{
  "compose": 1,
  "id": "ch6-quant",
  "title": "Quantifiers (§6.4)",
  "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": [
        "runs",
        "run"
      ],
      "denotation": "Lx.run(x)"
    },
    {
      "words": [
        "sings",
        "sing"
      ],
      "denotation": "Lx.sing(x)"
    },
    {
      "words": [
        "sleeps",
        "sleep"
      ],
      "denotation": "Lx.sleep(x)"
    },
    {
      "words": [
        "loves",
        "love"
      ],
      "denotation": "Ly.Lx.love(x,y)"
    },
    {
      "words": [
        "fears",
        "fear"
      ],
      "denotation": "Ly.Lx.fear(x,y)"
    },
    {
      "words": [
        "trusts",
        "trust"
      ],
      "denotation": "Ly.Lx.trust(x,y)"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "dwarf"
      ],
      "denotation": "Lx.dwarf(x)"
    },
    {
      "words": [
        "orc"
      ],
      "denotation": "Lx.orc(x)"
    },
    {
      "words": [
        "ranger"
      ],
      "denotation": "Lx.ranger(x)"
    },
    {
      "words": [
        "magical"
      ],
      "denotation": "Lx.magical(x)"
    },
    {
      "words": [
        "beautiful"
      ],
      "denotation": "Lx.beautiful(x)"
    },
    {
      "words": [
        "late"
      ],
      "denotation": "Lx.late(x)"
    },
    {
      "words": [
        "drinks",
        "drink"
      ],
      "denotation": "Lx.drink(x)"
    },
    {
      "words": [
        "is",
        "are"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "doesnt",
        "does-not"
      ],
      "denotation": "LP.Lx.~P(x)"
    },
    {
      "words": [
        "not"
      ],
      "denotation": "LP.Lx.~P(x)"
    },
    {
      "words": [
        "everything",
        "everyone"
      ],
      "denotation": "LP.Ax[P(x)]"
    },
    {
      "words": [
        "something",
        "someone"
      ],
      "denotation": "LP.Ex[P(x)]"
    },
    {
      "words": [
        "nothing",
        "no-one"
      ],
      "denotation": "LP.~Ex[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)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true
    }
  },
  "exercises": [
    {
      "title": "A. Simple quantifiers",
      "instructions": "\"Everything\", \"something\" and \"nothing\" are complete generalized quantifiers of type ⟨⟨e,t⟩,t⟩. They apply directly to the predicate — the first uses a copular predicate, the second a transitive verb, the third adds negation.",
      "items": [
        {
          "sentence": "Everything is beautiful.",
          "tree": "[.S [.DP everything ] [.VP [.V is ] [.AP beautiful ] ] ]",
          "reading": {
            "section": "6.4"
          }
        },
        {
          "sentence": "Nothing loves Sauron.",
          "tree": "[.S [.DP nothing ] [.VP [.V loves ] [.DP Sauron ] ] ]",
          "reading": {
            "section": "6.4"
          }
        },
        {
          "sentence": "Something is not beautiful.",
          "tree": "[.S [.DP something ] [.VP [.Neg not ] [.VP [.V is ] [.AP beautiful ] ] ] ]",
          "reading": {
            "section": "6.4"
          }
        }
      ]
    },
    {
      "title": "B. Generalized quantifiers",
      "instructions": "A determiner combines with a noun to form a generalized quantifier ⟨⟨e,t⟩,t⟩, which takes the VP as its argument. The third example layers in VP negation.",
      "items": [
        {
          "sentence": "Every hobbit drinks.",
          "tree": "[.S [.DP [.D every ] [.NP hobbit ] ] [.VP drinks ] ]",
          "reading": {
            "section": "6.4.2"
          }
        },
        {
          "sentence": "No elf fears Sauron.",
          "tree": "[.S [.DP [.D no ] [.NP elf ] ] [.VP [.V fears ] [.DP Sauron ] ] ]",
          "reading": {
            "section": "6.4.2"
          }
        },
        {
          "sentence": "Some elf doesn't drink.",
          "tree": "[.S [.DP [.D some ] [.NP elf ] ] [.VP [.Neg doesnt ] [.VP drinks ] ] ]",
          "reading": {
            "section": "6.4.2"
          }
        }
      ]
    },
    {
      "title": "C. Quantifiers with transitive verbs",
      "instructions": "A subject quantifier scopes over the clause. The later items fold in tools from earlier sections: VP negation, and a copular predicate nominal.",
      "items": [
        {
          "sentence": "Every dwarf trusts Aragorn.",
          "tree": "[.S [.DP [.D every ] [.NP dwarf ] ] [.VP [.V trusts ] [.DP Aragorn ] ] ]",
          "reading": {
            "section": "6.4.2"
          }
        },
        {
          "sentence": "Some elf doesn't fear Sauron.",
          "tree": "[.S [.DP [.D some ] [.NP elf ] ] [.VP [.Neg doesnt ] [.VP [.V fears ] [.DP Sauron ] ] ] ]",
          "reading": {
            "section": "6.4.2"
          }
        },
        {
          "sentence": "No orc is a ranger.",
          "tree": "[.S [.DP [.D no ] [.NP orc ] ] [.VP [.V is ] [.NP [.D a ] [.NP ranger ] ] ] ]",
          "reading": {
            "section": "6.4.2"
          }
        }
      ]
    },
    {
      "title": "D. Scope ambiguity",
      "instructions": "Negation and a universal quantifier can interact in two ways. Sentential (wide-scope) negation ¬∀ says the universal statement is false; VP (narrow-scope) negation ∀¬ says every individual fails. Derive the reading shown.",
      "items": [
        {
          "sentence": "Not every hobbit loves Gandalf.",
          "instructions": "Surface-scope reading only: negation takes wide scope.\n∀x[hobbit(x) → love(x,g)]",
          "tree": "[.S [.NegP [.Neg not ] [.S [.DP [.D every ] [.NP hobbit ] ] [.VP [.V loves ] [.DP Gandalf ] ] ] ] ]",
          "targets": [
            "wide scope: ¬∀x[hobbit(x) → love(x, g)]"
          ],
          "reading": {
            "section": "6.6"
          }
        },
        {
          "sentence": "Every hobbit doesn't love Gandalf.",
          "instructions": "Two scope readings are possible. Derive each using the tree provided for that reading.",
          "tree": "[.S [.DP [.D every ] [.NP hobbit ] ] [.VP [.Neg doesnt ] [.VP [.V love ] [.DP Gandalf ] ] ] ]",
          "targets": [
            "surface scope (∀ > ¬): ∀x[hobbit(x) → ¬love(x, g)]",
            "inverse scope (¬ > ∀): ¬∀x[hobbit(x) → love(x, g)]"
          ],
          "scopeTrees": {
            "0": "[.S [.DP [.D every ] [.NP hobbit ] ] [.VP [.Neg doesnt ] [.VP [.V love ] [.DP Gandalf ] ] ] ]",
            "1": "[.S [.NegP [.Neg not ] [.S [.DP [.D every ] [.NP hobbit ] ] [.VP [.V loves ] [.DP Gandalf ] ] ] ] ]"
          },
          "reading": {
            "section": "6.6"
          }
        },
        {
          "sentence": "Some elf doesn't trust Sauron.",
          "instructions": "Two scope readings are possible.",
          "tree": "[.S [.DP [.D some ] [.NP elf ] ] [.VP [.Neg doesnt ] [.VP [.V trust ] [.DP Sauron ] ] ] ]",
          "targets": [
            "surface scope (∃ > ¬): ∃x[elf(x) ∧ ¬trust(x, n)]",
            "inverse scope (¬ > ∃): ¬∃x[elf(x) ∧ trust(x, n)]"
          ],
          "scopeTrees": {
            "0": "[.S [.DP [.D some ] [.NP elf ] ] [.VP [.Neg doesnt ] [.VP [.V trust ] [.DP Sauron ] ] ] ]",
            "1": "[.S [.NegP [.Neg not ] [.S [.DP [.D some ] [.NP elf ] ] [.VP [.V trusts ] [.DP Sauron ] ] ] ] ]"
          },
          "reading": {
            "section": "6.6"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 6 · Quantifiers\n\nA name denotes an individual — type $e$. But *everyone* does not pick out any single individual: no one person *is* everyone. Quantifiers therefore live one type up, as functions over predicates.\n\n## 6.4 Generalized quantifiers\n\nA quantifier is a **property of properties**: type $<<e,t>,t>$. It takes a verb-phrase meaning of type $<e,t>$ and returns a truth value.\n\n\\begin{derivation}\n[[everything]] = lambda P.forall x[P(x)]  : <<e,t>,t>\n[[something]]  = lambda P.exists x[P(x)]  : <<e,t>,t>\n[[nothing]]    = lambda P.~exists x[P(x)] : <<e,t>,t>\n\\end{derivation}\n\nBecause it is already a complete quantifier, it applies **directly** to the predicate by Function Application:\n\n\\begin{forest}\n[S{forall x[beautiful(x)]}\n  [DP{lambda P.forall x[P(x)]} everything]\n  [VP{lambda x.beautiful(x)} is beautiful]]\n\\end{forest}\n\nSo the sentences in (\\ref{simple}) come out as $forall x[beautiful(x)]$ and $~exists x[love(x,n)]$ respectively — exactly the truth conditions we want.\n\n\\ex<simple> Everything is beautiful.\n\\xe\n\n\\ex Nothing loves Sauron.\n\\xe\n\n\\ex Something is not beautiful.\n\\xe\n\n## 6.4.2 Quantificational determiners\n\nA determiner like *every*, *some*, or *no* builds a quantifier out of a noun. It is type $<<e,t>,<<e,t>,t>>$: it takes the **restrictor** noun first, then the **scope** verb phrase.\n\n\\begin{derivation}\n[[every]] = lambda X.lambda Y.forall x[X(x) -> Y(x)] : <<e,t>,<<e,t>,t>>\n[[some]]  = lambda X.lambda Y.exists x[X(x) & Y(x)]  : <<e,t>,<<e,t>,t>>\n[[no]]    = lambda X.lambda Y.~exists x[X(x) & Y(x)] : <<e,t>,<<e,t>,t>>\n\\end{derivation}\n\nNote the connective: *every* joins restrictor and scope with the conditional $->$, while *some* and *no* use conjunction $&$.[^donkey] The determiner first combines with the noun to form a generalized quantifier of type $<<e,t>,t>$, which then takes the VP:\n\n\\begin{forest}\n[S{forall x[hobbit(x) -> drink(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  [VP{lambda x.drink(x)} drinks]]\n\\end{forest}\n\nThe same recipe scales to transitive verbs and to predicate nominals, as in (\\ref{det}) — only the VP that the quantifier consumes changes.\n\n\\ex<det> Every hobbit drinks.\n\\xe\n\n\\ex No elf fears Sauron.\n\\xe\n\n\\ex No orc is a ranger.\n\\xe\n\n[^donkey]: This restrictor–scope split is the heart of *generalized quantifier theory*: a determiner is a relation between two sets, the noun's denotation and the VP's.\n\n## 6.6 Negation and scope ambiguity\n\nWhen negation meets a quantifier, two readings can arise, depending on which element scopes over the other.[^scope] *Every hobbit doesn't love Gandalf* can mean either of (\\ref{scope}):\n\n\\ex<scope> Surface, $forall > ~$ : $forall x[hobbit(x) -> ~love(x,g)]$ — every hobbit fails to love him.\n\\xe\n\n\\ex Inverse, $~ > forall$ : $~forall x[hobbit(x) -> love(x,g)]$ — not all hobbits do.\n\\xe\n\nSentential negation sits **above** the clause — $not [S ...]$ — giving the wide-scope $~forall$ reading; VP negation sits **inside** the clause, giving $forall~$. COMPOSE supplies a separate tree for each reading, so derive the one named in the exercise.\n\n[^scope]: A *scope ambiguity* is one string with two distinct logical forms. The surface tree fixes one reading; the other requires negation to take scope it does not have on the surface."
  }
}