{
  "compose": 1,
  "id": "ch10-fragment",
  "title": "10.5 The Full Fragment (L∗ with starred verbs)",
  "subtitle": "The book's §10.5.4 translations: once the domain has sums, every verb is starred (holds distributively of sums). Capstone, incl. cumulative readings via the relational star.",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "mx l pt j su my t1 t2 t3 t4",
      "<e,t>": "singer toothbrush fish",
      "<e,<e,t>>": "catch eat own"
    },
    "variables": {
      "e": "x y z w Z",
      "et": "P Q X Y"
    }
  },
  "lexicon": [
    { "words": ["Maxine"], "denotation": "mx" },
    { "words": ["LaVerne"], "denotation": "l" },
    { "words": ["Patty"], "denotation": "pt" },
    { "words": ["John"], "denotation": "j" },
    { "words": ["Sue"], "denotation": "su" },
    { "words": ["Mary"], "denotation": "my" },

    { "words": ["smokes", "smoke"], "denotation": "Lx.*smoke(x)" },
    { "words": ["drinks", "drink"], "denotation": "Lx.*drink(x)" },

    { "words": ["caught", "catch"], "denotation": "Ly.Lx.*catch(x,y)" },
    { "words": ["ate", "eat"], "denotation": "Ly.Lx.*eat(x,y)" },
    { "words": ["own", "owns"], "denotation": "Ly.Lx.*own(x,y)" },

    { "words": ["two"], "denotation": "Lx.card(x)=2" },
    { "words": ["three"], "denotation": "Lx.card(x)=3" },
    { "words": ["four"], "denotation": "Lx.card(x)=4" },

    { "words": ["singer"], "denotation": "singer" },
    { "words": ["toothbrush"], "denotation": "toothbrush" },
    { "words": ["fish"], "denotation": "fish" },

    { "words": ["-s", "PL"], "denotation": "LP.*P" },
    { "words": ["andE", "andSUM"], "denotation": "Ly.Lz.z⊕y" },
    { "words": ["andS"], "denotation": "Lq.Lp.[p & q]" },
    { "words": ["orS"], "denotation": "Lq.Lp.[p ∨ q]" },
    { "words": ["andVP"], "denotation": "LY.LX.Lx.[X(x) & Y(x)]" },
    { "words": ["andV"], "denotation": "LR.LS.Ly.Lx.[S(y)(x) & R(y)(x)]" },
    { "words": ["is", "are", "a", "be"], "denotation": "LP.P" },
    { "words": ["the", "theSUPR"], "denotation": "LP.Iz.[P(z) & Ax[P(x) -> x≤z]]" },
    { "words": ["∅D", "emptyD"], "denotation": "LP.LQ.Ex[P(x) & Q(x)]" },
    { "words": ["some", "Some"], "denotation": "LX.LY.Ex[X(x) & Y(x)]" }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true,
      "predicateAbstraction": true
    },
    "typeShifts": ["raiseO"],
    "quantifierRaising": true
  },
  "exercises": [
    {
      "title": "A. Why verbs are starred — distributivity over sums",
      "instructions": "Once plural individuals live in the domain, a verb like smoke must hold of a SUM whenever it holds of each atom. So the fragment stars every verb: smokes ↝ λx.∗smoke(x) (entry §10.5.4). Form the plural subject with andE (λyλz.z⊕y), which sums two entities, and apply the starred verb. ∗smoke(mx⊕l) is true exactly when both Maxine and LaVerne smoke — the distributive reading falls out of the star, with no extra distributivity operator.",
      "items": [
        {
          "sentence": "Maxine and LaVerne smoke.",
          "tree": "[.S [.DP [.DP Maxine ] [.Coord' [.Coord andE ] [.DP LaVerne ] ] ] [.VP smoke ] ]",
          "targets": [
            "∗smoke(mx⊕l)"
          ]
        },
        {
          "sentence": "Maxine, LaVerne and Patty drink.",
          "tree": "[.S [.DP [.DP Maxine ] [.Coord' [.Coord andE ] [.DP [.DP LaVerne ] [.Coord' [.Coord andE ] [.DP Patty ] ] ] ] ] [.VP drink ] ]",
          "targets": [
            "∗drink(mx⊕(l⊕pt))"
          ]
        }
      ]
    },
    {
      "title": "B. Starred transitive verbs",
      "instructions": "Transitive verbs are starred too: caught ↝ λyλx.∗catch(x,y), own ↝ λyλx.∗own(x,y) (§10.5.4). The object is taken first, then the subject, giving a flat ∗catch(subject, object). With singular arguments the star is vacuous, but the entry is uniform across singular and plural.",
      "items": [
        {
          "sentence": "Maxine caught Sue.",
          "tree": "[.S [.DP Maxine ] [.VP [.V caught ] [.DP Sue ] ] ]",
          "targets": [
            "∗catch(mx,su)"
          ]
        },
        {
          "sentence": "Maxine and LaVerne caught Mary.",
          "tree": "[.S [.DP [.DP Maxine ] [.Coord' [.Coord andE ] [.DP LaVerne ] ] ] [.VP [.V caught ] [.DP Mary ] ] ]",
          "targets": [
            "∗catch(mx⊕l,my)"
          ]
        }
      ]
    },
    {
      "title": "C. The plural morpheme with a starred verb",
      "instructions": "Combine the pieces: the plural noun is [.N singer ] + [.Pl -s ] (-s ↝ λP.∗P), the silent ∅D ↝ λPλQ.∃x[P(x) ∧ Q(x)] turns it into a quantifier, and the starred verb supplies the predicate. \"Some singers smoke\" existentially quantifies over singer-sums that smoke.",
      "items": [
        {
          "sentence": "Some singers smoke.",
          "tree": "[.S [.DP [.D some ] [.NP [.N singer ] [.Pl -s ] ] ] [.VP smoke ] ]",
          "targets": [
            "∃x[∗singer(x) ∧ ∗smoke(x)]"
          ]
        }
      ]
    },
    {
      "title": "D. Cumulative readings via the relational star (Exercise 41)",
      "instructions": "The relational star generalizes ∗ to two-place predicates: ∗own holds of a pair of sums when they divide into atom-pairs each in the own-relation (entries 39–41). \"Maxine, LaVerne and Patty own four toothbrushes\" gets a CUMULATIVE reading: a total of four toothbrushes are collectively owned. The object \"four toothbrushes\" is ∅D(four ⊓ ∗toothbrush), a quantifier; let it take scope over the starred verb (apply RaiseO to the verb so the object combines in situ, or raise the object). The result is the book's (41).",
      "items": [
        {
          "sentence": "Maxine, LaVerne and Patty own four toothbrushes. (cumulative)",
          "tree": "[.S [.DP [.DP Maxine ] [.Coord' [.Coord andE ] [.DP [.DP LaVerne ] [.Coord' [.Coord andE ] [.DP Patty ] ] ] ] ] [.VP [.V own ] [.DP [.D ∅D ] [.NP [.Card four ] [.NP [.N toothbrush ] [.Pl -s ] ] ] ] ] ]",
          "targets": [
            "∃x[[card(x)=4 ∧ ∗toothbrush(x)] ∧ ∗own(mx⊕(l⊕pt),x)]"
          ]
        }
      ]
    }
  ]
}