{
  "compose": 1,
  "id": "hk6-quantifiers",
  "title": "Quantifiers: Their Semantic Type (§6.3–6.4, §6.7)",
  "subtitle": "Heim & Kratzer — generalized quantifiers, quantifying determiners, and presuppositional both/neither",
  "notation": "hk",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "a jn"
    },
    "variables": {
      "e": "x y z",
      "et": "f g P Q X Y",
      "t": "p q"
    }
  },
  "lexicon": [
    {
      "words": [
        "cat"
      ],
      "denotation": "Lx.cat(x)"
    },
    {
      "words": [
        "dog"
      ],
      "denotation": "Lx.dog(x)"
    },
    {
      "words": [
        "vanished",
        "vanish"
      ],
      "denotation": "Lx.vanish(x)"
    },
    {
      "words": [
        "purrs",
        "purr"
      ],
      "denotation": "Lx.purr(x)"
    },
    {
      "words": [
        "is"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "has"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "striped"
      ],
      "denotation": "Lx.striped(x)"
    },
    {
      "words": [
        "stripes"
      ],
      "denotation": "Lx.striped(x)"
    },
    {
      "words": [
        "something"
      ],
      "denotation": "Lf.Ex[f(x)]"
    },
    {
      "words": [
        "nothing"
      ],
      "denotation": "Lf.~Ex[f(x)]"
    },
    {
      "words": [
        "everything"
      ],
      "denotation": "Lf.Ax[f(x)]"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "Lf.Lg.Ax[f(x) -> g(x)]"
    },
    {
      "words": [
        "no",
        "No"
      ],
      "denotation": "Lf.Lg.~Ex[f(x) ∧ g(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "Lf.Lg.Ex[f(x) ∧ g(x)]"
    },
    {
      "words": [
        "both",
        "Both"
      ],
      "denotation": "Lf : card2(f) . Lg.Ax[f(x) -> g(x)]",
      "displayAs": "both"
    },
    {
      "words": [
        "neither",
        "Neither"
      ],
      "denotation": "Lf : card2(f) . Lg.~Ex[f(x) ∧ g(x)]",
      "displayAs": "neither"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true
    }
  },
  "exercises": [
    {
      "title": "A. Generalized quantifiers (§6.3)",
      "instructions": "A quantificational DP is neither an individual (e) nor a set (⟨e,t⟩): it is a SECOND-ORDER property of type ⟨⟨e,t⟩,t⟩, taking the VP predicate as argument.",
      "items": [
        {
          "sentence": "Something vanished.",
          "tree": "[.S [.DP something ] [.VP vanished ] ]",
          "targets": [
            "∃x[vanish(x)]"
          ],
          "reading": {
            "section": "6.3"
          }
        },
        {
          "sentence": "Nothing vanished.",
          "tree": "[.S [.DP nothing ] [.VP vanished ] ]",
          "targets": [
            "¬∃x[vanish(x)]"
          ],
          "reading": {
            "section": "6.3"
          }
        },
        {
          "sentence": "Everything vanished.",
          "tree": "[.S [.DP everything ] [.VP vanished ] ]",
          "targets": [
            "∀x[vanish(x)]"
          ],
          "reading": {
            "section": "6.3"
          }
        }
      ]
    },
    {
      "title": "B. Quantifying determiners (§6.4)",
      "instructions": "A determiner is type ⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩: it takes the noun restrictor first, then the VP predicate. FA applies twice.",
      "items": [
        {
          "sentence": "Every cat vanished.",
          "tree": "[.S [.DP [.D every ] [.NP cat ] ] [.VP vanished ] ]",
          "targets": [
            "∀x[cat(x) → vanish(x)]"
          ],
          "reading": {
            "section": "6.4"
          }
        },
        {
          "sentence": "No dog purrs.",
          "tree": "[.S [.DP [.D no ] [.NP dog ] ] [.VP purrs ] ]",
          "targets": [
            "¬∃x[dog(x) ∧ purr(x)]"
          ],
          "reading": {
            "section": "6.4"
          }
        },
        {
          "sentence": "Some cat purrs.",
          "tree": "[.S [.DP [.D some ] [.NP cat ] ] [.VP purrs ] ]",
          "targets": [
            "∃x[cat(x) ∧ purr(x)]"
          ],
          "reading": {
            "section": "6.4"
          }
        }
      ]
    },
    {
      "title": "C. Presuppositional both & neither (§6.7)",
      "instructions": "Both and neither PRESUPPOSE that the restrictor has exactly two members — a partial function (Chapter 4's colon), presupposing |f|=2. The presupposition projects up to the whole sentence. They cannot be captured by a purely relational (total) theory.",
      "items": [
        {
          "sentence": "neither cat",
          "tree": "[.DP [.D neither ] [.NP cat ] ]",
          "targets": [
            "card2(λx.cat(x)) : λg.¬∃x[cat(x) ∧ g(x)]"
          ],
          "reading": {
            "section": "6.7"
          }
        },
        {
          "sentence": "Neither cat has stripes.",
          "tree": "[.S [.DP [.D neither ] [.NP cat ] ] [.VP [.V has ] [.NP stripes ] ] ]",
          "targets": [
            "card2(λx.cat(x)) : ¬∃x[cat(x) ∧ striped(x)]"
          ],
          "reading": {
            "section": "6.7"
          }
        },
        {
          "sentence": "Both cats purr.",
          "tree": "[.S [.DP [.D both ] [.NP cat ] ] [.VP purr ] ]",
          "targets": [
            "card2(λx.cat(x)) : ∀x[cat(x) → purr(x)]"
          ],
          "reading": {
            "section": "6.7"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 6 · Quantifiers: Their Semantic Type\n\nQuantificational DPs denote neither individuals (§6.1 — the entailment patterns of\n*no man*, *only John* diverge from names) nor sets of individuals (§6.2). The\nFregean answer:\n\n## 6.3 Generalized quantifiers\n\nIn *Nothing vanished* the VP is `⟨e,t⟩` and the sentence must be `t`; if\ncomposition is FA, the subject is a function from `⟨e,t⟩` to `t` — type\n`⟨⟨e,t⟩,t⟩`, a **generalized quantifier** (\\ref{gq}). The book's entry: ⟦nothing⟧ =\n\"λf ∈ D⟨e,t⟩ . there is no x ∈ Dₑ such that f(x) = 1\" (p. 141); in the bundle's\nnotation:\n\n\\begin{derivation}\n[[something]]   = λf . ∃x[f(x)=1]    : <<e,t>,t>\n[[nothing]]     = λf . ¬∃x[f(x)=1]   : <<e,t>,t>\n[[everything]]  = λf . ∀x[f(x)=1]    : <<e,t>,t>\n\\end{derivation}\n\n\\begin{derivation}\n[[Nothing vanished]]  = ¬∃x[vanish(x)=1]    : t\n\\end{derivation}\n\n\\ex<gq> Something vanished.\n\\xe\n\n\\ex Nothing vanished.\n\\xe\n\n\\ex Everything vanished.\n\\xe\n\n## 6.4 Quantifying determiners\n\nA determiner takes the noun restrictor first: type `⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩` (\\ref{det}).\nThe book's ⟦every⟧: \"λf ∈ D⟨e,t⟩ . [λg ∈ D⟨e,t⟩ . for all x ∈ Dₑ such that\nf(x) = 1, g(x) = 1]\" (p. 146); \"in each case\" the two nodes compose by FA (p. 146).\n\n\\begin{derivation}\n[[every]]  = λf . λg . ∀x[f(x)=1 → g(x)=1]    : <<e,t>,<<e,t>,t>>\n[[no]]     = λf . λg . ¬∃x[f(x)=1 and g(x)=1]  : <<e,t>,<<e,t>,t>>\n[[some]]   = λf . λg . ∃x[f(x)=1 and g(x)=1]   : <<e,t>,<<e,t>,t>>\n\\end{derivation}\n\n\\begin{derivation}\n[[every cat]]           = λg . ∀x[cat(x)=1 → g(x)=1]    : <<e,t>,t>\n[[Every cat vanished]]  = ∀x[cat(x)=1 → vanish(x)=1]    : t\n\\end{derivation}\n\n\\ex<det> Every cat vanished.\n\\xe\n\n\\ex No cat vanished.\n\\xe\n\n## 6.7 Presuppositional quantifier phrases: *both* and *neither*\n\nWith one cat or three, *Neither cat has stripes* is neither true nor false. H&K's\nentry is partial — \"λA : A ∈ Pow(D) and |A| = 2 . [λB ∈ Pow(D) . A ∩ B = ∅]\"\n(p. 154 (2)) — and \"predicts that (1) presupposes there to be exactly two cats\"\n(p. 154). Our `card2(f)` / `|f|=2` is the same condition on the characteristic\nfunction (engine notation):\n\n\\begin{derivation}\n[[neither]]  = λf : |f|=2 . λg . ¬∃x[f(x)=1 and g(x)=1]   : <<e,t>,<<e,t>,t>>\n[[both]]     = λf : |f|=2 . λg . ∀x[f(x)=1 → g(x)=1]      : <<e,t>,<<e,t>,t>>\n\\end{derivation}\n\n\\begin{derivation}\n[[neither cat]]              = |cat|=2 : λg . ¬∃x[cat(x)=1 and g(x)=1]   : <<e,t>,t>\n[[Neither cat has stripes]]  = |cat|=2 : ¬∃x[cat(x)=1 and striped(x)=1]  : t\n\\end{derivation}\n\n§6.7.2: presupposition is \"actually incompatible with a strictly relational theory\"\n(pp. 154–155) — a relation between sets is total and cannot leave the\nnot-exactly-two-cats case truth-valueless. Partiality is essential.\n\n\\ex<both> Neither cat has stripes.\n\\xe\n\n\\ex Both cats have stripes.\n\\xe\n\n> **Beyond this chapter (§6.8).** §6.7's DP presuppositions (*both*, *neither*,\n> definites) are \"rather apparent and uncontroversial\" (p. 159); the contested\n> cases are Strawson's strong determiners (*every*, *all*, *no*, pp. 159–162) and\n> the §6.8.2 weak determiners (*a*, *few*, *most*, p. 163). Separately, *most* is\n> not definable from 1-place quantifiers — ch. 7's argument (§7.4.2, pp. 191–193)."
  }
}