{
  "compose": 1,
  "id": "hk2-fa",
  "title": "Function Application & Semantic Types (§2.1–2.4)",
  "subtitle": "Heim & Kratzer — the first Fregean fragment: FA, types e/t, transitive verbs, connectives",
  "notation": "hk",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "a jn mr"
    },
    "variables": {
      "e": "x y z",
      "et": "f g P Q X Y",
      "t": "p q"
    }
  },
  "lexicon": [
    {
      "words": [
        "Ann"
      ],
      "denotation": "a"
    },
    {
      "words": [
        "Jan"
      ],
      "denotation": "jn"
    },
    {
      "words": [
        "Maria"
      ],
      "denotation": "mr"
    },
    {
      "words": [
        "smokes",
        "smoke"
      ],
      "denotation": "Lx.smoke(x)"
    },
    {
      "words": [
        "works",
        "work"
      ],
      "denotation": "Lx.work(x)"
    },
    {
      "words": [
        "is"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "boring"
      ],
      "denotation": "Lx.boring(x)"
    },
    {
      "words": [
        "likes",
        "like"
      ],
      "denotation": "Ly.Lx.like(x,y)"
    },
    {
      "words": [
        "knows",
        "know"
      ],
      "denotation": "Ly.Lx.know(x,y)"
    },
    {
      "words": [
        "and"
      ],
      "denotation": "Lp.Lq.q ∧ p",
      "displayAs": "and"
    },
    {
      "words": [
        "itisnotthecasethat",
        "not"
      ],
      "denotation": "Lp.~p",
      "displayAs": "it is not the case that"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true
    }
  },
  "exercises": [
    {
      "title": "A. Intransitive verbs: FA, types e and ⟨e,t⟩ (§2.1)",
      "instructions": "A name denotes an individual (type e); an intransitive verb denotes a characteristic function (type ⟨e,t⟩). Functional Application feeds the subject to the verb. The copula is is vacuous (λX.X) — strictly §4.1's entry (p. 62), previewed here; ch. 2's own fragment stops at names + (in)transitive verbs.",
      "items": [
        {
          "sentence": "Ann smokes.",
          "tree": "[.S [.NP [.N Ann ] ] [.VP [.V smokes ] ] ]",
          "targets": [
            "smoke(a)"
          ],
          "reading": {
            "section": "2.1"
          }
        },
        {
          "sentence": "Maria works.",
          "tree": "[.S [.NP [.N Maria ] ] [.VP [.V works ] ] ]",
          "targets": [
            "work(mr)"
          ],
          "reading": {
            "section": "2.1"
          }
        },
        {
          "sentence": "Ann is boring.",
          "tree": "[.S [.NP [.N Ann ] ] [.VP [.V is ] [.AP boring ] ] ]",
          "targets": [
            "boring(a)"
          ],
          "reading": {
            "section": "2.1"
          }
        }
      ]
    },
    {
      "title": "B. Transitive verbs: Schönfinkelization (§2.3)",
      "instructions": "A transitive verb is a Schönfinkeled (curried) function of type ⟨e,⟨e,t⟩⟩ that takes its OBJECT first, then its subject. FA applies twice: verb + object, then VP + subject.",
      "items": [
        {
          "sentence": "Ann likes Jan.",
          "tree": "[.S [.NP Ann ] [.VP [.V likes ] [.NP Jan ] ] ]",
          "targets": [
            "like(a,jn)"
          ],
          "reading": {
            "section": "2.3"
          }
        },
        {
          "sentence": "Jan knows Maria.",
          "tree": "[.S [.NP Jan ] [.VP [.V knows ] [.NP Maria ] ] ]",
          "targets": [
            "know(jn,mr)"
          ],
          "reading": {
            "section": "2.3"
          }
        },
        {
          "sentence": "Maria likes Ann.",
          "tree": "[.S [.NP Maria ] [.VP [.V likes ] [.NP Ann ] ] ]",
          "targets": [
            "like(mr,a)"
          ],
          "reading": {
            "section": "2.3"
          }
        }
      ]
    },
    {
      "title": "C. Sentential connectives (§2.4)",
      "instructions": "Negation (it is not the case that) is type ⟨t,t⟩; sentential and is type ⟨t,⟨t,t⟩⟩. They take whole sentences (type t) as arguments.",
      "items": [
        {
          "sentence": "It is not the case that Jan smokes.",
          "tree": "[.S [.Neg not ] [.S [.NP Jan ] [.VP smokes ] ] ]",
          "targets": [
            "¬smoke(jn)"
          ],
          "reading": {
            "section": "2.4"
          }
        },
        {
          "sentence": "Jan works, and it is not the case that Jan smokes.",
          "tree": "[.S [.S [.NP Jan ] [.VP works ] ] [.Cnj [.Con and ] [.S [.Neg not ] [.S [.NP Jan ] [.VP smokes ] ] ] ] ]",
          "targets": [
            "work(jn) ∧ ¬smoke(jn)"
          ],
          "reading": {
            "section": "2.4"
          }
        },
        {
          "sentence": "Ann smokes, and Maria works.",
          "tree": "[.S [.S [.NP Ann ] [.VP smokes ] ] [.Cnj [.Con and ] [.S [.NP Maria ] [.VP works ] ] ] ]",
          "targets": [
            "smoke(a) ∧ work(mr)"
          ],
          "reading": {
            "section": "2.4"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 2 · Executing the Fregean Program\n\nA lexicon assigns each word a denotation; composition rules combine them up the\ntree. Ch. 2's fragment: proper names and (in)transitive verbs (p. 13).\n\n## 2.1 Denotations and semantic types\n\nNames denote individuals (type `e`); sentences denote truth values (type `t`); an\nintransitive verb denotes the characteristic function of a set, type `⟨e,t⟩`:\n\n\\begin{derivation}\n[[Ann]]      = a                       : e\n[[smokes]]   = λx . x smokes  =  λx . smoke(x)=1    : <e,t>\n\\end{derivation}\n\n**Functional Application (FA).** \"If α is a branching node with daughters β and γ,\nand ⟦β⟧ is a function whose domain contains ⟦γ⟧, then ⟦α⟧ = ⟦β⟧(⟦γ⟧).\"\n\n**Non-branching Nodes (NN).** If α has a single daughter β, then `⟦α⟧ = ⟦β⟧`.\n\nCh. 2 states these only as the anonymous rules (S1)–(S6); the names FA/NN, the\ngeneral statements, and the type labels are §3.1's (p. 44). The bundle uses them\nfrom the start.\n\n\\begin{derivation}\n[[Ann smokes]]  = [[smokes]]([[Ann]])  =  smoke(a)=1    : t\n\\end{derivation}\n\n\\ex<smokes> Ann smokes.\n\\xe\n\n*Ann is boring* looks ahead: the ch. 2 fragment has no adjectives (p. 13); the\nvacuous copula `⟦is⟧ = λf . f` is §4.1's entry (p. 62), previewed here.\n\n\\ex Ann is boring.\n\\xe\n\n## 2.3 Transitive verbs and Schönfinkelization\n\nA transitive verb takes its two arguments one at a time — Schönfinkelized (§2.4),\nobject first:\n\n\\begin{derivation}\n[[likes]]      = λy . λx . x likes y           : <e,<e,t>>\n[[likes Jan]]  = λx . x likes Jan  =  λx . like(x,jn)=1    : <e,t>   (FA)\n\\end{derivation}\n\nIn *Ann likes Jan* (\\ref{likes}) FA applies twice: object into the verb, subject\ninto the VP.\n\n\\begin{derivation}\n[[Ann likes Jan]]  = like(a,jn)=1    : t\n\\end{derivation}\n\n\\ex<likes> Ann likes Jan.\n\\xe\n\n## 2.4 Sentential connectives\n\nThe entries solve the book's own exercises — the p. 32 sentence is Exercise 2\nverbatim, the `⟨t,⟨t,t⟩⟩` *and* is Exercise 3 (p. 40); H&K call the task \"the\nexercise on connectives in section 2.1\" (p. 45).\n\n\\begin{derivation}\n[[it is not the case that]]  = λp . p = 0   =  λp . ¬p    : <t,t>\n[[and]]                      = λp . λq . q = 1 and p = 1  =  λp . λq . q ∧ p    : <t,<t,t>>\n\\end{derivation}\n\n\\begin{derivation}\n[[it is not the case that Jan smokes]]  = ¬smoke(jn)=1    : t\n[[Jan works and ... Jan smokes]]        = work(jn)=1 and ¬smoke(jn)=1    : t\n\\end{derivation}\n\n\\ex<conj> Jan works, and it is not the case that Jan smokes.\n\\xe\n\n---\n\n### A note on Chapter 3: type-driven interpretation\n\nCh. 3 recasts composition as type-driven: the types decide which rule applies —\nFA fires when one sister is a function defined on the other's denotation. The\nengine works the same way."
  }
}