{
  "compose": 1,
  "id": "ch10-mereology",
  "title": "10.2 Mereology — Sums and Parthood",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w"
    },
    "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": [
        "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": [
        "are",
        "is"
      ],
      "denotation": "LP.P"
    },
    {
      "words": [
        "two_"
      ],
      "displayAs": "two",
      "denotation": "Lx.|x|=2"
    },
    {
      "words": [
        "three_"
      ],
      "displayAs": "three",
      "denotation": "Lx.|x|=3"
    },
    {
      "words": [
        "two"
      ],
      "denotation": "LP.LQ.Ex[P(x) & |x|=2 & Q(x)]"
    },
    {
      "words": [
        "three"
      ],
      "denotation": "LP.LQ.Ex[P(x) & |x|=3 & Q(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    }
  },
  "exercises": [
    {
      "title": "A. Sum-formation (andE — name coordination)",
      "instructions": "andE ↝ λy.λz.z⊕y (type ⟨e,⟨e,e⟩⟩) takes the right name first, then FA with the left, forming a sum of type e. The result is a new individual — the mereological sum of the two names — that can serve as subject or object of a predicate.",
      "items": [
        {
          "sentence": "Frodo and Sam (the sum f⊕s).",
          "tree": "[.DP [.DP Frodo ] [.ConjP [.Conj andE ] [.DP Sam ] ] ]",
          "targets": [
            "f⊕s"
          ]
        },
        {
          "sentence": "Frodo, Sam and Aragorn (the sum f⊕s⊕a).",
          "tree": "[.DP [.DP [.DP Frodo ] [.ConjP [.Conj andE ] [.DP Sam ] ] ] [.ConjP [.Conj andE ] [.DP Aragorn ] ] ]",
          "targets": [
            "f⊕s⊕a"
          ]
        },
        {
          "sentence": "Gandalf and Aragorn carry the ring (collective subject).",
          "tree": "[.S [.DP [.DP Gandalf ] [.ConjP [.Conj andE ] [.DP Aragorn ] ] ] [.VP [.V carry_coll ] [.DP [.D the ] [.NP ring ] ] ] ]",
          "targets": [
            "carry_coll(g⊕a,ιz[ring(z)])"
          ]
        }
      ]
    },
    {
      "title": "B. Parthood (≤)",
      "instructions": "The parthood relation ≤ holds between an individual and a sum that contains it: f≤f⊕s. In a sentence \"Frodo is part of Frodo and Sam\", derive the sum first via andE, then apply the parthood predicate. The truth conditions contain ≤ directly.",
      "items": [
        {
          "sentence": "Frodo is part of Frodo and Sam. (f ≤ f⊕s)",
          "tree": "[.S [.DP Frodo ] [.VP [.V part ] [.DP [.DP Frodo ] [.ConjP [.Conj andE ] [.DP Sam ] ] ] ] ]",
          "targets": [
            "f≤f⊕s"
          ]
        },
        {
          "sentence": "Sam and Gandalf carry the sword together; Frodo is part of their sum.",
          "tree": "[.S [.DP Frodo ] [.VP [.V part ] [.DP [.DP Sam ] [.ConjP [.Conj andE ] [.DP Gandalf ] ] ] ] ]",
          "targets": [
            "f≤s⊕g"
          ]
        }
      ]
    },
    {
      "title": "C. Cardinality of sums (|·|)",
      "instructions": "The cardinality operator |x| counts the atomic parts of an individual: |f⊕s| = 2. The copula \"are\" (λP.P) passes the numeral predicate to the plural subject, which then applies to the sum. Because ⊕ is commutative, associative and idempotent, the parts may be summed in any order — |s⊕f| = 2 is accepted just as readily.",
      "items": [
        {
          "sentence": "Frodo and Sam are two. (|f⊕s| = 2)",
          "tree": "[.S [.DP [.DP Frodo ] [.ConjP [.Conj andE ] [.DP Sam ] ] ] [.VP [.V are ] [.NumP two_ ] ] ]",
          "targets": [
            "|f⊕s|=2"
          ]
        },
        {
          "sentence": "Frodo, Sam and Gandalf are three. (|f⊕s⊕g| = 3)",
          "tree": "[.S [.DP [.DP [.DP Frodo ] [.ConjP [.Conj andE ] [.DP Sam ] ] ] [.ConjP [.Conj andE ] [.DP Gandalf ] ] ] [.VP [.V are ] [.NumP three_ ] ] ]",
          "targets": [
            "|f⊕s⊕g|=3"
          ]
        }
      ]
    },
    {
      "title": "D. Cardinal numerals as quantifiers",
      "instructions": "A numeral determiner \"two\" ⇝ λP.λQ.∃x[P(x) ∧ |x|=2 ∧ Q(x)] existentially quantifies over a sum of two atoms that satisfies the pluralized noun (∗) and the predicate. It is type ⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩, a generalized quantifier, and combines with the VP by Function Application.",
      "items": [
        {
          "sentence": "Two hobbits ride.",
          "tree": "[.S [.DP [.D two ] [.NP hobbits ] ] [.VP ride ] ]",
          "targets": [
            "Ex[*hobbit(x) & |x|=2 & ride(x)]"
          ]
        },
        {
          "sentence": "Three elves sing.",
          "tree": "[.S [.DP [.D three ] [.NP elves ] ] [.VP sing ] ]",
          "targets": [
            "Ex[*elf(x) & |x|=3 & sing(x)]"
          ]
        }
      ]
    }
  ]
}