{
  "compose": 1,
  "id": "ch10-cumulative",
  "title": "10.4 Cumulative Readings",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w",
      "<e,<e,t>>": "carry guard read find"
    },
    "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": "Lx.hobbit(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "dwarf"
      ],
      "denotation": "Lx.dwarf(x)"
    },
    {
      "words": [
        "ring"
      ],
      "denotation": "Lx.ring(x)"
    },
    {
      "words": [
        "sword"
      ],
      "denotation": "Lx.sword(x)"
    },
    {
      "words": [
        "map"
      ],
      "denotation": "Lx.map(x)"
    },
    {
      "words": [
        "hobbits"
      ],
      "denotation": "*Lx.hobbit(x)"
    },
    {
      "words": [
        "elves"
      ],
      "denotation": "*Lx.elf(x)"
    },
    {
      "words": [
        "dwarves"
      ],
      "denotation": "*Lx.dwarf(x)"
    },
    {
      "words": [
        "rings"
      ],
      "denotation": "*Lx.ring(x)"
    },
    {
      "words": [
        "swords"
      ],
      "denotation": "*Lx.sword(x)"
    },
    {
      "words": [
        "maps"
      ],
      "denotation": "*Lx.map(x)"
    },
    {
      "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": [
        "cum_carry"
      ],
      "denotation": "Ly.Lx.*carry(x,y)"
    },
    {
      "words": [
        "cum_read"
      ],
      "denotation": "Ly.Lx.*read(x,y)"
    },
    {
      "words": [
        "cum_guard"
      ],
      "denotation": "Ly.Lx.*guard(x,y)"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    },
    "typeShifts": [
      "raiseO"
    ]
  },
  "exercises": [
    {
      "title": "A. Cumulative predicate applied to plural sums",
      "instructions": "The cumulative verb is the relational star ∗carry — the cumulative closure of the carry relation. ∗carry(X,Y) holds of two sums X and Y when X and Y can be divided into parts that pair up under carry, so every part of X carries some part of Y and vice versa. The verb is λy.λx.∗carry(x,y); the star operator now applies to a two-place relation ⟨e,⟨e,t⟩⟩, just as it applies to a one-place predicate ⟨e,t⟩ in the plural nouns. Build the sentence with theSUPR-formed plural definites as subject and object.",
      "items": [
        {
          "sentence": "The hobbits carried the rings. (cumulative)",
          "tree": "[.S [.DP [.D theSUPR ] [.NP hobbits ] ] [.VP [.V cum_carry ] [.DP [.D theSUPR ] [.NP rings ] ] ] ]",
          "targets": [
            "∗carry(ιx.[∗(λz.hobbit(z))(x) ∧ ∀y[∗(λz.hobbit(z))(y) → y≤x]], ιx.[∗(λz.ring(z))(x) ∧ ∀y[∗(λz.ring(z))(y) → y≤x]])"
          ]
        },
        {
          "sentence": "The elves read the maps. (cumulative)",
          "tree": "[.S [.DP [.D theSUPR ] [.NP elves ] ] [.VP [.V cum_read ] [.DP [.D theSUPR ] [.NP maps ] ] ] ]",
          "targets": [
            "∗read(ιx.[∗(λz.elf(z))(x) ∧ ∀y[∗(λz.elf(z))(y) → y≤x]], ιx.[∗(λz.map(z))(x) ∧ ∀y[∗(λz.map(z))(y) → y≤x]])"
          ]
        }
      ]
    },
    {
      "title": "B. Distributive vs. cumulative contrast",
      "instructions": "\"Every hobbit carried every ring\" (distributive — every individual to every individual) vs. \"The hobbits carried the rings\" (cumulative — sums to sums). Build both and compare. The distributive reading uses every+every and gives ∀x∀y[…carry(x,y)]; the cumulative reading uses theSUPR+theSUPR and the relational star ∗carry over the two plural sums.",
      "items": [
        {
          "sentence": "Every hobbit carried every ring. (distributive)",
          "tree": "[.S [.DP [.D every ] [.NP hobbit ] ] [.VP [.V carries ] [.DP [.D every ] [.NP ring ] ] ] ]",
          "targets": [
            "∀x[hobbit(x) → ∀y[ring(y) → carry(x,y)]]"
          ]
        },
        {
          "sentence": "The hobbits carried the rings. (cumulative — theSUPR + cum_carry)",
          "tree": "[.S [.DP [.D theSUPR ] [.NP hobbits ] ] [.VP [.V cum_carry ] [.DP [.D theSUPR ] [.NP rings ] ] ] ]",
          "targets": [
            "∗carry(ιx.[∗(λz.hobbit(z))(x) ∧ ∀y[∗(λz.hobbit(z))(y) → y≤x]], ιx.[∗(λz.ring(z))(x) ∧ ∀y[∗(λz.ring(z))(y) → y≤x]])"
          ]
        }
      ]
    }
  ]
}