Moreno Mitrović. The Saarland Lectures on Formal Semantics

Transcription

1 ,, 3 Moreno Mitrović The Saarland Lectures on Formal Semantics

2 λ-

3 λ-

4 λ- ( λ- ) Before we move onto this, let's recall our f -notation for intransitive verbs 1/33

5 λ- ( λ- ) Before we move onto this, let's recall our f -notation for intransitive verbs (1) Ann snores = snores (Ann) = (iff Ann actually snores) What about transitive verbs? 1/33

6 λ- ( λ- ) Before we move onto this, let's recall our f -notation for intransitive verbs (1) Ann snores = snores (Ann) = (iff Ann actually snores) What about transitive verbs? (2) Ann loves Jan = (two notations) a loves (Ann)(Jan) = (iff Ann actually loves Jan) 1/33

7 λ- ( λ- ) Before we move onto this, let's recall our f -notation for intransitive verbs (1) Ann snores = snores (Ann) = (iff Ann actually snores) What about transitive verbs? (2) Ann loves Jan = (two notations) a loves (Ann)(Jan) = (iff Ann actually loves Jan) b loves (Ann, Jan) = (iff Ann actually loves Jan) 1/33

8 λ- ( λ- ) Imagine we were interpreting an expression containing just the two words: noun Maggie and verb love(s) We first need to construct a tree In our case, there are two possible trees since something is missing (3) (4) Maggie λx (Mary, x ) loves denotes the characteristic function of the set of individuals that Maggie loves?? loves λx ( x, Maggie) denotes the characteristic function of the set of individuals that love Maggie Maggie 2/33

9 Very loosely, a λ-formula specifies the conditions that need to be met under which the function is true 3/33

10 Very loosely, a λ-formula specifies the conditions that need to be met under which the function is true A verb like smoke makes sense (it is or can be true) only if there a single argument which can saturate its meaning 3/33

11 Very loosely, a λ-formula specifies the conditions that need to be met under which the function is true A verb like smoke makes sense (it is or can be true) only if there a single argument which can saturate its meaning (5) smokes = 3/33

12 Very loosely, a λ-formula specifies the conditions that need to be met under which the function is true A verb like smoke makes sense (it is or can be true) only if there a single argument which can saturate its meaning (5) smokes = (6) λx smokes (x) = 3/33

13 Very loosely, a λ-formula specifies the conditions that need to be met under which the function is true A verb like smoke makes sense (it is or can be true) only if there a single argument which can saturate its meaning (5) smokes = (6) λx smokes (x) = The last notation can be read 'if there was an x, smoke could be true' 3/33

14 If φ is an expression denoting a function, and x is an expression that is of the right type to be used as an argument to φ, then φ(x) denotes the result of applying φ to x (saturation) 4/33

15 If φ is an expression denoting a function, and x is an expression that is of the right type to be used as an argument to φ, then φ(x) denotes the result of applying φ to x (saturation) For example Expression (x) denotes the result of applying the function denoted by bored to the value of x 4/33

16 5/33

17 Another example (7) [λx (Maggie)(x)] 5/33

18 Another example (7) [λx (Maggie)(x)](Bill) 7 denotes the result of applying the function is loved by Maggie to Bill This is then equivalent to (8) (8) (Maggie)(Bill) where Bill replaced the placeholder x 5/33

19 Another example (7) [λx (Maggie)(x)](Bill) 7 denotes the result of applying the function is loved by Maggie to Bill This is then equivalent to (8) (8) (Maggie)(Bill) where Bill replaced the placeholder x This 'conversion' process is known as β-conversion or β-reduction 5/33

20 λ- : We all remember formulae like (9) from high school (9) f(x) = x + a Now let x = b Then we have: f(x) = x + f( ) = + (9) is the same as (11) (10) a f(x) = x + λxx + b [λxx + ] 6/33

21 λ- : We all remember formulae like (9) from high school (9) f(x) = x + a Now let x = b Then we have: f(x) = x + f( ) = + (9) is the same as (11) (10) a f(x) = x + λxx + b [λxx + ]( ) 6/33

22 λ- : We all remember formulae like (9) from high school (9) f(x) = x + a Now let x = b Then we have: f(x) = x + f( ) = + (9) is the same as (11) (10) a f(x) = x + λxx + b [λxx + ]( ) + 6/33

23 λ- : (11) a f(x) = x + λxx + b [λxx + ] 7/33

24 λ- : (11) a f(x) = x + λxx + b [λxx + ]( ) 7/33

25 λ- : (11) a f(x) = x + λxx + b [λxx + ]( ) + That's all λ-abstraction is: abstraction with a λ-clause specifies the conditions under which the value description (11a) β-reduction (β-conversion), reduces or converts the variable x into whatever value we feed it in our case, number 5 7/33

26 ? 7/33

27 7/33

28 : λ- (12) {x N x } iff (13) Massachusetts {x D California is a western state} = D iff California is a Western state (14) {x D California is a western state} = D if California is a western state (15) {x D California is a western state} = if California is not a western state (16) {x N x } = {y N y } 8/33

29 : λ- (17) [λx D[λy D[λz Dz introduced x to y]]](ann)(sue) (18) [λx N[λy Ny > and y < ](x)]]] 9/33

30 λ- We've written λ-functions in the following way: (19) λx φ ''We need x to saturate φ'' (x 'completes' φ) We can write a λ-abstracted function in more detail: (20) λα γ φ 10/33

31 λ- (21) λα γ φ The colon here provides the domain condition and the full stop the value description For instance, this applies to (23a), which we can reduce to (23b) (22) a λx x D e x b λx D e x 11/33

32 λ- (23) a λx x D e x ''the function which maps every x, such that x is an individual, to if x smokes, and to otherwise'' ''any x, such that x is an individual, is suitable to make the function true if x smokes and untrue if x does not smoke'' ''find an x, such that x is an individual, which can make the function true if x smokes and untrue if x does not smoke b λx D e x ''map any x which is an individual to if x smokes, and to otherwise'' 12/33

33 λ So far, we've dealt with examples where a λ is needed to 'meet the condition' of a truth-value denoting predicate (like smoke) It doesn't have to 13/33

34 λ : (24) Read [λα γ φ] as either (i) or (ii), whichever makes sense i ''the function which maps every α, such that γ, to 1, if φ, and to 0 otherwise'' (eg, intr-v give an ex) ii ''the function which maps every α, such that γ, to φ'' (eg, tr-v give an ex) 14/33

35 λ- (25) [λf f D e,t there is some x D e such that f(x) = ] 15/33

36 λ- (25) [λf f D e,t there is some x D e such that f(x) = ] Can you think of an example that would work with (25)? 15/33

37 λ- λ

38 A transitive verb like likes takes two arguments, hence is a 2-place function In λ-abstracted form, it looks something like (26) (26) likes = [λx D[λy D y likes x]] We know both x, y are in D so we could drop '' D'' Since want to be precise, we are not going to 16/33

39 S NP N Ann V VP likes N N Bill 17/33

40 S NP N Ann V VP likes N N Bill 17/33

41 S NP Ann Ann V VP likes N N Bill 17/33

42 S Ann Ann Ann V VP likes N N Bill 17/33

43 S Ann Ann Ann V VP likes N N Bill 17/33

44 S Ann VP Ann Ann V [λx D[λy D y likes x]] N Bill Bill 17/33

45 S Ann VP Ann Ann V [λx D[λy D y likes x]] Bill Bill Bill 17/33

46 S Ann VP Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

47 S Ann VP Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

48 S Ann VP Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

49 Ann S [λx D[λy D y likes x]](bill) Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

50 Ann S [λx D[λy D y likes x]](bill) Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

51 S Ann [λy D y likes Bill] Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

52 S Ann [λy D y likes Bill] Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

53 S Ann [λy D y likes Bill] Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

54 [λy D y likes x](ann) Ann [λy D y likes Bill] Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

55 [λy D y likes x](ann) Ann [λy D y likes Bill] Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

56 Ann Ann likes Bill [λy D y likes Bill] Ann Ann [λx D[λy D y likes x]] [λx D[λy D y likes x]] Bill Bill Bill 17/33

57 So far, the left-most λ-term combined with the right-most argument (in brackets) When drawing trees, this is much easier, since a λ-term will apply only to its syntactic sister (locally) 18/33

58 So far, the left-most λ-term combined with the right-most argument (in brackets) When drawing trees, this is much easier, since a λ-term will apply only to its syntactic sister (locally) Think of the problem with the example below (27) λx D [λy D y loves x](sue) 18/33

59 So far, the left-most λ-term combined with the right-most argument (in brackets) When drawing trees, this is much easier, since a λ-term will apply only to its syntactic sister (locally) Think of the problem with the example below (27) λx D [λy D y loves x](sue) a [λx D [λy D y loves x](sue)] = λx D Sue loves y b [λx D [λy D y loves x]](sue) = λy D y loves Sue 18/33

60 18/33

61 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) 19/33

62 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] 19/33

63 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] (30) [λx D [λy D [λz D z introduced x to y](ann)]](sue) 19/33

64 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] (30) [λx D [λy D [λz D z introduced x to y](ann)]](sue) (31) [λx D [λy D [λz D z introduced x to y]](ann)](sue) 19/33

65 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] (30) [λx D [λy D [λz D z introduced x to y](ann)]](sue) (31) [λx D [λy D [λz D z introduced x to y]](ann)](sue) (32) [λf D e,t [λx D e f(x) = and x is grey]] (λy D e y is a cat) 19/33

66 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] (30) [λx D [λy D [λz D z introduced x to y](ann)]](sue) (31) [λx D [λy D [λz D z introduced x to y]](ann)](sue) (32) [λf D e,t [λx D e f(x) = and x is grey]] (λy D e y is a cat) (33) [λf D e, e,t [λx D e f(x)(ann) = ]] ([λy D e [λz D e z saw y]]) 19/33

67 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] (30) [λx D [λy D [λz D z introduced x to y](ann)]](sue) (31) [λx D [λy D [λz D z introduced x to y]](ann)](sue) (32) [λf D e,t [λx D e f(x) = and x is grey]] (λy D e y is a cat) (33) [λf D e, e,t [λx D e f(x)(ann) = ]] ([λy D e [λz D e z saw y]]) (34) [λx N [λy N y > and y < ](x)] 19/33

68 (28) [λx D [λy D [λz D z introduced x to y]]](ann)(sue) (29) [λx D [λy D [λz D z introduced x to y](ann)](sue)] (30) [λx D [λy D [λz D z introduced x to y](ann)]](sue) (31) [λx D [λy D [λz D z introduced x to y]](ann)](sue) (32) [λf D e,t [λx D e f(x) = and x is grey]] (λy D e y is a cat) (33) [λf D e, e,t [λx D e f(x)(ann) = ]] ([λy D e [λz D e z saw y]]) (34) [λx N [λy N y > and y < ](x)] (35) [λz N [λy N [λx N x > and x < ](y)](z)] 19/33

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70 20/33

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72 Some lexical items make no semantic contribution to the structures in which they occur 21/33

73 Some lexical items make no semantic contribution to the structures in which they occur Can you think of some? 21/33

74 Some lexical items make no semantic contribution to the structures in which they occur Can you think of some? (36) a of John = John b be rich = rich c a cat = cat 21/33

75 Some lexical items make no semantic contribution to the structures in which they occur Can you think of some? (36) a of John = John b be rich = rich c a cat = cat What is the meaning of items of, be, and a above so that the equivalences hold? (3mins) 21/33

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77 Not only verbs are functions 22/33

78 Not only verbs are functions So are predicates (we discussed this yesterday) 22/33

79 Not only verbs are functions So are predicates (we discussed this yesterday) Can you think of any 1-place (monadic) non-verbal predicates? Can you think of any 2-place (dyadic) non-verbal predicates? 22/33

80 Not only verbs are functions So are predicates (we discussed this yesterday) Can you think of any 1-place (monadic) non-verbal predicates? Can you think of any 2-place (dyadic) non-verbal predicates? Monadic: a cat b student c bored d gray 22/33

81 Not only verbs are functions So are predicates (we discussed this yesterday) Can you think of any 1-place (monadic) non-verbal predicates? Can you think of any 2-place (dyadic) non-verbal predicates? Monadic: a cat b student c bored d gray Dyadic: a part b fond c in d from 22/33

82 Not only verbs are functions So are predicates (we discussed this yesterday) Can you think of any 1-place (monadic) non-verbal predicates? Can you think of any 2-place (dyadic) non-verbal predicates? Monadic: Dyadic: a cat a part b student b fond c bored c in d gray d from What are the meanings of each of these? 22/33

83 : What is the denotation of in Texas? 23/33

84 : What is the denotation of in Texas? Let's calculate the truth-conditions of the following: (37) a Joes is in Texas b Joe is fond of Kaline c Kaline is a cat 23/33

85 What about the meaning of a grey cat? Try composing it (2mins) 24/33

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87 : We need a new rule that can handle cases where Functional Application (FA) cannot apply Predicate Modification (PM) If α is a branching node, {β, γ} is the set of α's daughters, and β and γ are both in D e,t, then α = λx D e β (x) = γ (x) = 25/33

88 : We need a new rule that can handle cases where Functional Application (FA) cannot apply Predicate Modification (PM) If α is a branching node, {β, γ} is the set of α's daughters, and β and γ are both in D e,t, then α = λx D e β (x) = γ (x) = (= λx D e x is β and x is γ ) 25/33

89 We can now calculate the truth conditions of (38) (38) city in Texas = 26/33

90 We can now calculate the truth conditions of (38) (38) city in Texas = (39) Denver is a city in Texas 26/33

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92 Common nouns, like 'cat', denote the characteristic functions of sets of individuals (What type are they, again?) 27/33

93 Common nouns, like 'cat', denote the characteristic functions of sets of individuals (What type are they, again?) What does this imply for the semantic analysis of determiners? Let's just focus on the for now 27/33

94 Basic intuition: 28/33

95 Basic intuition: ''the NP'' denotes individuals, just like proper names student denotes a ( f of a) set of students the student denotes a unique individual (Russell, cf Frege who thought so too) (40) a the ( German chancellor ) =Angela Merkel b the ( opera by Beethoven ) =Fidelio c the ( negative square root of 4 ) = 28/33

96 What, then, does the denote? 29/33

97 What, then, does the denote? The is a function What is its domain? Its range? 29/33

98 What, then, does the denote? The is a function What is its domain? Its range? (41) For ant f D e,t such that there is exactly one x for which f(x) =, the (f) = the unique x for which f(x) = 29/33

99 What about fs that do not maps exactly one individual to 1? What would the yield? (42) the escalator in South College (43) the stairway in South College 30/33

100 What about fs that do not maps exactly one individual to 1? What would the yield? (42) the escalator in South College (43) the stairway in South College NB There are no escalators in South College and there is more than one stairway 30/33

101 What about fs that do not maps exactly one individual to 1? What would the yield? (42) the escalator in South College (43) the stairway in South College NB There are no escalators in South College and there is more than one stairway What objects do (44) and (45) denote? 30/33

102 What about fs that do not maps exactly one individual to 1? What would the yield? (42) the escalator in South College (43) the stairway in South College NB There are no escalators in South College and there is more than one stairway What objects do (44) and (45) denote? 30/33

103 (44) the escalator in South College (45) the stairway in South College NB There are no escalators in South College and there is more than one stairway 31/33

104 (44) the escalator in South College (45) the stairway in South College NB There are no escalators in South College and there is more than one stairway No objects are denoted The function cannot apply since the escalator in South College and the stairway in South College are not in the domain of the 31/33

105 (44) the escalator in South College (45) the stairway in South College NB There are no escalators in South College and there is more than one stairway No objects are denoted The function cannot apply since the escalator in South College and the stairway in South College are not in the domain of the If they are not in the domain of the, then the cannot apply to them we cannot apply FA to calculate a semantic value for (44) and (45) 31/33

106 The generalisation that emerges regarding the domain of the is the following: 32/33

107 The generalisation that emerges regarding the domain of the is the following: (46) The domain of the contains just those functions f D e,t which satisfy the condition that there is exactly one x for which f(x) 32/33

108 The generalisation that emerges regarding the domain of the is the following: (46) The domain of the contains just those functions f D e,t which satisfy the condition that there is exactly one x for which f(x) The semantics of the is then the following (recall the colon in λ-terms): (47) the = λf f D e,t and there is exactly one x such that f(x) the unique y such that f(y) = 32/33

109 What, then, is the type of the? 33/33

110 What, then, is the type of the? Does it map totally? Are all x D e,t in the domain of the? 33/33

111 What, then, is the type of the? Does it map totally? Are all x D e,t in the domain of the? A partial function from A to B is a function from a subset of A to B 33/33