{
  "compose": 1,
  "id": "ch7-adj",
  "title": "Adjectives & Predicate Modification (§7.2)",
  "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": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(x)"
    },
    {
      "words": [
        "wizard"
      ],
      "denotation": "Lx.wizard(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "creature"
      ],
      "denotation": "Lx.creature(x)"
    },
    {
      "words": [
        "ranger"
      ],
      "denotation": "Lx.ranger(x)"
    },
    {
      "words": [
        "human"
      ],
      "denotation": "Lx.human(x)"
    },
    {
      "words": [
        "king"
      ],
      "denotation": "Lx.king(x)"
    },
    {
      "words": [
        "brave"
      ],
      "denotation": "Lx.brave(x)"
    },
    {
      "words": [
        "mischievous"
      ],
      "denotation": "Lx.mischievous(x)"
    },
    {
      "words": [
        "mundane"
      ],
      "denotation": "Lx.mundane(x)"
    },
    {
      "words": [
        "magical"
      ],
      "denotation": "Lx.magical(x)"
    },
    {
      "words": [
        "traveling"
      ],
      "denotation": "Lx.traveling(x)"
    },
    {
      "words": [
        "wise"
      ],
      "denotation": "Lx.wise(x)"
    },
    {
      "words": [
        "former"
      ],
      "denotation": "LF.Lx.former(F)(x)"
    },
    {
      "words": [
        "alleged"
      ],
      "denotation": "LF.Lx.alleged(F)(x)"
    },
    {
      "words": [
        "sees",
        "see"
      ],
      "denotation": "Ly.Lx.see(x,y)"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "no",
        "No"
      ],
      "denotation": "LX.LY.~Ex[X(x) & Y(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true
    }
  },
  "exercises": [
    {
      "title": "A. Intersective adjectives with copula",
      "instructions": "An intersective adjective is a predicate ⟨e,t⟩. Two predicates sisters in NP combine by Predicate Modification: λx.[F(x) ∧ G(x)]. The copula passes the result up unchanged. The third adds a quantifier as complication.",
      "items": [
        {
          "sentence": "Pippin is a mischievous hobbit.",
          "tree": "[.S [.DP Pippin ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mischievous ] [.NP hobbit ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "mischievous(pi) ∧ hobbit(pi)"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote pi and λx.mischievous(x) ∧ hobbit(x). Combine them and β-reduce."
          ],
          "id": "d1"
        },
        {
          "sentence": "Tom is not a mundane creature.",
          "tree": "[.S [.DP Tom ] [.VP [.Neg not ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mundane ] [.NP creature ] ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "¬(mundane(tm) ∧ creature(tm))"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote tm and λx.¬(mundane(x) ∧ creature(x)). Combine them and β-reduce."
          ],
          "id": "d2"
        },
        {
          "sentence": "Every traveling hobbit is a brave creature.",
          "tree": "[.S [.DP [.D every ] [.NP [.AP traveling ] [.NP hobbit ] ] ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP brave ] [.NP creature ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "∀x.traveling(x) ∧ hobbit(x) → brave(x) ∧ creature(x)"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote λY.∀x.traveling(x) ∧ hobbit(x) → Y(x) and λx.brave(x) ∧ creature(x). Combine them and β-reduce."
          ],
          "id": "d3"
        }
      ],
      "id": "g1"
    },
    {
      "title": "B. Stacked adjectives",
      "instructions": "Multiple intersective adjectives stack by repeated Predicate Modification. Each AP adds its predicate conjunctively. The third example adds sentential negation.",
      "items": [
        {
          "sentence": "Tom is a mischievous magical creature.",
          "tree": "[.S [.DP Tom ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mischievous ] [.NP [.AP magical ] [.NP creature ] ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "mischievous(tm) ∧ magical(tm) ∧ creature(tm)"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote tm and λx.mischievous(x) ∧ magical(x) ∧ creature(x). Combine them and β-reduce."
          ],
          "id": "d1"
        },
        {
          "sentence": "Strider is not a human ranger.",
          "tree": "[.S [.DP Strider ] [.VP [.Neg not ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP human ] [.NP ranger ] ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "¬(human(st) ∧ ranger(st))"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote st and λx.¬(human(x) ∧ ranger(x)). Combine them and β-reduce."
          ],
          "id": "d2"
        },
        {
          "sentence": "No traveling hobbit is a mischievous brave creature.",
          "tree": "[.S [.DP [.D no ] [.NP [.AP traveling ] [.NP hobbit ] ] ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mischievous ] [.NP [.AP brave ] [.NP creature ] ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "¬(∃x.traveling(x) ∧ hobbit(x) ∧ mischievous(x) ∧ brave(x) ∧ creature(x))"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote λY.¬(∃x.traveling(x) ∧ hobbit(x) ∧ Y(x)) and λx.mischievous(x) ∧ brave(x) ∧ creature(x). Combine them and β-reduce."
          ],
          "id": "d3"
        }
      ],
      "id": "g2"
    },
    {
      "title": "C. Non-intersective adjectives",
      "instructions": "Non-intersective adjectives like \"alleged\" and \"former\" are predicate modifiers ⟨⟨e,t⟩,⟨e,t⟩⟩ — they take a property and return a property, without entailing the base noun. They combine by Function Application, not PM.",
      "items": [
        {
          "sentence": "Strider is an alleged king.",
          "tree": "[.S [.DP Strider ] [.VP [.V is ] [.NP [.D an ] [.NP [.AP alleged ] [.NP king ] ] ] ] ]",
          "reading": {
            "section": "7.2.1"
          },
          "targets": [
            "alleged(λx.king(x), st)"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote st and λx.alleged(λx.king(x), x). Combine them and β-reduce."
          ],
          "id": "d1"
        },
        {
          "sentence": "Bilbo is a former traveling hobbit.",
          "tree": "[.S [.DP Bilbo ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP former ] [.NP [.AP traveling ] [.NP hobbit ] ] ] ] ] ]",
          "reading": {
            "section": "7.2.1"
          },
          "targets": [
            "former(λx.traveling(x) ∧ hobbit(x), bi)"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote bi and λx.former(λx.traveling(x) ∧ hobbit(x), x). Combine them and β-reduce."
          ],
          "id": "d2"
        },
        {
          "sentence": "Some alleged wise wizard sees Sauron.",
          "tree": "[.S [.DP [.D some ] [.NP [.AP alleged ] [.NP [.AP wise ] [.NP wizard ] ] ] ] [.VP [.V sees ] [.DP Sauron ] ] ]",
          "reading": {
            "section": "7.2.1"
          },
          "targets": [
            "∃x.alleged(λx.wise(x) ∧ wizard(x), x) ∧ see(x, n)"
          ],
          "hints": [
            "Work bottom-up: resolve every leaf from the lexicon first. The root node is resolved by Function Application — one sister denotes a function whose type matches the other sister.",
            "At the root, the sisters denote λY.∃x.alleged(λx.wise(x) ∧ wizard(x), x) ∧ Y(x) and λx.see(x, n). Combine them and β-reduce."
          ],
          "id": "d3"
        }
      ],
      "id": "g3"
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 7 · Predicate Modification\n\nChapter 6 established **Function Application** as the core composition rule. Chapter 7 adds a second rule — **Predicate Modification** — for adjective–noun combinations, and widens the type inventory to include predicate modifiers ⟨⟨e,t⟩,⟨e,t⟩⟩.\n\n## 7.2 Predicate Modification\n\n**Predicate Modification (PM).** If two sisters α and β are both type ⟨e,t⟩, their mother denotes λx.[α(x) ∧ β(x)].\n\nAn *intersective* adjective is a plain one-place predicate. It shares type ⟨e,t⟩ with the noun, so the two sisters combine by PM. The determiner *a/an* and copula *is/are* are identity functions that pass their sister's denotation straight up.\n\n\\ex<pm-ex> Pippin is a mischievous hobbit.\n\\xe\n\n\\ex Tom is not a mundane creature.\n\\xe\n\n\\ex Every traveling hobbit is a brave creature.\n\\xe\n\n\\begin{derivation}\n[[mischievous]]                     = lambda x.mischievous(x)                      : <e,t>\n[[hobbit]]                          = lambda x.hobbit(x)                           : <e,t>\n[[mischievous hobbit]]              = lambda x.mischievous(x) /\\ hobbit(x)         : <e,t>\n[[a]]                               = lambda X.X                                   : <<e,t>,<e,t>>\n[[a mischievous hobbit]]            = lambda x.mischievous(x) /\\ hobbit(x)         : <e,t>\n[[is]]                              = lambda X.X                                   : <<e,t>,<e,t>>\n[[is a mischievous hobbit]]         = lambda x.mischievous(x) /\\ hobbit(x)         : <e,t>\n[[Pippin is a mischievous hobbit]]  = mischievous(pi) /\\ hobbit(pi)                : t\n\\end{derivation}\n\n\\begin{forest}\n[S{mischievous(pi) /\\ hobbit(pi)}\n  [DP{pi} Pippin]\n  [VP{lambda x.mischievous(x) /\\ hobbit(x)}\n    [V{lambda X.X} is]\n    [NP{lambda x.mischievous(x) /\\ hobbit(x)}\n      [D{lambda X.X} a]\n      [NP{lambda x.mischievous(x) /\\ hobbit(x)}\n        [AP{lambda x.mischievous(x)} mischievous]\n        [NP{lambda x.hobbit(x)} hobbit]]]]]\n\\end{forest}\n\nNegation applies after the VP is assembled: `⟦not⟧` = λP.λx.¬P(x), so *is not a mundane creature* = λx.¬[mundane(x) ∧ creature(x)].\n\nMultiple intersective adjectives stack by repeated PM. Each AP sister adds one conjunct (\\ref{stack}):\n\n\\ex<stack> Tom is a mischievous magical creature.\n\\xe\n\n\\ex Strider is not a human ranger.\n\\xe\n\n\\ex No traveling hobbit is a mischievous brave creature.\n\\xe\n\n\\begin{derivation}\n[[mischievous magical creature]] = lambda x.mischievous(x) /\\ magical(x) /\\ creature(x) : <e,t>\n\\end{derivation}\n\nWhen the subject is a quantifier, PM assembles the predicate first; then the determiner applies by FA:\n\n\\begin{derivation}\n[[every]]                               = lambda X.lambda Y.forall x[X(x) -> Y(x)]              : <<e,t>,<<e,t>,t>>\n[[traveling hobbit]]                    = lambda x.traveling(x) /\\ hobbit(x)                    : <e,t>\n[[every traveling hobbit]]              = lambda Y.forall x[(traveling(x) /\\ hobbit(x)) -> Y(x)] : <<e,t>,t>\n[[brave creature]]                      = lambda x.brave(x) /\\ creature(x)                      : <e,t>\n[[Every traveling hobbit is a brave creature]] = forall x[(traveling(x) /\\ hobbit(x)) -> (brave(x) /\\ creature(x))] : t\n\\end{derivation}\n\n## 7.2.1 Non-intersective adjectives\n\n*Alleged* and *former* are not intersective: an alleged king need not be a king; a former hobbit is no longer one. These adjectives are **predicate modifiers** of type ⟨⟨e,t⟩,⟨e,t⟩⟩ — they take a property and return a (possibly weaker) property. Because one daughter is ⟨⟨e,t⟩,⟨e,t⟩⟩ and the other ⟨e,t⟩, they combine by **FA** — PM would fail since the daughters are not the same type.\n\n\\ex<nonintersect> Strider is an alleged king.\n\\xe\n\n\\ex Bilbo is a former traveling hobbit.\n\\xe\n\n\\ex Some alleged wise wizard sees Sauron.\n\\xe\n\n\\begin{derivation}\n[[alleged]]                    = lambda F.lambda x.alleged(F)(x)    : <<e,t>,<e,t>>\n[[king]]                       = lambda x.king(x)                   : <e,t>\n[[alleged king]]               = lambda x.alleged(king)(x)          : <e,t>\n[[Strider is an alleged king]] = alleged(king)(st)                  : t\n\n[[former]]                           = lambda F.lambda x.former(F)(x)               : <<e,t>,<e,t>>\n[[traveling hobbit]]                 = lambda x.traveling(x) /\\ hobbit(x)           : <e,t>\n[[former traveling hobbit]]          = lambda x.former(lambda x.traveling(x) /\\ hobbit(x))(x) : <e,t>\n[[Bilbo is a former traveling hobbit]] = former(lambda x.traveling(x) /\\ hobbit(x))(bi) : t\n\\end{derivation}\n\nThe notation `alleged(king)(x)` treats *alleged* as mapping the property `king` to a new property, then applying to an individual. No entailment that the individual satisfies `king` is triggered.[^nonintersect]\n\n[^nonintersect]: *Former* works the same way but adds a temporal flavour: `former(P)(x)` means x once had property P but no longer does. Both predicates are opaque to their nominal argument."
  }
}