upping the ante the full lambek calculus
play

Upping the ante: the full Lambek Calculus Robert Levine Ohio State - PowerPoint PPT Presentation

Dec 01, 2022 •179 likes •1.82k views

Upping the ante: the full Lambek Calculus Robert Levine Ohio State University levine.1@osu.edu Robert Levine The Lambek Calculus 1 / 24 Coordination Coordination is a cross-categorial phenomenon Robert Levine The Lambek Calculus 2 / 24

  1. Cross-categorial analysis of coordination ◮ Can we do that? How can we get λx. walk ( x ) � λz. talk ( z ) ⇒ λw. walk ( w ) ∧ talk ( w ) ◮ (where � is the name of the ‘mystery operation’ required) Generalized conjunction � P ∧ Q if P and Q are of type t P ⊓ Q = def λh.P ( h ) ⊓ Q ( h ) otherwise Examples: (3) walk ⊓ talk = λx. walk ( x ) ⊓ λz. talk ( z ) � �� � � �� � P Q Robert Levine The Lambek Calculus 5 / 24

  2. Cross-categorial analysis of coordination ◮ Can we do that? How can we get λx. walk ( x ) � λz. talk ( z ) ⇒ λw. walk ( w ) ∧ talk ( w ) ◮ (where � is the name of the ‘mystery operation’ required) Generalized conjunction � P ∧ Q if P and Q are of type t P ⊓ Q = def λh.P ( h ) ⊓ Q ( h ) otherwise Examples: (3) walk ⊓ talk = λx. walk ( x ) ⊓ λz. talk ( z ) � �� � � �� � P Q = λw [ λx [ walk ( x )]( w ) ⊓ λz [ talk ( z )]( w )] Robert Levine The Lambek Calculus 5 / 24

  3. Cross-categorial analysis of coordination ◮ Can we do that? How can we get λx. walk ( x ) � λz. talk ( z ) ⇒ λw. walk ( w ) ∧ talk ( w ) ◮ (where � is the name of the ‘mystery operation’ required) Generalized conjunction � P ∧ Q if P and Q are of type t P ⊓ Q = def λh.P ( h ) ⊓ Q ( h ) otherwise Examples: (3) walk ⊓ talk = λx. walk ( x ) ⊓ λz. talk ( z ) � �� � � �� � P Q = λw [ λx [ walk ( x )]( w ) ⊓ λz [ talk ( z )]( w )] = λw [ walk ( w ) ∧ talk ( w )] Robert Levine The Lambek Calculus 5 / 24

  4. Cross-categorial analysis of coordination ◮ Can we do that? How can we get λx. walk ( x ) � λz. talk ( z ) ⇒ λw. walk ( w ) ∧ talk ( w ) ◮ (where � is the name of the ‘mystery operation’ required) Generalized conjunction � P ∧ Q if P and Q are of type t P ⊓ Q = def λh.P ( h ) ⊓ Q ( h ) otherwise Examples: (3) walk ⊓ talk = λx. walk ( x ) ⊓ λz. talk ( z ) � �� � � �� � P Q = λw [ λx [ walk ( x )]( w ) ⊓ λz [ talk ( z )]( w )] = λw [ walk ( w ) ∧ talk ( w )] Robert Levine The Lambek Calculus 5 / 24

  5. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) Robert Levine The Lambek Calculus 6 / 24

  6. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) Robert Levine The Lambek Calculus 6 / 24

  7. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) Robert Levine The Lambek Calculus 6 / 24

  8. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) Robert Levine The Lambek Calculus 6 / 24

  9. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . Robert Levine The Lambek Calculus 6 / 24

  10. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with Robert Levine The Lambek Calculus 6 / 24

  11. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) Robert Levine The Lambek Calculus 6 / 24

  12. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) Robert Levine The Lambek Calculus 6 / 24

  13. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) Robert Levine The Lambek Calculus 6 / 24

  14. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] Robert Levine The Lambek Calculus 6 / 24

  15. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) Robert Levine The Lambek Calculus 6 / 24

  16. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) ◮ The VP talked to the members of the jury passionately and eloquently can now apply to John Robert Levine The Lambek Calculus 6 / 24

  17. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) ◮ The VP talked to the members of the jury passionately and eloquently can now apply to John ◮ to give Robert Levine The Lambek Calculus 6 / 24

  18. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) ◮ The VP talked to the members of the jury passionately and eloquently can now apply to John ◮ to give ◮ passionately ( talked ( mj ))( j ) ∧ eloquently ( talked ( mj ))( j ) Robert Levine The Lambek Calculus 6 / 24

  19. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) ◮ The VP talked to the members of the jury passionately and eloquently can now apply to John ◮ to give ◮ passionately ( talked ( mj ))( j ) ∧ eloquently ( talked ( mj ))( j ) ◮ How about auxiliaries? Robert Levine The Lambek Calculus 6 / 24

  20. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) ◮ The VP talked to the members of the jury passionately and eloquently can now apply to John ◮ to give ◮ passionately ( talked ( mj ))( j ) ∧ eloquently ( talked ( mj ))( j ) ◮ How about auxiliaries? Robert Levine The Lambek Calculus 6 / 24

  21. ◮ passionately �� e , t � , � e , t �� ⊓ eloquently �� e , t � , � e , t �� = λRλx. passionately ( R )( x ) ⊓ λSλy. eloquently ( S )( y ) = λT.λR [ λx. passionately ( R )( x )]( T ) ⊓ λS [ λy. eloquently ( S )( y )]( T ) = λT λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y ) ◮ Then talked to the members of the jury passionately and eloquently will apply λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )] to the semantics of the VP talked to the members of the jury , talked ( mj ) . . . ◮ . . . and we wind up with ◮ λT [ λx. passionately ( T )( x ) ⊓ λy. eloquently ( T )( y )]( talked ( mj )) = λx. passionately ( talked ( mj ))( x ) ⊓ λy. eloquently ( talked ( mj ))( y ) = λv [ λx [ passionately ( talked ( mj ))( x )]( v ) ⊓ λy [ eloquently ( talked ( mj ))( y )]( v )] = λv. passionately ( talked ( mj ))( v ) ∧ eloquently ( talked ( mj ))( v ) ◮ The VP talked to the members of the jury passionately and eloquently can now apply to John ◮ to give ◮ passionately ( talked ( mj ))( j ) ∧ eloquently ( talked ( mj ))( j ) ◮ How about auxiliaries? (4) Mary can and should apply for the job. Robert Levine The Lambek Calculus 6 / 24

  22. Syntax and semantics of coordination (5) and ; λ P λ Q . Q ⊓ P ; ( X \ X ) / X and ; talked ; λ P λ Q . Q ⊓ P ; ( X \ X ) / X talk ; VP [ fin ] walked ; / E talk ; VP [ fin ] and • talked ; λ Q . Q ⊓ talk ; VP fin \ VP fin / E walked • and • talked ; walk ⊓ talk ; VP fin ............................................. john ; walked • and • talked ; j ; NP λy. walk ( y ) ⊓ talk ( y ); VP fin \ E john • walked • and • talked ; λy [ walk ( y ) ∧ talk ( y )]( j ) ; S fin ................................................................. john • walked • and • talked ; walk ( r ) ∧ talk ( j ) ; S fin Robert Levine The Lambek Calculus 7 / 24

  23. But wait, there’s more: nonconstituent coordination Robert Levine The Lambek Calculus 8 / 24

  24. But wait, there’s more: nonconstituent coordination Right node raising Robert Levine The Lambek Calculus 8 / 24

  25. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Robert Levine The Lambek Calculus 8 / 24

  26. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination Robert Levine The Lambek Calculus 8 / 24

  27. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers]. Robert Levine The Lambek Calculus 8 / 24

  28. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers]. ◮ In RNR, what is coordinated? Robert Levine The Lambek Calculus 8 / 24

  29. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers]. ◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne . Robert Levine The Lambek Calculus 8 / 24

  30. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers]. ◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne . ◮ to Anne is an PP Robert Levine The Lambek Calculus 8 / 24

  31. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers]. ◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne . ◮ to Anne is an PP ◮ Bill gave the pair of pliers to Anne is an S. Robert Levine The Lambek Calculus 8 / 24

  32. But wait, there’s more: nonconstituent coordination Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers]. ◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne . ◮ to Anne is an PP ◮ Bill gave the pair of pliers to Anne is an S. ◮ So what is the category we have to posit for Bill gave the pair of pliers ? Robert Levine The Lambek Calculus 8 / 24

  33. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: Robert Levine The Lambek Calculus 9 / 24

  34. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: Robert Levine The Lambek Calculus 9 / 24

  35. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: Robert Levine The Lambek Calculus 9 / 24

  36. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: (8) a. I put the teapot, and John carefully placed the vase, on the shelf. Robert Levine The Lambek Calculus 9 / 24

  37. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: (8) a. I put the teapot, and John carefully placed the vase, on the shelf. Robert Levine The Lambek Calculus 9 / 24

  38. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: (8) a. I put the teapot, and John carefully placed the vase, on the shelf. b. John carefully placed the vase, and I put the teapot, on the shelf. Robert Levine The Lambek Calculus 9 / 24

  39. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: (8) a. I put the teapot, and John carefully placed the vase, on the shelf. b. John carefully placed the vase, and I put the teapot, on the shelf. Robert Levine The Lambek Calculus 9 / 24

  40. ◮ Examples such as (8) show that the ‘raised’ material must in some sense count as a constituent of both conjuncts: (8) a. I put the teapot, and John carefully placed the vase, on the shelf. b. John carefully placed the vase, and I put the teapot, on the shelf. ◮ What do we know about put ? Robert Levine The Lambek Calculus 9 / 24

  41. (9) a. The Nobel Committee awarded–and the King of Sweden presented—the Physics prize to Peter Higgs. b. John wanted to wager—and Mary insisted on betting–a ton of money with the bookies on the outcome of the game. Robert Levine The Lambek Calculus 10 / 24

  42. (9) a. The Nobel Committee awarded–and the King of Sweden presented—the Physics prize to Peter Higgs. b. John wanted to wager—and Mary insisted on betting–a ton of money with the bookies on the outcome of the game. ◮ In (9a), we have a RNRed string corresponding to an NP and a PP; in b, to a sequence NP PP with PP on . Robert Levine The Lambek Calculus 10 / 24

  43. (9) a. The Nobel Committee awarded–and the King of Sweden presented—the Physics prize to Peter Higgs. b. John wanted to wager—and Mary insisted on betting–a ton of money with the bookies on the outcome of the game. ◮ In (9a), we have a RNRed string corresponding to an NP and a PP; in b, to a sequence NP PP with PP on . ◮ How are we going to get all of these examples to go through? Robert Levine The Lambek Calculus 10 / 24

  44. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . Robert Levine The Lambek Calculus 11 / 24

  45. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned Robert Levine The Lambek Calculus 11 / 24

  46. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon Robert Levine The Lambek Calculus 11 / 24

  47. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. Robert Levine The Lambek Calculus 11 / 24

  48. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. ◮ Consider again (10): Robert Levine The Lambek Calculus 11 / 24

  49. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. ◮ Consider again (10): Robert Levine The Lambek Calculus 11 / 24

  50. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. ◮ Consider again (10): (10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Robert Levine The Lambek Calculus 11 / 24

  51. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. ◮ Consider again (10): (10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. ◮ We are conjoining John gave the pair of pliers and Mary offered the hammar drill . Let’s examine a bit more closely what the problem with these is. Robert Levine The Lambek Calculus 11 / 24

  52. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. ◮ Consider again (10): (10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. ◮ We are conjoining John gave the pair of pliers and Mary offered the hammar drill . Let’s examine a bit more closely what the problem with these is. Robert Levine The Lambek Calculus 11 / 24

  53. What needs doing? ◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type is assigned ◮ either via the lexicon ◮ or by our proof theory. ◮ Consider again (10): (10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. ◮ We are conjoining John gave the pair of pliers and Mary offered the hammar drill . Let’s examine a bit more closely what the problem with these is. gave ; give ; VP / PP / NP the • pair • of • pliers ; pl ; NP gave • the • pair • of • pliers ; john ; offer ( pl ); VP / PP j ; NP FAIL Robert Levine The Lambek Calculus 11 / 24

  54. ◮ There’s no way forward here. Robert Levine The Lambek Calculus 12 / 24

  55. ◮ There’s no way forward here. ◮ But there HAS to be, no? Robert Levine The Lambek Calculus 12 / 24

  56. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: Robert Levine The Lambek Calculus 12 / 24

  57. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. Robert Levine The Lambek Calculus 12 / 24

  58. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, Robert Levine The Lambek Calculus 12 / 24

  59. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, Robert Levine The Lambek Calculus 12 / 24

  60. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , Robert Levine The Lambek Calculus 12 / 24

  61. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, Robert Levine The Lambek Calculus 12 / 24

  62. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. Robert Levine The Lambek Calculus 12 / 24

  63. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type we need, Robert Levine The Lambek Calculus 12 / 24

  64. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type we need, ◮ and then paying it back. Robert Levine The Lambek Calculus 12 / 24

  65. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type we need, ◮ and then paying it back. ◮ But we can’t do that, Robert Levine The Lambek Calculus 12 / 24

  66. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type we need, ◮ and then paying it back. ◮ But we can’t do that, ◮ because our proof theory gives us no way to do the ignoring-and-remembering that we need to do. Robert Levine The Lambek Calculus 12 / 24

  67. ◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable: ◮ We know that John gave the pair of pliers is really a sentence lacking a PP on the right, i.e. ◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP combine with John , ◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type we need, ◮ and then paying it back. ◮ But we can’t do that, ◮ because our proof theory gives us no way to do the ignoring-and-remembering that we need to do. ◮ But it SHOULD , if standard logics are any guide. Robert Levine The Lambek Calculus 12 / 24

  68. Implication introduction in logic. ◮ Let’s look at classical logic again for a clue as to why: Robert Levine The Lambek Calculus 13 / 24

  69. Implication introduction in logic. ◮ Let’s look at classical logic again for a clue as to why: Robert Levine The Lambek Calculus 13 / 24

  70. Implication introduction in logic. ◮ Let’s look at classical logic again for a clue as to why: φ ⊃ ψ φ (11) ⊃ Elim ψ Robert Levine The Lambek Calculus 13 / 24

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Complexity of the Lambek Calculus and Its Fragments Mati Pentus http://lpcs.math.msu.su/~pentus/

Complexity of the Lambek Calculus and Its Fragments Mati Pentus http://lpcs.math.msu.su/~pentus/

Complexity of the Lambek Calculus and Its Fragments Mati Pentus http://lpcs.math.msu.su/~pentus/ The Lambek calculus (denoted L) is a mathematical tool for formal language specification. It generates the class of all context-free languages

773 views • 34 slides
Upping the Ante: Affordable, Multifamily Passive Housing through Integrated Design Dylan Lamar,

Upping the Ante: Affordable, Multifamily Passive Housing through Integrated Design Dylan Lamar,

7 th Annual North American Passive House Conference September 27-30, 2012 Denver CO Upping the Ante: Affordable, Multifamily Passive Housing through Integrated Design Dylan Lamar, Intern Architect, CPHC Alex Boetzel, Construction Manager, CPHC

559 views • 30 slides
Upping the Ante: Equilibrium Effects of Unconditional School Grants Jishnu Das (World Bank,

Upping the Ante: Equilibrium Effects of Unconditional School Grants Jishnu Das (World Bank,

Upping the Ante: Equilibrium Effects of Unconditional School Grants Jishnu Das (World Bank, Washington DC) Tahir Andrabi (Pomona College), Asim Khwaja (Harvard University), Selcuk Ozyurt (Sabanci University) and Niharika Singh (Harvard

602 views • 49 slides
Syntax and Semantics in Generalized Lambek Calculus Michael Moortgat LIRa seminar, January 24,

Syntax and Semantics in Generalized Lambek Calculus Michael Moortgat LIRa seminar, January 24,

Syntax and Semantics in Generalized Lambek Calculus Michael Moortgat LIRa seminar, January 24, 2011, Amsterdam Abstract Lambeks Syntactic Calculus (1961) is a logic completely without structural rules: rules af- fecting multiplicity

941 views • 65 slides
Beyond the context-free boundary: generalizing Lambek calculus Michael Moortgat Flowincat

Beyond the context-free boundary: generalizing Lambek calculus Michael Moortgat Flowincat

Beyond the context-free boundary: generalizing Lambek calculus Michael Moortgat Flowincat 2010 Oxford Abstract Lambeks syntactic calculi, both the associative and the non-associative variant, are strictly contextfree. A well-tried

862 views • 82 slides
NP-completeness of Lambek calculus and multiplicative noncommutative linear logic Mati Pentus

NP-completeness of Lambek calculus and multiplicative noncommutative linear logic Mati Pentus

L Languages L H L Grammars Models MCLL Complexity Proof nets Equivalence PNCL NP-completeness of Lambek calculus and multiplicative noncommutative linear logic Mati Pentus http://markov.math.msu.ru/~pentus/ L Languages L H L

810 views • 49 slides
A Lambek Calculus with Dependent Types Zhaohui Luo Dept of Computer Science Royal Holloway, Univ

A Lambek Calculus with Dependent Types Zhaohui Luo Dept of Computer Science Royal Holloway, Univ

A Lambek Calculus with Dependent Types Zhaohui Luo Dept of Computer Science Royal Holloway, Univ of London This talk Background and motivation Categorial grammars and Montague semantics NL semantics in Modern TTs syntactic

618 views • 18 slides
Lambek Calculus Extended with Subexponential and Bracket Modalities Max Kanovich, Stepan

Lambek Calculus Extended with Subexponential and Bracket Modalities Max Kanovich, Stepan

Lambek Calculus Extended with Subexponential and Bracket Modalities Max Kanovich, Stepan Kuznetsov, Andre Scedrov Basic Categorial Grammar John loves Mary Basic Categorial Grammar John loves Mary np ( np \ s ) / np np Basic Categorial

1.7k views • 95 slides
1 Side Track: Evaluation Order Evaluation Order, II Full -reduction: is Redex: Term

1 Side Track: Evaluation Order Evaluation Order, II Full -reduction: is Redex: Term

calculus Course 2D1453, 2006-07 Alonzo Church, 1903-1995 Church-Turing thesis First undecidability results Advanced Formal Methods Invented -calculus in 30s Lecture 2: Lambda calculus -Calculus

391 views • 6 slides
Full reduction at full throttle Extension to the Calculus of Inductive Constructions Gerco van

Full reduction at full throttle Extension to the Calculus of Inductive Constructions Gerco van

Full reduction at full throttle Extension to the Calculus of Inductive Constructions Gerco van Heerdt December 19, 2014 1 / 18 Symbolic CIC Term t , T , P ::= x | t 1 t 2 | v | C i ( t ) | case P t of ( C i ( x i ) t i )

225 views • 18 slides
Revolutionary and New Category in Skin Health 2019 Micah Chemistry ante ante- a prefix

Revolutionary and New Category in Skin Health 2019 Micah Chemistry ante ante- a prefix

Revolutionary and New Category in Skin Health 2019 Micah Chemistry ante ante- a prefix meaning before, of Latin origin Micah ante oxidant Micah anteoxidant TM - to prevent the visible signs of photoaging, of Hallstar origin Micah

501 views • 48 slides
-Calculus Christine Rizkallah CSE, UNSW (and data61) Term3 2019 1 -Calculus Church

-Calculus Christine Rizkallah CSE, UNSW (and data61) Term3 2019 1 -Calculus Church

-Calculus Church Encodings -Calculus Christine Rizkallah CSE, UNSW (and data61) Term3 2019 1 -Calculus Church Encodings -Calculus The term language we defined for Higher Order Abstract Syntax is almost a full featured programming

620 views • 32 slides
= = = f f BOB BOB not does like not like = Alice Bob Alice Bob not Bob Coecke

= = = f f BOB BOB not does like not like = Alice Bob Alice Bob not Bob Coecke

picture logic for info-flow in language and physics (incl. the Lambek vs. Lambek battle) That flowin phyling October 2010 ALICE ALICE f f f f = = = f f BOB BOB not does like not like = Alice Bob Alice Bob not Bob

1.06k views • 94 slides
Higher-Order Categorical Substructural Logics Yoshihiro Maruyama Oxford University and Kyoto

Higher-Order Categorical Substructural Logics Yoshihiro Maruyama Oxford University and Kyoto

Full Lambek Hyperdoctrine Full Lambek Tripos Higher-Order Categorical Substructural Logics Yoshihiro Maruyama Oxford University and Kyoto University CT 2015, Aveiro, 18 June Yoshihiro Maruyama Higher-Order Categorical Substructural Logics CT

611 views • 27 slides
The full -calculus: simple and expressive -Refresh. What we know so far? We have studied

The full -calculus: simple and expressive -Refresh. What we know so far? We have studied

The full -calculus: simple and expressive -Refresh. What we know so far? We have studied the asynchronous monadic -calculus no continuation on the output (asynchronous) u v .P only one value is communicated (monadic) u

426 views • 16 slides
Harmony, the Curry-Howard(-Lambek) Correspondence, and Higher Proof Theory Yoshihiro Maruyama

Harmony, the Curry-Howard(-Lambek) Correspondence, and Higher Proof Theory Yoshihiro Maruyama

Proof-Theoretic Harmony The Curry-Howard(-Lambek) Corrspondence Higher Proof Theory Harmony, the Curry-Howard(-Lambek) Correspondence, and Higher Proof Theory Yoshihiro Maruyama The Hakubi Centre for Advanced Research, Kyoto University Sojin,

325 views • 15 slides
an Open Source RISC-V Microarchitecture CARRV 2019 June 22 nd , 2019 - Phoenix, Arizona Abraham

an Open Source RISC-V Microarchitecture CARRV 2019 June 22 nd , 2019 - Phoenix, Arizona Abraham

Replicating and Mitigating Spectre Attacks on an Open Source RISC-V Microarchitecture CARRV 2019 June 22 nd , 2019 - Phoenix, Arizona Abraham Gonzalez , Ben Korpan, Jerry Zhao , Ed Younis Krste Asanovi University of California, Berkeley

709 views • 32 slides
Semantic Spaces for Zero-Shot Behaviour Analysis Xun Xu Computer Vision and Interactive Media

Semantic Spaces for Zero-Shot Behaviour Analysis Xun Xu Computer Vision and Interactive Media

Semantic Spaces for Zero-Shot Behaviour Analysis Xun Xu Computer Vision and Interactive Media Lab, NUS Singapore 1 Collaborators Prof. Shaogang Gong Dr. Timothy Hospedales 2 Outline Background Transductive Zero-Shot Action

764 views • 73 slides
Lossless or Quantized Boosting with Integer Arithmetic Richard Nock Robert C. Williamson

Lossless or Quantized Boosting with Integer Arithmetic Richard Nock Robert C. Williamson

Lossless or Quantized Boosting with Integer Arithmetic Richard Nock Robert C. Williamson The big picture Many constraints in today-ML (e.g. for privacy, at-the-edge, distributed or deep) generic : integer encoding,

467 views • 8 slides
Integration of general-purpose automated theorem provers in Lean Gabriel Ebner Formal Methods in

Integration of general-purpose automated theorem provers in Lean Gabriel Ebner Formal Methods in

Integration of general-purpose automated theorem provers in Lean Gabriel Ebner Formal Methods in Mathematics 2020-01-08 Vrije Universiteit Amsterdam Introduction Premise selection Applicative translation to FOL Monomorphizing translation to

455 views • 34 slides
The Big Problem with Meta-Learning and How Bayesians Can Fix It Chelsea Finn Stanford

The Big Problem with Meta-Learning and How Bayesians Can Fix It Chelsea Finn Stanford

The Big Problem with Meta-Learning and How Bayesians Can Fix It Chelsea Finn Stanford training data test datapoint Braque Cezanne By Braque or Cezanne? How did you accomplish this? Through previous experience. How might you get a

264 views • 25 slides
Hammering towards Qed Cezary Kaliszyk Josef Urban University of Innsbruck Radboud University

Hammering towards Qed Cezary Kaliszyk Josef Urban University of Innsbruck Radboud University

Hammering towards Qed Cezary Kaliszyk Josef Urban University of Innsbruck Radboud University July 18, 2014 1 / 21 Outline Automation for Interactive Proof Translations Evaluation Machine Learning Reconstruction Towards Qed Strength

628 views • 29 slides
Decomposition of effect algebras and the Hammer-Sobczyk theorem A report of the joint paper by

Decomposition of effect algebras and the Hammer-Sobczyk theorem A report of the joint paper by

Decomposition of effect algebras and the Hammer-Sobczyk theorem A report of the joint paper by Anna Avallone, Giuseppina Barbieri Paolo Vitolo and Hans Weber on Algebra Universalis E. Marczewski Centennial Conference, 2007 DEFINITION A measure

595 views • 16 slides
Lecture 4 (Part 3): Hierarchical 3D Models Prof Emmanuel Agu Computer Science Dept. Worcester

Lecture 4 (Part 3): Hierarchical 3D Models Prof Emmanuel Agu Computer Science Dept. Worcester

Computer Graphics (CS 543) Lecture 4 (Part 3): Hierarchical 3D Models Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Instance Transformation Start with unique object (a symbol ) Each appearance of object

767 views • 42 slides

More recommend