{
  "compose": 1,
  "id": "ch7-relcl",
  "title": "Relative Clauses (§7.3)",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w bi st pi tm el sr mr x1 x2"
    },
    "variables": {
      "e": "x y z",
      "et": "X Y Z P Q R F G H",
      "t": "p q"
    }
  },
  "lexicon": [
    {
      "words": [
        "Frodo"
      ],
      "denotation": "f"
    },
    {
      "words": [
        "Sam"
      ],
      "denotation": "s"
    },
    {
      "words": [
        "Gandalf"
      ],
      "denotation": "g"
    },
    {
      "words": [
        "Sauron"
      ],
      "denotation": "n"
    },
    {
      "words": [
        "Aragorn"
      ],
      "denotation": "a"
    },
    {
      "words": [
        "Boromir"
      ],
      "denotation": "b"
    },
    {
      "words": [
        "Legolas"
      ],
      "denotation": "l"
    },
    {
      "words": [
        "Gimli"
      ],
      "denotation": "m"
    },
    {
      "words": [
        "Galadriel"
      ],
      "denotation": "w"
    },
    {
      "words": [
        "Bilbo"
      ],
      "denotation": "bi"
    },
    {
      "words": [
        "Strider"
      ],
      "denotation": "st"
    },
    {
      "words": [
        "Pippin"
      ],
      "denotation": "pi"
    },
    {
      "words": [
        "Tom"
      ],
      "denotation": "tm"
    },
    {
      "words": [
        "Elrond"
      ],
      "denotation": "el"
    },
    {
      "words": [
        "Saruman"
      ],
      "denotation": "sr"
    },
    {
      "words": [
        "Moria"
      ],
      "denotation": "mr"
    },
    {
      "words": [
        "is",
        "are"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "not"
      ],
      "denotation": "LP.Lx.~P(x)"
    },
    {
      "words": [
        "doesnt",
        "does-not"
      ],
      "denotation": "LP.Lx.~P(x)"
    },
    {
      "words": [
        "who"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(x)"
    },
    {
      "words": [
        "wizard"
      ],
      "denotation": "Lx.wizard(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "ranger"
      ],
      "denotation": "Lx.ranger(x)"
    },
    {
      "words": [
        "human"
      ],
      "denotation": "Lx.human(x)"
    },
    {
      "words": [
        "creature"
      ],
      "denotation": "Lx.creature(x)"
    },
    {
      "words": [
        "brave"
      ],
      "denotation": "Lx.brave(x)"
    },
    {
      "words": [
        "wise"
      ],
      "denotation": "Lx.wise(x)"
    },
    {
      "words": [
        "drinks",
        "drink"
      ],
      "denotation": "Lx.drink(x)"
    },
    {
      "words": [
        "sings",
        "sing"
      ],
      "denotation": "Lx.sing(x)"
    },
    {
      "words": [
        "fears",
        "fear"
      ],
      "denotation": "Ly.Lx.fear(x,y)"
    },
    {
      "words": [
        "loves",
        "love"
      ],
      "denotation": "Ly.Lx.love(x,y)"
    },
    {
      "words": [
        "trusts",
        "trust"
      ],
      "denotation": "Ly.Lx.trust(x,y)"
    },
    {
      "words": [
        "visits",
        "visit"
      ],
      "denotation": "Ly.Lx.visit(x,y)"
    },
    {
      "words": [
        "with"
      ],
      "denotation": "Ly.Lx.with(x,y)"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    }
  },
  "exercises": [
    {
      "title": "A. Subject gap relative clauses",
      "instructions": "A relative clause with a subject gap: \"who drinks\" is λx.drink(x), combining with the head noun by PM. Items 2 and 3 add copular predicates and negation.",
      "items": [
        {
          "sentence": "wizard who drinks.",
          "tree": "[.NP [.NP wizard ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP drinks ] ] ] ] ]",
          "reading": {
            "section": "7.3.1"
          },
          "targets": [
            "λx.wizard(x) ∧ drink(x)"
          ],
          "id": "d1"
        },
        {
          "sentence": "hobbit who is brave.",
          "tree": "[.NP [.NP hobbit ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP [.V is ] [.AP brave ] ] ] ] ] ]",
          "reading": {
            "section": "7.3.1"
          },
          "targets": [
            "λx.hobbit(x) ∧ brave(x)"
          ],
          "id": "d2"
        },
        {
          "sentence": "wizard who doesn't fear Sauron.",
          "tree": "[.NP [.NP wizard ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP [.Neg doesnt ] [.VP [.V fears ] [.DP Sauron ] ] ] ] ] ] ]",
          "reading": {
            "section": "7.3.1"
          },
          "targets": [
            "λx.wizard(x) ∧ ¬fear(x, n)"
          ],
          "id": "d3"
        }
      ],
      "id": "g1"
    },
    {
      "title": "B. Object gap relative clauses",
      "instructions": "A relative clause with an object gap uses Predicate Abstraction over the trace. Items 2 and 3 fold in negation and a prepositional phrase.",
      "items": [
        {
          "sentence": "hobbit who Gandalf loves.",
          "tree": "[.NP [.NP hobbit ] [.CP [.C who ] [.LP 1 [.S [.DP Gandalf ] [.VP [.V loves ] [.DP t1 ] ] ] ] ] ]",
          "reading": {
            "section": "7.3.2"
          },
          "targets": [
            "λx.hobbit(x) ∧ love(g, x)"
          ],
          "id": "d1"
        },
        {
          "sentence": "human who Strider doesn't trust.",
          "tree": "[.NP [.NP human ] [.CP [.C who ] [.LP 1 [.S [.DP Strider ] [.VP [.Neg doesnt ] [.VP [.V trusts ] [.DP t1 ] ] ] ] ] ] ]",
          "reading": {
            "section": "7.3.2"
          },
          "targets": [
            "λx.human(x) ∧ ¬trust(st, x)"
          ],
          "id": "d2"
        },
        {
          "sentence": "ranger who Frodo is with.",
          "tree": "[.NP [.NP ranger ] [.CP [.C who ] [.LP 1 [.S [.DP Frodo ] [.VP [.V is ] [.PP [.P with ] [.DP t1 ] ] ] ] ] ] ]",
          "reading": {
            "section": "7.3.2"
          },
          "targets": [
            "λx.ranger(x) ∧ with(f, x)"
          ],
          "id": "d3"
        }
      ],
      "id": "g2"
    },
    {
      "title": "C. Full sentences with relative clauses",
      "instructions": "A relative clause restricts the domain of a quantifier or identifies an individual via the copula. Items 2 and 3 introduce quantifiers.",
      "items": [
        {
          "sentence": "Gandalf is a wizard who is wise.",
          "tree": "[.S [.DP Gandalf ] [.VP [.V is ] [.NP [.D a ] [.NP [.NP wizard ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP [.V is ] [.AP wise ] ] ] ] ] ] ] ] ]",
          "reading": {
            "section": "7.3.2"
          },
          "targets": [
            "wizard(g) ∧ wise(g)"
          ],
          "id": "d1"
        },
        {
          "sentence": "Every hobbit who drinks sings.",
          "tree": "[.S [.DP [.D every ] [.NP [.NP hobbit ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP drinks ] ] ] ] ] ] [.VP sings ] ]",
          "reading": {
            "section": "7.3.2"
          },
          "targets": [
            "∀x.hobbit(x) ∧ drink(x) → sing(x)"
          ],
          "id": "d2"
        },
        {
          "sentence": "Some wise wizard who fears Sauron visits Saruman.",
          "tree": "[.S [.DP [.D some ] [.NP [.AP wise ] [.NP [.NP wizard ] [.CP [.C who ] [.LP 1 [.S [.DP t1 ] [.VP [.V fears ] [.DP Sauron ] ] ] ] ] ] ] ] [.VP [.V visits ] [.DP Saruman ] ] ]",
          "reading": {
            "section": "7.3.2"
          },
          "targets": [
            "∃x.wise(x) ∧ wizard(x) ∧ fear(x, n) ∧ visit(x, sr)"
          ],
          "id": "d3"
        }
      ],
      "id": "g3"
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 7 · Relative Clauses\n\nA relative clause like *who drinks* restricts the set of individuals denoted by the head noun. Compositionally, the clause must produce a predicate of type ⟨e,t⟩ so it can combine with the noun by **Predicate Modification**. The mechanism is **Predicate Abstraction**, which turns a clause containing a gap into a lambda-abstract.\n\n## 7.3 Predicate Abstraction\n\nA relative clause leaves a **trace** — a silent pronoun — in the gap position. The trace is co-indexed with the relative pronoun. A **λ-phrase node** (LP) abstracts over the index, turning the full clause denotation into a function:\n\n**Predicate Abstraction (PA).** ⟦[LP n φ]⟧ = λx. ⟦φ⟧ where every free occurrence of trace t_n in φ is replaced by x.\n\nThe relative pronoun *who* is an identity function λX.X — it passes the LP denotation up unchanged.\n\n## 7.3.1 Subject gap relative clauses\n\nWhen the trace is in subject position, the S-internal predicate is already a one-place property once PA is applied. The resulting CP combines with the head noun by PM.\n\n\\ex<subj-gap> wizard who drinks\n\\xe\n\n\\ex hobbit who is brave\n\\xe\n\n\\ex wizard who doesn't fear Sauron\n\\xe\n\n\\begin{derivation}\n[[drinks]]               = lambda x.drink(x)                        : <e,t>\n[[LP 1 [S t1 drinks]]]   = lambda x.drink(x)                        : <e,t>   (PA)\n[[who]]                  = lambda X.X                                : <<e,t>,<e,t>>\n[[who drinks]]           = lambda x.drink(x)                        : <e,t>\n[[wizard]]               = lambda x.wizard(x)                       : <e,t>\n[[wizard who drinks]]    = lambda x.wizard(x) /\\ drink(x)           : <e,t>   (PM)\n\\end{derivation}\n\n## 7.3.2 Object gap relative clauses\n\nWhen the trace is in **object position**, the clause contains a complete subject and a VP with the trace filling the object slot. PA abstracts over the trace index, producing a predicate that identifies which individuals stand in the relevant relation to the subject.\n\n\\ex<obj-gap> hobbit who Gandalf loves\n\\xe\n\n\\ex human who Strider doesn't trust\n\\xe\n\n\\ex ranger who Frodo is with\n\\xe\n\n\\begin{derivation}\n[[loves]]                         = lambda y.lambda x.love(x,y)          : <e,<e,t>>\n[[loves t1]]                      = lambda x.love(x,t_1)                 : <e,t>\n[[Gandalf loves t1]]              = love(g,t_1)                          : t\n[[LP 1 [S Gandalf loves t1]]]     = lambda x.love(g,x)                   : <e,t>   (PA)\n[[who [LP 1 ...]]]                = lambda x.love(g,x)                   : <e,t>\n[[hobbit]]                        = lambda x.hobbit(x)                   : <e,t>\n[[hobbit who Gandalf loves]]      = lambda x.hobbit(x) /\\ love(g,x)      : <e,t>   (PM)\n\\end{derivation}\n\n\\begin{forest}\n[NP{lambda x.hobbit(x) /\\ love(g,x)}\n  [NP{lambda x.hobbit(x)} hobbit]\n  [CP{lambda x.love(g,x)}\n    [C{lambda X.X} who]\n    [LP{lambda x.love(g,x)} 1\n      [S{love(g,t_1)}\n        [DP{g} Gandalf]\n        [VP{lambda x.love(x,t_1)}\n          [V{lambda y.lambda x.love(x,y)} loves]\n          [DP{t_1} t_1]]]]]]\n\\end{forest}\n\nThe full-sentence exercises in Group C embed relative clauses inside DPs. A restricted quantifier like *every hobbit who drinks* has the restrictor λx.[hobbit(x) ∧ drink(x)] built by PM inside the NP, then the determiner *every* applies as usual:\n\n\\begin{derivation}\n[[hobbit who drinks]]          = lambda x.hobbit(x) /\\ drink(x)                 : <e,t>\n[[every hobbit who drinks]]    = lambda Y.forall x[(hobbit(x) /\\ drink(x)) -> Y(x)] : <<e,t>,t>\n[[every hobbit who drinks sings]] = forall x[(hobbit(x) /\\ drink(x)) -> sing(x)] : t\n\\end{derivation}"
  }
}