{
  "compose": 1,
  "id": "ch10-plural",
  "title": "10.3 Plurals & the Star Operator",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w bi st pi tm el sr mr",
      "<e,t>": "hobbit elf dwarf orc ring sword map singer"
    },
    "variables": {
      "e": "x y z",
      "et": "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": [
        "rides",
        "ride"
      ],
      "denotation": "Lx.ride(x)"
    },
    {
      "words": [
        "sings",
        "sing"
      ],
      "denotation": "Lx.sing(x)"
    },
    {
      "words": [
        "rests",
        "rest"
      ],
      "denotation": "Lx.rest(x)"
    },
    {
      "words": [
        "meets",
        "meet"
      ],
      "denotation": "Lx.meet(x)"
    },
    {
      "words": [
        "carries",
        "carry"
      ],
      "denotation": "Ly.Lx.carry(x,y)"
    },
    {
      "words": [
        "guards",
        "guard"
      ],
      "denotation": "Ly.Lx.guard(x,y)"
    },
    {
      "words": [
        "reads",
        "read"
      ],
      "denotation": "Ly.Lx.read(x,y)"
    },
    {
      "words": [
        "finds",
        "find"
      ],
      "denotation": "Ly.Lx.find(x,y)"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "hobbit"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "elf"
    },
    {
      "words": [
        "dwarf"
      ],
      "denotation": "dwarf"
    },
    {
      "words": [
        "ring"
      ],
      "denotation": "ring"
    },
    {
      "words": [
        "sword"
      ],
      "denotation": "sword"
    },
    {
      "words": [
        "map"
      ],
      "denotation": "map"
    },
    {
      "words": [
        "hobbits"
      ],
      "denotation": "*hobbit"
    },
    {
      "words": [
        "elves"
      ],
      "denotation": "*elf"
    },
    {
      "words": [
        "dwarves"
      ],
      "denotation": "*dwarf"
    },
    {
      "words": [
        "rings"
      ],
      "denotation": "*ring"
    },
    {
      "words": [
        "swords"
      ],
      "denotation": "*sword"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "LX.Ix.X(x)"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "a"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "theSUPR"
      ],
      "denotation": "LX.Ix.[X(x) & Ay[X(y) -> y≤x]]"
    },
    {
      "words": [
        "andS"
      ],
      "denotation": "Lq.Lp.[p & q]"
    },
    {
      "words": [
        "orS"
      ],
      "denotation": "Lq.Lp.[p ∨ q]"
    },
    {
      "words": [
        "andVP"
      ],
      "denotation": "LQ.LP.Lx.[P(x) & Q(x)]"
    },
    {
      "words": [
        "orVP"
      ],
      "denotation": "LQ.LP.Lx.[P(x) ∨ Q(x)]"
    },
    {
      "words": [
        "andV"
      ],
      "denotation": "LR.LS.Ly.Lx.[S(y)(x) & R(y)(x)]"
    },
    {
      "words": [
        "andDP"
      ],
      "denotation": "LT.LU.LP.[U(P) & T(P)]"
    },
    {
      "words": [
        "orDP"
      ],
      "denotation": "LT.LU.LP.[U(P) ∨ T(P)]"
    },
    {
      "words": [
        "andE"
      ],
      "denotation": "Ly.Lz.z⊕y"
    },
    {
      "words": [
        "part"
      ],
      "denotation": "Ly.Lx.x≤y"
    },
    {
      "words": [
        "carry_coll"
      ],
      "denotation": "Ly.Lx.carry_coll(x,y)"
    },
    {
      "words": [
        "singer"
      ],
      "denotation": "singer"
    },
    {
      "words": [
        "-s",
        "PL"
      ],
      "denotation": "LP.*P"
    },
    {
      "words": [
        "are",
        "is",
        "be"
      ],
      "denotation": "LP.P"
    },
    {
      "words": [
        "two"
      ],
      "denotation": "Lx.card(x)=2"
    },
    {
      "words": [
        "theSUM"
      ],
      "denotation": "LP.⊕P"
    },
    {
      "words": [
        "∅D",
        "emptyD"
      ],
      "denotation": "LP.LP'.Ex[P(x) & P'(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    }
  },
  "exercises": [
    {
      "title": "A. Plural predicate (∗P)",
      "instructions": "The plural noun \"hobbits\" denotes ∗hobbit of type ⟨e,t⟩. It is true of any individual that is either a hobbit or a sum of hobbits. Build the NP and check its type — it should be ⟨e,t⟩, same as the singular \"hobbit\".",
      "items": [
        {
          "sentence": "hobbits (∗hobbit)",
          "tree": "[.NP [.N hobbit ] [.Pl -s ] ]",
          "targets": [
            "∗hobbit"
          ]
        },
        {
          "sentence": "elves (∗elf)",
          "tree": "[.NP [.N elf ] [.Pl -s ] ]",
          "targets": [
            "∗elf"
          ]
        },
        {
          "sentence": "dwarves (∗dwarf)",
          "tree": "[.NP [.N dwarf ] [.Pl -s ] ]",
          "targets": [
            "∗dwarf"
          ]
        }
      ]
    },
    {
      "title": "B. Plural definite (theSUPR)",
      "instructions": "The plural definite \"the hobbits\" uses the supremum iota: ιx.[∗hobbit(x) ∧ ∀y[∗hobbit(y)→y≤x]] — the maximal sum of all hobbits. theSUPR ↝ λX.ιx.[X(x) ∧ ∀y[X(y)→y≤x]] takes the plural predicate (∗hobbit) and returns its supremum — an individual of type e.",
      "items": [
        {
          "sentence": "the hobbits",
          "tree": "[.DP [.D theSUPR ] [.NP hobbits ] ]",
          "targets": [
            "ιx.[∗hobbit(x) ∧ ∀y[∗hobbit(y) → y≤x]]"
          ]
        },
        {
          "sentence": "the elves",
          "tree": "[.DP [.D theSUPR ] [.NP elves ] ]",
          "targets": [
            "ιx.[∗elf(x) ∧ ∀y[∗elf(y) → y≤x]]"
          ]
        }
      ]
    },
    {
      "title": "C. Plural definite as subject",
      "instructions": "A plural definite description is a type-e individual (the maximal sum). Apply it as a subject to a VP by FA — the VP predicate holds of the entire sum, expressing a collective or distributive claim depending on the predicate.",
      "items": [
        {
          "sentence": "The hobbits ride.",
          "tree": "[.S [.DP [.D theSUPR ] [.NP hobbits ] ] [.VP rides ] ]",
          "targets": [
            "ride(ιx.[∗hobbit(x) ∧ ∀y[∗hobbit(y) → y≤x]])"
          ]
        },
        {
          "sentence": "The elves sing.",
          "tree": "[.S [.DP [.D theSUPR ] [.NP elves ] ] [.VP sings ] ]",
          "targets": [
            "sing(ιx.[∗elf(x) ∧ ∀y[∗elf(y) → y≤x]])"
          ]
        }
      ]
    },
    {
      "title": "D. The plural morpheme -s (Exercise 8)",
      "instructions": "Instead of treating \"singers\" as an unanalyzed starred predicate, derive it compositionally. The plural morpheme is -s ⇝ λP.∗P (entry 30): it takes a singular noun and returns its algebraic closure. Build the noun by composing [.N singer ] with [.Pl -s ] via Function Application. For coordinated proper names, use andE (λyλz.z⊕y) to form the sum, and treat \"are\"/\"is\" as the identity function λP.P. These reproduce the book's (31) and (32b).",
      "items": [
        {
          "sentence": "singers ( [singer -s] = ∗singer )",
          "tree": "[.NP [.N singer ] [.Pl -s ] ]",
          "targets": [
            "∗singer"
          ]
        },
        {
          "sentence": "Frodo and Sam are singers. (32b)",
          "tree": "[.S [.DP [.DP Frodo ] [.Coord' [.Coord andE ] [.DP Sam ] ] ] [.VP [.V are ] [.NP [.N singer ] [.Pl -s ] ] ] ]",
          "targets": [
            "∗singer(f⊕s)"
          ]
        },
        {
          "sentence": "Some singers met. (31b)",
          "tree": "[.S [.DP [.D some ] [.NP [.N singer ] [.Pl -s ] ] ] [.VP meet ] ]",
          "targets": [
            "∃x[∗singer(x) ∧ meet(x)]"
          ]
        },
        {
          "sentence": "Frodo, Sam and Pippin met. (31a)",
          "tree": "[.S [.DP [.DP Frodo ] [.Coord' [.Coord andE ] [.DP [.DP Sam ] [.Coord' [.Coord andE ] [.DP Pippin ] ] ] ] ] [.VP meet ] ]",
          "targets": [
            "meet(f⊕(s⊕pi))"
          ]
        }
      ]
    },
    {
      "title": "E. theSUM — the sum analysis of plural the (Exercise 10)",
      "instructions": "Before the supremum analysis, the book first tries treating plural the as the generalized sum: theSUM ⇝ λP.⊕P (entry 33), where ⊕P is the sum of all the P's (type e). For \"the singers\" this gives the right answer — the maximal singer-sum. But notice the second item: with \"two singers\", theSUM sums up ALL the two-singer pluralities, collapsing them back to the total sum — so \"theSUM two singers\" wrongly denotes the sum of all singers, not a pair. This is exactly why the book replaces theSUM with theSUPR. (Here two ⇝ λx.card(x)=2 combines with singers by Predicate Modification.)",
      "items": [
        {
          "sentence": "theSUM singers met.",
          "tree": "[.S [.DP [.D theSUM ] [.NP [.N singer ] [.Pl -s ] ] ] [.VP meet ] ]",
          "targets": [
            "meet(⊕∗singer)"
          ]
        },
        {
          "sentence": "theSUM two singers met. (the over-summing problem)",
          "tree": "[.S [.DP [.D theSUM ] [.NP [.Card two ] [.NP [.N singer ] [.Pl -s ] ] ] ] [.VP meet ] ]",
          "targets": [
            "meet(⊕(λx.[card(x)=2 ∧ ∗singer(x)]))"
          ]
        }
      ]
    },
    {
      "title": "F. The silent determiner ∅D (Exercise 9)",
      "instructions": "Bare plurals like \"Two singers met\" have no overt determiner, but combine with the VP like a quantifier. The book posits a silent determiner with existential-quantifier semantics: ∅D ⇝ λPλP'.∃x[P(x) ∧ P'(x)] (Exercise 9). Include the ∅D node explicitly in the tree. \"two singers\" is formed by Predicate Modification, then ∅D binds it existentially.",
      "items": [
        {
          "sentence": "Two singers met.",
          "tree": "[.S [.DP [.D ∅D ] [.NP [.Card two ] [.NP [.N singer ] [.Pl -s ] ] ] ] [.VP meet ] ]",
          "targets": [
            "∃x[[card(x)=2 ∧ ∗singer(x)] ∧ meet(x)]"
          ]
        }
      ]
    }
  ]
}