{
  "compose": 1,
  "id": "ch10-lift",
  "title": "10.1 DP Coordination & Montague Lift",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w"
    },
    "variables": {
      "e": "x y z",
      "et": "P Q"
    }
  },
  "lexicon": [
    {
      "words": [
        "Frodo"
      ],
      "denotation": "f"
    },
    {
      "words": [
        "Sam"
      ],
      "denotation": "s"
    },
    {
      "words": [
        "Gandalf"
      ],
      "denotation": "g"
    },
    {
      "words": [
        "Sauron"
      ],
      "denotation": "n"
    },
    {
      "words": [
        "Aragorn"
      ],
      "denotation": "a"
    },
    {
      "words": [
        "Boromir"
      ],
      "denotation": "b"
    },
    {
      "words": [
        "Legolas"
      ],
      "denotation": "l"
    },
    {
      "words": [
        "Gimli"
      ],
      "denotation": "m"
    },
    {
      "words": [
        "Galadriel"
      ],
      "denotation": "w"
    },
    {
      "words": [
        "Bilbo"
      ],
      "denotation": "bi"
    },
    {
      "words": [
        "Strider"
      ],
      "denotation": "st"
    },
    {
      "words": [
        "Pippin"
      ],
      "denotation": "pi"
    },
    {
      "words": [
        "Tom"
      ],
      "denotation": "tm"
    },
    {
      "words": [
        "Elrond"
      ],
      "denotation": "el"
    },
    {
      "words": [
        "Saruman"
      ],
      "denotation": "sr"
    },
    {
      "words": [
        "Moria"
      ],
      "denotation": "mr"
    },
    {
      "words": [
        "rides",
        "ride"
      ],
      "denotation": "Lx.ride(x)"
    },
    {
      "words": [
        "sings",
        "sing"
      ],
      "denotation": "Lx.sing(x)"
    },
    {
      "words": [
        "rests",
        "rest"
      ],
      "denotation": "Lx.rest(x)"
    },
    {
      "words": [
        "meets",
        "meet"
      ],
      "denotation": "Lx.meet(x)"
    },
    {
      "words": [
        "carries",
        "carry"
      ],
      "denotation": "Ly.Lx.carry(x,y)"
    },
    {
      "words": [
        "guards",
        "guard"
      ],
      "denotation": "Ly.Lx.guard(x,y)"
    },
    {
      "words": [
        "reads",
        "read"
      ],
      "denotation": "Ly.Lx.read(x,y)"
    },
    {
      "words": [
        "finds",
        "find"
      ],
      "denotation": "Ly.Lx.find(x,y)"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "dwarf"
      ],
      "denotation": "Lx.dwarf(x)"
    },
    {
      "words": [
        "ring"
      ],
      "denotation": "Lx.ring(x)"
    },
    {
      "words": [
        "sword"
      ],
      "denotation": "Lx.sword(x)"
    },
    {
      "words": [
        "map"
      ],
      "denotation": "Lx.map(x)"
    },
    {
      "words": [
        "hobbits"
      ],
      "denotation": "*Lx.hobbit(x)"
    },
    {
      "words": [
        "elves"
      ],
      "denotation": "*Lx.elf(x)"
    },
    {
      "words": [
        "dwarves"
      ],
      "denotation": "*Lx.dwarf(x)"
    },
    {
      "words": [
        "rings"
      ],
      "denotation": "*Lx.ring(x)"
    },
    {
      "words": [
        "swords"
      ],
      "denotation": "*Lx.sword(x)"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "LX.Ix.X(x)"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "a"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "theSUPR"
      ],
      "denotation": "LX.Ix.[X(x) & Ay[X(y) -> y≤x]]"
    },
    {
      "words": [
        "andS"
      ],
      "denotation": "Lq.Lp.[p & q]"
    },
    {
      "words": [
        "orS"
      ],
      "denotation": "Lq.Lp.[p ∨ q]"
    },
    {
      "words": [
        "andVP"
      ],
      "denotation": "LQ.LP.Lx.[P(x) & Q(x)]"
    },
    {
      "words": [
        "orVP"
      ],
      "denotation": "LQ.LP.Lx.[P(x) ∨ Q(x)]"
    },
    {
      "words": [
        "andV"
      ],
      "denotation": "LR.LS.Ly.Lx.[S(y)(x) & R(y)(x)]"
    },
    {
      "words": [
        "andDP"
      ],
      "denotation": "LT.LU.LP.[U(P) & T(P)]"
    },
    {
      "words": [
        "orDP"
      ],
      "denotation": "LT.LU.LP.[U(P) ∨ T(P)]"
    },
    {
      "words": [
        "andE"
      ],
      "denotation": "Ly.Lz.z⊕y"
    },
    {
      "words": [
        "part"
      ],
      "denotation": "Ly.Lx.x≤y"
    },
    {
      "words": [
        "carry_coll"
      ],
      "denotation": "Ly.Lx.carry_coll(x,y)"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    },
    "typeShifts": [
      "lift"
    ]
  },
  "exercises": [
    {
      "title": "A. DP coordination (two quantified DPs)",
      "instructions": "andDP ↝ λT.λU.λX.[T(λz.X(z)) ∧ U(λz.X(z))] takes two GQs (type ⟨⟨e,t⟩,t⟩) and returns a GQ. Place it in a ConjP with the right DP; FA with the left DP gives the coordinated GQ. The result distributes the VP over both DP denotations.",
      "items": [
        {
          "sentence": "Every hobbit and some elf ride.",
          "tree": "[.S [.DP [.DP [.D every ] [.NP hobbit ] ] [.ConjP [.Conj andDP ] [.DP [.D some ] [.NP elf ] ] ] ] [.VP rides ] ]",
          "targets": [
            "∀x[hobbit(x) → ride(x)] ∧ ∃x[elf(x) ∧ ride(x)]"
          ]
        },
        {
          "sentence": "Some dwarf or every elf sings.",
          "tree": "[.S [.DP [.DP [.D some ] [.NP dwarf ] ] [.ConjP [.Conj orDP ] [.DP [.D every ] [.NP elf ] ] ] ] [.VP sings ] ]",
          "targets": [
            "∃x[dwarf(x) ∧ sing(x)] ∨ ∀x[elf(x) → sing(x)]"
          ]
        }
      ]
    },
    {
      "title": "B. Mixed coordination — name + quantifier (Lift required)",
      "instructions": "A proper name (type e) cannot directly coordinate with a generalized quantifier. First apply the Montague Lift type-shift to the name's DP node, turning e.g. Frodo (type e) into λP.P(f) of type ⟨⟨e,t⟩,t⟩. Then andDP combines the two quantifiers in a ConjP, and the coordinated GQ applies to the VP. Lift the name yourself — there is no pre-lifted entry.",
      "items": [
        {
          "sentence": "Frodo and every hobbit ride.",
          "tree": "[.S [.DP [.DP Frodo ] [.ConjP [.Conj andDP ] [.DP [.D every ] [.NP hobbit ] ] ] ] [.VP rides ] ]",
          "targets": [
            "ride(f) ∧ ∀x[hobbit(x) → ride(x)]"
          ]
        },
        {
          "sentence": "Sam and some elf guard the ring.",
          "tree": "[.S [.DP [.DP Sam ] [.ConjP [.Conj andDP ] [.DP [.D some ] [.NP elf ] ] ] ] [.VP [.V guards ] [.DP [.D the ] [.NP ring ] ] ] ]",
          "targets": [
            "guard(s,ιz[ring(z)]) ∧ ∃x[elf(x) ∧ guard(x,ιz[ring(z)])]"
          ]
        },
        {
          "sentence": "Gandalf or every dwarf sings.",
          "tree": "[.S [.DP [.DP Gandalf ] [.ConjP [.Conj orDP ] [.DP [.D every ] [.NP dwarf ] ] ] ] [.VP sings ] ]",
          "targets": [
            "sing(g) ∨ ∀x[dwarf(x) → sing(x)]"
          ]
        }
      ]
    }
  ]
}