{
  "compose": 1,
  "id": "hk4-definites",
  "title": "Predicates, Modifiers & the Definite Article (§4.3–4.5)",
  "subtitle": "Heim & Kratzer — Predicate Modification and Frege's partial 'the'",
  "notation": "hk",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "k j o tx a jn"
    },
    "variables": {
      "e": "x y z",
      "et": "f g P Q X Y",
      "t": "p q"
    }
  },
  "lexicon": [
    {
      "words": [
        "Kaline"
      ],
      "denotation": "k"
    },
    {
      "words": [
        "Julius"
      ],
      "denotation": "j"
    },
    {
      "words": [
        "Joe"
      ],
      "denotation": "o"
    },
    {
      "words": [
        "Texas"
      ],
      "denotation": "tx"
    },
    {
      "words": [
        "Ann"
      ],
      "denotation": "a"
    },
    {
      "words": [
        "Jan"
      ],
      "denotation": "jn"
    },
    {
      "words": [
        "is",
        "are"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "of"
      ],
      "denotation": "Lx.x"
    },
    {
      "words": [
        "cat"
      ],
      "denotation": "Lx.cat(x)"
    },
    {
      "words": [
        "house"
      ],
      "denotation": "Lx.house(x)"
    },
    {
      "words": [
        "book"
      ],
      "denotation": "Lx.book(x)"
    },
    {
      "words": [
        "gray"
      ],
      "denotation": "Lx.gray(x)"
    },
    {
      "words": [
        "empty"
      ],
      "denotation": "Lx.empty(x)"
    },
    {
      "words": [
        "fond"
      ],
      "denotation": "Ly.Lx.fond(x,y)"
    },
    {
      "words": [
        "in"
      ],
      "denotation": "Ly.Lx.in(x,y)"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "Lf : Ex[f(x) ∧ Ay[f(y) -> y=x]] . Ix.f(x)",
      "displayAs": "the"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "predicateModification": true,
      "nonBranchingNodes": true
    }
  },
  "exercises": [
    {
      "title": "A. Predicate Modification (§4.3)",
      "instructions": "Two ⟨e,t⟩ sisters cannot combine by Functional Application. Predicate Modification (PM) conjoins them: ⟦gray cat⟧ = λx. gray(x)=1 ∧ cat(x)=1. The copula is and the article a are vacuous (λX.X).",
      "items": [
        {
          "sentence": "gray cat",
          "tree": "[.NP [.AP gray ] [.NP cat ] ]",
          "reading": {
            "section": "4.3"
          }
        },
        {
          "sentence": "Julius is a gray cat.",
          "tree": "[.S [.DP Julius ] [.VP [.V is ] [.DP [.D a ] [.NP [.AP gray ] [.NP cat ] ] ] ] ]",
          "targets": [
            "gray(j) ∧ cat(j)"
          ],
          "reading": {
            "section": "4.3"
          }
        },
        {
          "sentence": "Kaline is a cat in Texas.",
          "tree": "[.S [.DP Kaline ] [.VP [.V is ] [.DP [.D a ] [.NP [.NP cat ] [.PP [.P in ] [.DP Texas ] ] ] ] ] ]",
          "targets": [
            "cat(k) ∧ in(k,tx)"
          ],
          "reading": {
            "section": "4.3"
          }
        }
      ]
    },
    {
      "title": "B. The definite article (§4.4)",
      "instructions": "Following Frege, the is a PARTIAL function: ⟦the⟧ = λf : ∃!x[f(x)=1] . ιx[f(x)=1]. The condition after the colon (before the dot) is presupposed — a unique satisfier; the value after the dot is the unique individual. In a sentence, the presupposition projects to the top.",
      "items": [
        {
          "sentence": "the cat",
          "tree": "[.DP [.D the ] [.NP cat ] ]",
          "targets": [
            "∃x[cat(x) ∧ ∀y[cat(y) → y=x]] : ιx.cat(x)"
          ],
          "reading": {
            "section": "4.4"
          }
        },
        {
          "sentence": "The cat is gray.",
          "tree": "[.S [.DP [.D the ] [.NP cat ] ] [.VP [.V is ] [.AP gray ] ] ]",
          "targets": [
            "∃x[cat(x) ∧ ∀y[cat(y) → y=x]] : gray(ιx.cat(x))"
          ],
          "reading": {
            "section": "4.4"
          }
        }
      ]
    },
    {
      "title": "C. Modifiers in definite descriptions (§4.5)",
      "instructions": "PM builds the modified predicate first; then the takes it as argument, so the uniqueness presupposition is about gray cats. Embedding the description as an argument projects the presupposition through the predicate.",
      "items": [
        {
          "sentence": "the gray cat",
          "tree": "[.DP [.D the ] [.NP [.AP gray ] [.NP cat ] ] ]",
          "targets": [
            "∃x[gray(x) ∧ cat(x) ∧ ∀y[gray(y) ∧ cat(y) → y=x]] : ιx.gray(x) ∧ cat(x)"
          ],
          "reading": {
            "section": "4.5"
          }
        },
        {
          "sentence": "The gray cat is fond of Joe.",
          "tree": "[.S [.DP [.D the ] [.NP [.AP gray ] [.NP cat ] ] ] [.VP [.V is ] [.AP [.A fond ] [.PP [.P of ] [.DP Joe ] ] ] ] ]",
          "targets": [
            "∃x[gray(x) ∧ cat(x) ∧ ∀y[gray(y) ∧ cat(y) → y=x]] : fond(ιx.gray(x) ∧ cat(x), o)"
          ],
          "reading": {
            "section": "4.5"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 4 · Nonverbal Predicates, Modifiers, and the Definite Article\n\n§4.1, \"Semantically vacuous words\": the copula *is* and the indefinite *a* carry no\ncontent — both are the identity `λX . X` (p. 62). *Is a cat* does the work of *cat*.\n\n## 4.3 Predicate Modification\n\n*Gray* and *cat* are both `⟨e,t⟩`; neither can take the other as argument, so FA\nfails. H&K's new rule (§4.3.1, p. 65):\n\n**Predicate Modification (PM).** If α is a branching node whose daughters β and γ\nare both of type `⟨e,t⟩`, then \"⟦α⟧ = λx ∈ Dₑ . ⟦β⟧(x) = ⟦γ⟧(x) = 1\" — equivalently\n(the book's own (7)): `λx . ⟦β⟧(x)=1 and ⟦γ⟧(x)=1`.\n\n\\begin{derivation}\n[[gray]]      = λx . gray(x)=1                : <e,t>\n[[cat]]       = λx . cat(x)=1                 : <e,t>\n[[gray cat]]  = λx . gray(x)=1 and cat(x)=1   : <e,t>   (PM)\n\\end{derivation}\n\n\\begin{forest}\n[S{gray(j)=1 and cat(j)=1}\n  [DP{j} Julius]\n  [VP{λx . gray(x)=1 and cat(x)=1}\n    [V{λX.X} is]\n    [DP{λx . gray(x)=1 and cat(x)=1}\n      [D{λX.X} a]\n      [NP{λx . gray(x)=1 and cat(x)=1}\n        [AP{λx . gray(x)=1} gray]\n        [NP{λx . cat(x)=1} cat]]]]]\n\\end{forest}\n\nPrepositional modifiers compose the same way: `⟦in⟧ = λy . λx . in(x,y)=1`, and\n*in Texas* combines with *cat* by PM.\n\n\\ex<gc> Julius is a gray cat.\n\\xe\n\n\\ex Kaline is a cat in Texas.\n\\xe\n\n§4.3.3 (p. 68): nonintersective adjectives — *former*, *alleged* — resist this\nanalysis. PM is for the intersective core.\n\n## 4.4 The definite article\n\n\"A lexical entry inspired by Frege\" (§4.4.1): *the* is a partial function of type\n`⟨⟨e,t⟩,e⟩`, defined only for a predicate with exactly one satisfier — the book's\nentry is \"λf : f ∈ D⟨e,t⟩ and there is exactly one x such that f(x) = 1 . the unique\ny such that f(y) = 1\" (p. 75). In the bundle's notation:\n\n\\begin{derivation}\n[[the]] = λf : ∃!x[f(x)=1] . ιx[f(x)=1]   : <<e,t>,e>\n\\end{derivation}\n\n\\begin{derivation}\n[[the cat]] = ∃!x[cat(x)=1] : ιx[cat(x)=1]   : e\n\\end{derivation}\n\nThe condition after the colon is presupposed; the value after the dot is asserted\n(§4.4.2). When the presupposition fails, a sentence containing the description is\n\"neither true nor false\" (pp. 75–76). With the vacuous copula, the condition rides\nto the top:\n\n\\begin{derivation}\n[[The cat is gray]] = ∃!x[cat(x)=1] : gray(ιx[cat(x)=1])=1   : t\n\\end{derivation}\n\n§4.4.4: presupposition failure is not uninterpretability — a type mismatch is\nvisible from the types alone; failure depends on the actual denotations.\n\n\\ex<thecat> the cat\n\\xe\n\n\\ex The cat is gray.\n\\xe\n\n## 4.5 Modifiers in definite descriptions\n\nCompose the modifier first (PM), then the article: the presupposition of *the gray\ncat* is uniqueness of *gray cats* (\\ref{tgc}).\n\n\\begin{derivation}\n[[gray cat]]      = λx . gray(x)=1 and cat(x)=1                        : <e,t>\n[[the gray cat]]  = ∃!x[gray(x)=1 and cat(x)=1] : ιx[gray(x)=1 and cat(x)=1]  : e\n\\end{derivation}\n\n\\begin{derivation}\n[[the gray cat is fond of Joe]]\n   = ∃!x[gray(x)=1 and cat(x)=1] : fond(ιx[gray(x)=1 and cat(x)=1], o)=1   : t\n\\end{derivation}\n\nDescriptions also take relative-clause restrictors — *the house which is empty*.\nHow a gapped clause comes to denote an `⟨e,t⟩` property is ch. 5's Predicate\nAbstraction.\n\n\\ex<tgc> the gray cat\n\\xe\n\n\\ex The gray cat is fond of Joe.\n\\xe\n\n\\ex the house which is empty\n\\xe"
  }
}