{
  "compose": 1,
  "id": "ch7-objraise",
  "title": "Quantifiers in Object Position (§7.4)",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w bi st pi tm el sr mr x1 x2"
    },
    "variables": {
      "e": "x y z",
      "et": "X Y Z P Q R F G H",
      "t": "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": [
        "loves",
        "love"
      ],
      "denotation": "Ly.Lx.love(x,y)"
    },
    {
      "words": [
        "trusts",
        "trust"
      ],
      "denotation": "Ly.Lx.trust(x,y)"
    },
    {
      "words": [
        "fears",
        "fear"
      ],
      "denotation": "Ly.Lx.fear(x,y)"
    },
    {
      "words": [
        "councils",
        "council"
      ],
      "denotation": "Ly.Lx.council(x,y)"
    },
    {
      "words": [
        "summons",
        "summon"
      ],
      "denotation": "Ly.Lx.summon(x,y)"
    },
    {
      "words": [
        "travels",
        "travel"
      ],
      "denotation": "Lx.travel(x)"
    },
    {
      "words": [
        "human"
      ],
      "denotation": "Lx.human(x)"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "dwarf"
      ],
      "denotation": "Lx.dwarf(x)"
    },
    {
      "words": [
        "creature"
      ],
      "denotation": "Lx.creature(x)"
    },
    {
      "words": [
        "brave"
      ],
      "denotation": "Lx.brave(x)"
    },
    {
      "words": [
        "wise"
      ],
      "denotation": "Lx.wise(x)"
    },
    {
      "words": [
        "good"
      ],
      "denotation": "Lx.good(x)"
    },
    {
      "words": [
        "evil"
      ],
      "denotation": "Lx.evil(x)"
    },
    {
      "words": [
        "doesnt",
        "does-not"
      ],
      "denotation": "LP.Lx.~P(x)"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "no",
        "No"
      ],
      "denotation": "LX.LY.~Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "LX.Ix.X(x)"
    },
    {
      "words": [
        "who"
      ],
      "denotation": "LX.X"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    },
    "typeShifts": [
      "raiseO"
    ],
    "quantifierRaising": true
  },
  "exercises": [
    {
      "title": "A. Simple object quantifiers via QR",
      "instructions": "Raise the object DP to sentence level, leaving a trace of type e. Resolve the LP node with Predicate Abstraction, then apply the quantifier.",
      "items": [
        {
          "sentence": "Gandalf loves every hobbit.",
          "tree": "[.S [.DP Gandalf ] [.VP [.V loves ] [.DP [.D every ] [.NP hobbit ] ] ] ]",
          "targets": [
            "∀x[hobbit(x) → love(g,x)]"
          ],
          "reading": {
            "section": "7.4"
          }
        },
        {
          "sentence": "Elrond summons every good wise creature.",
          "tree": "[.S [.DP Elrond ] [.VP [.V summons ] [.DP [.D every ] [.NP [.AP good ] [.NP [.AP wise ] [.NP creature ] ] ] ] ] ]",
          "targets": [
            "∀x[good(x) ∧ wise(x) ∧ creature(x) → summon(el,x)]"
          ],
          "reading": {
            "section": "7.4"
          }
        },
        {
          "sentence": "Legolas doesn't trust some brave dwarf.",
          "tree": "[.S [.DP Legolas ] [.VP [.Neg doesnt ] [.VP [.V trusts ] [.DP [.D some ] [.NP [.AP brave ] [.NP dwarf ] ] ] ] ] ]",
          "targets": [
            "∃x[brave(x) ∧ dwarf(x) ∧ ¬trust(l,x)]"
          ],
          "reading": {
            "section": "7.4"
          }
        }
      ]
    },
    {
      "title": "B. Object quantifiers with subject quantifiers",
      "instructions": "Both subject and object are quantifiers. Try both scope orders and derive the readings.",
      "items": [
        {
          "sentence": "No elf trusts every human.",
          "tree": "[.S [.DP [.D no ] [.NP elf ] ] [.VP [.V trusts ] [.DP [.D every ] [.NP human ] ] ] ]",
          "targets": [
            "every > some (linear): ∀x[elf(x) → ∃y[hobbit(y) ∧ trust(x,y)]]",
            "some > every (inverse): ∃y[hobbit(y) ∧ ∀x[elf(x) → trust(x,y)]]",
            "no > every: ¬∃x[elf(x) ∧ ∀y[human(y) → trust(x,y)]]",
            "every > no: ∀y[human(y) → ¬∃x[elf(x) ∧ trust(x,y)]]"
          ],
          "reading": {
            "section": "7.4.1"
          }
        },
        {
          "sentence": "Some elf councils every good wise creature.",
          "tree": "[.S [.DP [.D some ] [.NP elf ] ] [.VP [.V councils ] [.DP [.D every ] [.NP [.AP good ] [.NP [.AP wise ] [.NP creature ] ] ] ] ] ]",
          "targets": [
            "some > every (linear): ∃x[elf(x) ∧ ∀y[good(y) ∧ wise(y) ∧ creature(y) → council(x,y)]]",
            "every > some (inverse): ∀y[good(y) ∧ wise(y) ∧ creature(y) → ∃x[elf(x) ∧ council(x,y)]]"
          ],
          "reading": {
            "section": "7.4.1"
          }
        },
        {
          "sentence": "Every hobbit who travels fears some evil creature.",
          "tree": "[.S [.DP [.D every ] [.NP [.NP hobbit ] [.CP [.C who ] [.S [.DP t1 ] [.VP travels ] ] ] ] ] [.VP [.V fears ] [.DP [.D some ] [.NP [.AP evil ] [.NP creature ] ] ] ] ]",
          "targets": [
            "every > some (linear): ∀x[hobbit(x) ∧ travel(x) → ∃y[evil(y) ∧ creature(y) ∧ fear(x,y)]]",
            "some > every (inverse): ∃y[evil(y) ∧ creature(y) ∧ ∀x[hobbit(x) ∧ travel(x) → fear(x,y)]]"
          ],
          "reading": {
            "section": "7.4.1"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 7 · Quantifiers in Object Position\n\nWhen a transitive verb takes an individual (type e) as its object, FA applies cleanly. But a quantifier phrase like *every hobbit* has type ⟨⟨e,t⟩,t⟩ — it cannot serve as the direct argument of a verb expecting type e. **Quantifier Raising (QR)** resolves the mismatch by moving the quantifier out of its base position.\n\n## 7.4 Quantifier Raising\n\n**Quantifier Raising (QR).** The object DP moves covertly to adjoin at S-level, leaving a trace t_n of type e. A λ-phrase node (LP) hosts Predicate Abstraction over n, creating the ⟨e,t⟩ argument the quantifier needs.\n\n\\ex<qr-ex> Gandalf loves every hobbit.\n\\xe\n\n\\ex Elrond summons every good wise creature.\n\\xe\n\n\\ex Legolas doesn't trust some brave dwarf.\n\\xe\n\n\\begin{derivation}\n[[loves]]                        = lambda y.lambda x.love(x,y)                  : <e,<e,t>>\n[[loves t1]]                     = lambda x.love(x,t_1)                         : <e,t>\n[[Gandalf loves t1]]             = love(g,t_1)                                  : t\n[[LP 1 [S Gandalf loves t1]]]    = lambda y.love(g,y)                           : <e,t>   (PA)\n[[every hobbit]]                 = lambda Y.forall x[hobbit(x) -> Y(x)]         : <<e,t>,t>\n[[every hobbit loves Gandalf]]   = forall x[hobbit(x) -> love(g,x)]             : t\n\\end{derivation}\n\n\\begin{forest}\n[S{forall x[hobbit(x) -> love(g,x)]}\n  [DP{lambda Y.forall x[hobbit(x) -> Y(x)]}\n    [D{lambda X.lambda Y.forall x[X(x) -> Y(x)]} every]\n    [NP{lambda x.hobbit(x)} hobbit]]\n  [LP{lambda y.love(g,y)} 1\n    [S{love(g,t_1)}\n      [DP{g} Gandalf]\n      [VP{lambda x.love(x,t_1)}\n        [V{lambda y.lambda x.love(x,y)} loves]\n        [DP{t_1} t_1]]]]]\n\\end{forest}\n\n## 7.4.1 Two quantifiers and scope\n\nWhen both subject and object are quantifiers, QR can apply to either or both, generating different **scope orderings**. Raising the object above the subject yields object-wide scope; the base order (subject above raised object) yields subject-wide scope.\n\n\\ex<scope-ex> No elf trusts every human.\n\\xe\n\n\\ex Some elf councils every good wise creature.\n\\xe\n\n\\ex Every hobbit who travels fears some evil creature.\n\\xe\n\n**Subject wide scope** (no > every): the object is QR'd, the subject remains in situ.\n\n\\begin{derivation}\n[[LP 1 [S no-elf [VP trusts t1]]]] = lambda y.~exists x[elf(x) /\\ trust(x,y)] : <e,t>   (PA)\n[[every human applied]]            = forall y[human(y) -> ~exists x[elf(x) /\\ trust(x,y)]] : t\n\\end{derivation}\n\n**Object wide scope** (every > no): the object LP is embedded *within* a second LP that abstracts over the subject trace.\n\nThe two readings differ in truth conditions when the quantifiers are *some* and *every*: the subject-wide reading requires one witness to stand in the relation to all objects, while the object-wide reading only requires that for each object some (possibly different) subject qualifies.[^scope]\n\n[^scope]: For *no* and *every* the two readings are logically equivalent. The asymmetry becomes vivid with *some*: *some elf councils every creature* (∃-wide) entails a single elf does the counselling; the inverse (∀-wide) only requires each creature to be counselled by some elf, possibly different ones."
  }
}