{
  "compose": 1,
  "id": "hk12-intensions",
  "title": "First Steps Towards an Intensional Semantics (§12.1–12.3)",
  "subtitle": "Heim & Kratzer — opacity, intensions, and Intensional Functional Application",
  "notation": "hk",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "m jn d",
      "s": "w0"
    },
    "variables": {
      "e": "x y z",
      "s": "w' w''",
      "st": "p q",
      "et": "P Q X Y"
    }
  },
  "lexicon": [
    {
      "words": [
        "Mary"
      ],
      "denotation": "m"
    },
    {
      "words": [
        "Jan"
      ],
      "denotation": "jn"
    },
    {
      "words": [
        "Dick"
      ],
      "denotation": "d"
    },
    {
      "words": [
        "is"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "loyal"
      ],
      "denotation": "Lx.loyal(w0,x)"
    },
    {
      "words": [
        "deceitful"
      ],
      "denotation": "Lx.deceitful(w0,x)"
    },
    {
      "words": [
        "happy"
      ],
      "denotation": "Lx.happy(w0,x)"
    },
    {
      "words": [
        "believes",
        "believe"
      ],
      "denotation": "Lp.Lx.Aw'[dox(w0,x,w') -> p(w')]",
      "displayAs": "believes"
    },
    {
      "words": [
        "thinks",
        "think"
      ],
      "denotation": "Lp.Lx.Aw'[think(w0,x,w') -> p(w')]",
      "displayAs": "thinks"
    },
    {
      "words": [
        "hopes",
        "hope"
      ],
      "denotation": "Lp.Lx.Aw'[hope(w0,x,w') -> p(w')]",
      "displayAs": "hopes"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "intensionalFunctionApplication": true
    }
  },
  "exercises": [
    {
      "title": "A. Extensional clauses (FA) — §12.1",
      "instructions": "Each embedded sentence is, by itself, extensional: it denotes a truth value (type t) at the evaluation world w. Compose by Function Application. Predicates carry the world: ⟦loyal⟧^w = λx.loyal(w,x).",
      "items": [
        {
          "sentence": "Jan is loyal.",
          "tree": "[.S [.DP Jan ] [.VP [.V is ] [.AP loyal ] ] ]",
          "targets": [
            "loyal(w0,jn)"
          ],
          "reading": {
            "section": "12.1"
          }
        },
        {
          "sentence": "Dick is deceitful.",
          "tree": "[.S [.DP Dick ] [.VP [.V is ] [.AP deceitful ] ] ]",
          "targets": [
            "deceitful(w0,d)"
          ],
          "reading": {
            "section": "12.1"
          }
        }
      ]
    },
    {
      "title": "B. Attitude reports via IFA — §12.3",
      "instructions": "An attitude verb wants a PROPOSITION ⟨s,t⟩, but the embedded clause supplies only a truth value t — FA is stuck. Intensional Functional Application (IFA) feeds the verb the clause's INTENSION λw'.⟦clause⟧^w'. The embedded clause is then evaluated at the belief-worlds w'.",
      "items": [
        {
          "sentence": "Mary believes Jan is loyal.",
          "tree": "[.S [.DP Mary ] [.VP [.V believes ] [.S [.DP Jan ] [.VP [.V is ] [.AP loyal ] ] ] ] ]",
          "targets": [
            "∀w'[dox(w0,m,w') → loyal(w',jn)]"
          ],
          "reading": {
            "section": "12.3"
          }
        },
        {
          "sentence": "Mary believes Dick is deceitful.",
          "tree": "[.S [.DP Mary ] [.VP [.V believes ] [.S [.DP Dick ] [.VP [.V is ] [.AP deceitful ] ] ] ] ]",
          "targets": [
            "∀w'[dox(w0,m,w') → deceitful(w',d)]"
          ],
          "reading": {
            "section": "12.3"
          }
        },
        {
          "sentence": "Mary thinks Jan is happy.",
          "tree": "[.S [.DP Mary ] [.VP [.V thinks ] [.S [.DP Jan ] [.VP [.V is ] [.AP happy ] ] ] ] ]",
          "targets": [
            "∀w'[think(w0,m,w') → happy(w',jn)]"
          ],
          "reading": {
            "section": "12.3"
          }
        }
      ]
    },
    {
      "title": "C. Other attitude verbs — §12.3",
      "instructions": "The same IFA mechanism handles every attitude verb; only the accessibility relation changes. Each quantifies over the worlds compatible with the subject's attitude in w. (H&K's ch. 12 triple is believe/know/hope, p. 306; think is our addition on the same pattern.)",
      "items": [
        {
          "sentence": "Mary hopes Jan is happy.",
          "tree": "[.S [.DP Mary ] [.VP [.V hopes ] [.S [.DP Jan ] [.VP [.V is ] [.AP happy ] ] ] ] ]",
          "targets": [
            "∀w'[hope(w0,m,w') → happy(w',jn)]"
          ],
          "reading": {
            "section": "12.3"
          }
        },
        {
          "sentence": "Dick believes Mary is happy.",
          "tree": "[.S [.DP Dick ] [.VP [.V believes ] [.S [.DP Mary ] [.VP [.V is ] [.AP happy ] ] ] ] ]",
          "targets": [
            "∀w'[dox(w0,d,w') → happy(w',m)]"
          ],
          "reading": {
            "section": "12.3"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 12 · First Steps Towards an Intensional Semantics\n\nSo far every denotation has been an extension. Ch. 12 shows where that breaks and\nrelativises interpretation to a possible world: `⟦α⟧^w`.\n\n## 12.1 Where the extensional semantics breaks down\n\nIf composition sees only extensions, co-extensional parts are interchangeable.\nInside *believe* they are not. H&K's pair (\\ref{break}): with Jan loyal and Dick\ndeceitful in the actual world, `⟦Jan is loyal⟧^w = ⟦Dick is deceitful⟧^w = 1`, yet\nMary can believe one report and not the other (pp. 299–300). *Believe* is an opaque\ncontext: what matters is which worlds the complement is true in.\n\n\\ex<break> Mary believes Jan is loyal.\n\\xe\n\n\\ex Mary believes Dick is deceitful.\n\\xe\n\n## 12.2 What to do: intensions\n\nThe intension of `α` is `λw . ⟦α⟧^w`; for a sentence, a **proposition**, type\n`⟨s,t⟩` (p. 307). Names are rigid: `⟦Jan⟧^w = Jan` at every `w` (p. 304). Temporal\ndependence is set aside (p. 302).\n\nEngine notes: H&K's official types only build ⟨s,a⟩ (p. 303 (1d)) — no bare `Dₛ`;\nwe treat `s` as a type so worlds can be arguments, writing the world inside the\npredicate: `loyal(w,x)` for `⟦loyal⟧^w = λx . x is loyal in w` (p. 307). The matrix\nworld prints as `w`, shifted worlds as `w'`.\n\n\\begin{derivation}\n[[Jan is loyal]]^w   = loyal(w,jn)=1            : t\n(its intension)      = λw' . loyal(w',jn)=1     : <s,t>\n\\end{derivation}\n\n## 12.3 An intensional semantics\n\nFollowing Hintikka, ⟦believe⟧ quantifies over the subject's belief-worlds — the\nbook's entry: \"λp ∈ D⟨s,t⟩ . [λx ∈ D . for all w′ compatible with what x believes\nin w, p(w′) = 1]\" (p. 306 (7)); type `⟨⟨s,t⟩,⟨e,t⟩⟩`. In the bundle:\n`λp . λx . ∀w'[Dox(w)(x)(w') → p(w')=1]`.\n\nThe verb wants a proposition; the embedded clause supplies `t`. One new rule\n(p. 308 (9)):\n\n**Intensional Functional Application (IFA).** If α is a branching node with\ndaughters β and γ, and `⟦β⟧^w` is a function whose domain contains\n`λw' . ⟦γ⟧^w'`, then `⟦α⟧^w = ⟦β⟧^w(λw' . ⟦γ⟧^w')`.\n\nA composition rule, not an operator in the tree — H&K's endnote credits a rule of\nBittner's as an analogue; Montague is credited for the type system (p. 303).\n\n\\begin{derivation}\n[[believes [Jan is loyal]]]^w  = [[believes]]^w(λw' . loyal(w',jn)=1)            (IFA)\n[[Mary believes Jan is loyal]]^w = ∀w'[Dox(w)(m)(w') → loyal(w',jn)=1]           : t\n\\end{derivation}\n\nTrue at `w` iff Jan is loyal in every world compatible with Mary's beliefs in `w`.\nThe complement is evaluated at the belief-worlds `w'`, not at `w` — swapping in\n*Dick is deceitful* changes the result. Opacity captured (pp. 308–309).\n\n\\begin{derivation}\n[[Mary believes Dick is deceitful]]^w = ∀w'[Dox(w)(m)(w') → deceitful(w',d)=1]   : t\n\\end{derivation}\n\n> **Beyond these first steps (§12.4, \"Limitations and prospects\").** Carnap's\n> objection: propositions-as-world-sets are too coarse for attitudes — Bigelow's\n> pair, true in exactly the same worlds yet not believed together (p. 310);\n> structured meanings beckon. De re vs de dicto is the book's Exercise 1 of §12.3\n> (\"Mary hopes that a plumber is available\", p. 309). We stop where H&K do."
  }
}