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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

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