quntifiers, we hve n! mny different wys to perform QR up to n! different redings simple exmple Every mn loves womn. (My 15, 2012) emntics 1 Gerhrd Jäger 15 / 19

16 Multiple quntifiction -structure: object rising: P P i every mn V loves P womn P V P i womn every mn loves subject rising (= LF 1): P j every mn P i P i womn V P i loves (My 15, 2012) emntics 1 Gerhrd Jäger 16 / 19

17 Multiple quntifiction -structure: subject rising: P every mn V loves P womn every P j object rising (= LF 2): mn P j V P loves womn P i womn P j P j every mn V P i loves (My 15, 2012) emntics 1 Gerhrd Jäger 17 / 19

18 Multiple quntifiction Interprettion of LF1: λs y(mn (y) x(womn (x) love (s,y,x))) Pj λpλs x(mn (s, x) Q(s, x)) λs. x(womn (s, x) love (s, xj, x)) λpλqλs x(p(s, x) Q(s, x)) every λxλs.mn (s, x) mn Pi λqλs x(womn (s, x) Q(s, x)) λs.love (s, xj, xi) λpλqλs x(p(s, x) Q(s, x)) λxλs.womn (s, x) Pj xj λxλs.love (s, x, xi) womn V λyλxλs.love (s, x, y) Pi xi loves (My 15, 2012) emntics 1 Gerhrd Jäger 18 / 19

19 Multiple quntifiction Interprettion of LF2: λs x(womn (x) y(mn (x) love (s,y,x))) Pi λqλs x(womn (x) Q(s, x)) λs. y(mn (s, y) love (s, y, xi)) λpλqλs x(p(s, x) Q(s, x)) λxλs.womn (s, x) womn Pj λqλs x(mn (s, x) Q(s, x)) λs.love (s, xj, xi) λpλqλs x(p(s, x) Q(s, x)) λxλs.mn (s, x) Pj xj λxλs.love (s, x, xi) every mn V λyλxλs.love (s, x, y) Pi xi loves (My 15, 2012) emntics 1 Gerhrd Jäger 19 / 19