{
  "compose": 1,
  "id": "hk5-relatives",
  "title": "Relative Clauses & Predicate Abstraction (§5.1–5.3)",
  "subtitle": "Heim & Kratzer — traces, variable binding, and the third composition rule",
  "notation": "hk",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "j m o b"
    },
    "variables": {
      "e": "x y z",
      "et": "f g P Q X Y",
      "t": "p q"
    }
  },
  "lexicon": [
    {
      "words": [
        "John"
      ],
      "denotation": "j"
    },
    {
      "words": [
        "Mary"
      ],
      "denotation": "m"
    },
    {
      "words": [
        "Joe"
      ],
      "denotation": "o"
    },
    {
      "words": [
        "Bill"
      ],
      "denotation": "b"
    },
    {
      "words": [
        "is",
        "are"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "who",
        "which",
        "that"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "house"
      ],
      "denotation": "Lx.house(x)"
    },
    {
      "words": [
        "picture"
      ],
      "denotation": "Lx.picture(x)"
    },
    {
      "words": [
        "movie"
      ],
      "denotation": "Lx.movie(x)"
    },
    {
      "words": [
        "man"
      ],
      "denotation": "Lx.man(x)"
    },
    {
      "words": [
        "empty"
      ],
      "denotation": "Lx.empty(x)"
    },
    {
      "words": [
        "available"
      ],
      "denotation": "Lx.available(x)"
    },
    {
      "words": [
        "expensive"
      ],
      "denotation": "Lx.expensive(x)"
    },
    {
      "words": [
        "abandoned"
      ],
      "denotation": "Ly.Lx.abandon(x,y)"
    },
    {
      "words": [
        "saw"
      ],
      "denotation": "Ly.Lx.see(x,y)"
    },
    {
      "words": [
        "likes"
      ],
      "denotation": "Ly.Lx.like(x,y)"
    },
    {
      "words": [
        "bought"
      ],
      "denotation": "Ly.Lx.buy(x,y)"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "Lf : Ex[f(x) ∧ Ay[f(y) -> y=x]] . Ix.f(x)",
      "displayAs": "the"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "Lf.Lg.Ax[f(x) -> g(x)]"
    },
    {
      "words": [
        "no",
        "No"
      ],
      "denotation": "Lf.Lg.~Ex[f(x) ∧ g(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "predicateModification": true,
      "nonBranchingNodes": true,
      "predicateAbstraction": true
    }
  },
  "exercises": [
    {
      "title": "A. Subject-gap relatives (§5.1–5.2)",
      "instructions": "A relative clause is an intersective modifier of type ⟨e,t⟩, combining with the head noun by Predicate Modification. The trace t₁ denotes a variable; Predicate Abstraction over its index turns the clause into a property. The relative pronoun is vacuous (λX.X).",
      "items": [
        {
          "sentence": "which is empty",
          "tree": "[.CP [.C which ] [.LP 1 [.S [.DP t1 ] [.VP [.V is ] [.AP empty ] ] ] ] ]",
          "targets": [
            "λx.empty(x)"
          ],
          "reading": {
            "section": "5.2"
          }
        },
        {
          "sentence": "house which is empty",
          "tree": "[.NP [.NP house ] [.CP [.C which ] [.LP 1 [.S [.DP t1 ] [.VP [.V is ] [.AP empty ] ] ] ] ] ]",
          "targets": [
            "λx.house(x) ∧ empty(x)"
          ],
          "reading": {
            "section": "5.2"
          }
        },
        {
          "sentence": "man who is available",
          "tree": "[.NP [.NP man ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP [.V is ] [.AP available ] ] ] ] ] ]",
          "targets": [
            "λx.man(x) ∧ available(x)"
          ],
          "reading": {
            "section": "5.2"
          }
        }
      ]
    },
    {
      "title": "B. Object-gap relatives & Predicate Abstraction (§5.2.3)",
      "instructions": "When the trace is in OBJECT position, the desired value λx. John abandoned x is not the value of any subtree — Predicate Abstraction is essential. PA abstracts over the trace index after the subject and verb have composed around the trace.",
      "items": [
        {
          "sentence": "which John abandoned t",
          "tree": "[.CP [.C which ] [.LP 1 [.S [.DP John ] [.VP [.V abandoned ] [.DP t1 ] ] ] ] ]",
          "targets": [
            "λx.abandon(j,x)"
          ],
          "reading": {
            "section": "5.2.3"
          }
        },
        {
          "sentence": "picture which John abandoned t",
          "tree": "[.NP [.NP picture ] [.CP [.C which ] [.LP 1 [.S [.DP John ] [.VP [.V abandoned ] [.DP t1 ] ] ] ] ] ]",
          "targets": [
            "λx.picture(x) ∧ abandon(j,x)"
          ],
          "reading": {
            "section": "5.2.3"
          }
        },
        {
          "sentence": "movie which Mary saw t",
          "tree": "[.NP [.NP movie ] [.CP [.C which ] [.LP 1 [.S [.DP Mary ] [.VP [.V saw ] [.DP t1 ] ] ] ] ] ]",
          "targets": [
            "λx.movie(x) ∧ see(m,x)"
          ],
          "reading": {
            "section": "5.2.3"
          }
        }
      ]
    },
    {
      "title": "C. Relatives in full DPs (§5.1, §5.3)",
      "instructions": "The modified noun feeds a determiner. With the (Chapter 4's partial function), the uniqueness presupposition is now about the relative-clause-restricted set. With a quantifier, the relative clause restricts the quantifier's domain.",
      "items": [
        {
          "sentence": "The house which is empty is available.",
          "tree": "[.S [.DP [.D the ] [.NP [.NP house ] [.CP [.C which ] [.LP 1 [.S [.DP t1 ] [.VP [.V is ] [.AP empty ] ] ] ] ] ] ] [.VP [.V is ] [.AP available ] ] ]",
          "targets": [
            "∃x[house(x) ∧ empty(x) ∧ ∀y[house(y) ∧ empty(y) → y=x]] : available(ιx.house(x) ∧ empty(x))"
          ],
          "reading": {
            "section": "5.1"
          }
        },
        {
          "sentence": "Every picture which John abandoned is expensive.",
          "tree": "[.S [.DP [.D every ] [.NP [.NP picture ] [.CP [.C which ] [.LP 1 [.S [.DP John ] [.VP [.V abandoned ] [.DP t1 ] ] ] ] ] ] ] [.VP [.V is ] [.AP expensive ] ] ]",
          "targets": [
            "∀x[picture(x) ∧ abandon(j,x) → expensive(x)]"
          ],
          "reading": {
            "section": "5.3"
          }
        },
        {
          "sentence": "No movie which Mary saw is expensive.",
          "tree": "[.S [.DP [.D no ] [.NP [.NP movie ] [.CP [.C which ] [.LP 1 [.S [.DP Mary ] [.VP [.V saw ] [.DP t1 ] ] ] ] ] ] ] [.VP [.V is ] [.AP expensive ] ] ]",
          "targets": [
            "¬∃x[movie(x) ∧ see(m,x) ∧ expensive(x)]"
          ],
          "reading": {
            "section": "5.3"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 5 · Relative Clauses, Variables, and Variable Binding\n\nA restrictive relative clause is an intersective modifier: following Quine, *which\nis empty* denotes the same `⟨e,t⟩` property as *empty*, and combines with the head\nnoun by PM. The chapter's problem is how a clause containing a gap comes to denote a\nproperty at all.\n\n## 5.1 Relative clauses as predicates\n\nIn (\\ref{empty}), PM gives *house which is empty* the value \"λx ∈ Dₑ . x is a house\nand x is empty\" (p. 88), and ch. 4's partial *the* applies to that: defined iff\nexactly one empty house.\n\n\\ex<empty> The house which is empty is available.\n\\xe\n\nRestrictive vs non-restrictive: *The house, which is empty, is available*\npresupposes a unique **house**. Set aside (p. 88).\n\n## 5.2 Inside the relative clause: traces and variables\n\nAn object gap is the hard case: the target is \"λx ∈ D . John abandoned x\" (p. 90),\nand that is not the value of any subtree. The trace cannot pick up a referent\n(§5.2.1) — no individual-denoting constituent is available, and using the containing\nDP is circular. §5.2.2: the trace is a **variable**, interpreted relative to an\nassignment.\n\n**Traces and Pronouns.** Relative to an assignment `g`, a trace `t_i` (or pronoun)\ndenotes `g(i)` — type `e` (p. 111).\n\n\\ex<abandon> the picture which John abandoned t\n\\xe\n\n\\ex the movie which Mary saw t\n\\xe\n\n## 5.2.3 Predicate Abstraction\n\n**Predicate Abstraction (PA).** If α is a branching node whose daughters are an\nindex `i` and a node β, then `⟦α⟧^g = λx . ⟦β⟧^{g[i→x]}`.\n\n\\begin{derivation}\n[[John abandoned t1]]              = abandon(j,x)=1            : t\n[[1 [John abandoned t1]]]          = λx . abandon(j,x)=1       : <e,t>   (PA)\n[[which John abandoned t1]]        = λx . abandon(j,x)=1       : <e,t>\n[[picture]]                        = λx . picture(x)=1         : <e,t>\n[[picture which John abandoned]]   = λx . picture(x)=1 and abandon(j,x)=1   : <e,t>   (PM)\n\\end{derivation}\n\nThe relative pronoun passes the abstract up unchanged. Seam: in §5.2.3 itself the\npronoun is syncategorematic — \"not simply vacuous\" (pp. 96–98); the vacuous-pronoun\ntreatment implemented here is the ch. 7 revision (p. 186), back-applied so one rule\nset serves both chapters. The rule as stated above is likewise the book's final\nform (pp. 111–112, 186); §5.2.3's debut version abstracts directly over the\npronoun's index.\n\nSubject gaps compose the same way:\n\n\\begin{derivation}\n[[t1 is empty]]                = empty(x)=1            : t\n[[1 [t1 is empty]]]            = λx . empty(x)=1       : <e,t>   (PA)\n[[which is empty]]             = λx . empty(x)=1       : <e,t>\n[[house which is empty]]       = λx . house(x)=1 and empty(x)=1   : <e,t>   (PM)\n\\end{derivation}\n\n## 5.3 Multiple variables and such-that relatives\n\nPA scales to multiple variables and to *such that* relatives — *such* binds the\nclause-internal index (\"the book such that Joe bought it\", p. 107). Once indices\ndrive PA, the rule serves every binder in the grammar; ch. 7 reuses it for\nquantifier raising. Group C's quantified heads borrow ⟦every⟧/⟦no⟧ from §6.4\n(p. 146) — ch. 5 itself stops at definite heads.\n\n\\ex<such> the picture such that Mary saw it *(our example, after pp. 90/107)*\n\\xe"
  }
}