{
  "compose": 1,
  "id": "ch13-worlds-times",
  "title": "Worlds & Times Combined (§12–13)",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b",
      "i": "now dawn dusk"
    },
    "variables": {
      "e": "x y z",
      "i": "t t' t'' t1 t2",
      "s": "w0 w w' w''",
      "st": "p q",
      "it": "H J",
      "et": "P Q",
      "set": "R"
    }
  },
  "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": [
        "now"
      ],
      "denotation": "now"
    },
    {
      "words": [
        "dawn"
      ],
      "denotation": "dawn"
    },
    {
      "words": [
        "dusk"
      ],
      "denotation": "dusk"
    },
    {
      "words": [
        "rides",
        "ride"
      ],
      "denotation": "Lx.Lt.ride(w0,t,x)"
    },
    {
      "words": [
        "sings",
        "sing"
      ],
      "denotation": "Lx.Lt.sing(w0,t,x)"
    },
    {
      "words": [
        "reads",
        "read"
      ],
      "denotation": "Ly.Lx.Lt.read(w0,t,x,y)"
    },
    {
      "words": [
        "ring"
      ],
      "denotation": "Lx.ring(w0,x)"
    },
    {
      "words": [
        "map"
      ],
      "denotation": "Lx.map(w0,x)"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "LX.Ix.X(x)"
    },
    {
      "words": [
        "PFV"
      ],
      "denotation": "LH.Lt.Et'[t'⊆t & H(t')]"
    },
    {
      "words": [
        "PROG"
      ],
      "denotation": "LH.Lt.Et'[t⊆t' & H(t')]"
    },
    {
      "words": [
        "PAST"
      ],
      "denotation": "LH.Lt.[t=t1 & ∂(t1<now) & H(t1)]"
    },
    {
      "words": [
        "PRES"
      ],
      "denotation": "LH.Lt.[t=now & H(t)]"
    },
    {
      "words": [
        "PERFECT"
      ],
      "denotation": "LH.Lt.Et'[t'<t & H(t')]"
    },
    {
      "words": [
        "at"
      ],
      "denotation": "Lt.t"
    },
    {
      "words": [
        "necessarily"
      ],
      "denotation": "Lp.Aw'[acc(w0)(w') -> p(w')]"
    },
    {
      "words": [
        "possibly"
      ],
      "denotation": "Lp.Ew'[acc(w0)(w') & p(w')]"
    },
    {
      "words": [
        "believes",
        "believe"
      ],
      "denotation": "Lp.Lx.Aw'[dox(w0)(x,w') -> p(w')]"
    },
    {
      "words": [
        "wants",
        "want"
      ],
      "denotation": "Lp.Lx.Aw'[boul(w0)(x,w') -> p(w')]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateAbstraction": true
    }
  },
  "exercises": [
    {
      "title": "A. Tense with world-indexed predicates",
      "instructions": "With doubly-indexed predicates, the tense and aspect machinery from Ch. 12 applies without change — the world variable w0 is simply free throughout. Build the TenseP using a temporal modifier to saturate the time slot directly (no EC needed). The result is type t with w0 free: a claim about what happens at a particular time in the evaluation world.",
      "items": [
        {
          "sentence": "At dawn, Frodo rode.",
          "tree": "[.TenseP [.TenseP [.Tense PAST ] [.AspP [.Asp PFV ] [.vP [.DP Frodo ] [.VP rides ] ] ] ] [.PP [.P at ] [.DP dawn ] ] ]",
          "targets": [
            "dawn=t1 ∧ ∂(t1<now) ∧ ∃t'[t'⊆t1 ∧ ride(w0,t',f)]"
          ]
        },
        {
          "sentence": "At dusk, Gandalf read the map.",
          "tree": "[.TenseP [.TenseP [.Tense PAST ] [.AspP [.Asp PFV ] [.vP [.DP Gandalf ] [.VP [.V reads ] [.DP [.D the ] [.NP map ] ] ] ] ] ] [.PP [.P at ] [.DP dusk ] ] ]",
          "targets": [
            "dusk=t1 ∧ ∂(t1<now) ∧ ∃t'[t'⊆t1 ∧ read(w0,t',g,ιz[map(w0,z)])]"
          ]
        }
      ]
    },
    {
      "title": "B. Necessity over a tensed clause",
      "instructions": "A TenseP with a temporal modifier has type t with w0 free. Applying IFA with necessarily wraps this as λw0.φ — the temporal structure (dawn, t1, t′) stays intact inside the bound world. The truth conditions assert that across all accessible worlds, the temporal claim holds — but the time constants (dawn, t1) are rigid: the same time in every world.",
      "items": [
        {
          "sentence": "Necessarily, at dawn, Frodo rode.",
          "tree": "[.NecP [.Adv necessarily ] [.TenseP [.TenseP [.Tense PAST ] [.AspP [.Asp PFV ] [.vP [.DP Frodo ] [.VP rides ] ] ] ] [.PP [.P at ] [.DP dawn ] ] ] ]",
          "targets": [
            "∀w'[acc(w0)(w') → dawn=t1 ∧ ∂(t1<now) ∧ ∃t'[t'⊆t1 ∧ ride(w',t',f)]]"
          ]
        },
        {
          "sentence": "Possibly, at dusk, Gandalf read the map.",
          "tree": "[.PossP [.Adv possibly ] [.TenseP [.TenseP [.Tense PAST ] [.AspP [.Asp PFV ] [.vP [.DP Gandalf ] [.VP [.V reads ] [.DP [.D the ] [.NP map ] ] ] ] ] ] [.PP [.P at ] [.DP dusk ] ] ] ]",
          "targets": [
            "∃w'[acc(w0)(w') ∧ dusk=t1 ∧ ∂(t1<now) ∧ ∃t'[t'⊆t1 ∧ read(w',t',g,ιz[map(w',z)])]]"
          ]
        }
      ]
    },
    {
      "title": "C. Attitudes over tensed clauses",
      "instructions": "Attitude verbs interact with tense: \"Gandalf believes Frodo rode at dawn\" — the embedded TenseP has type t (after modifier saturation), w0 free. IFA with believes wraps it as λw0.φ and applies to the accessibility relation: in all of Gandalf's belief worlds, the temporal claim holds. The time constants and the tense variable t1 are the same across belief worlds.",
      "items": [
        {
          "sentence": "Gandalf believes Frodo rode at dawn.",
          "tree": "[.S [.DP Gandalf ] [.VP [.V believes ] [.CP [.TenseP [.TenseP [.Tense PAST ] [.AspP [.Asp PFV ] [.vP [.DP Frodo ] [.VP rides ] ] ] ] [.PP [.P at ] [.DP dawn ] ] ] ] ] ]",
          "targets": [
            "∀w'[dox(w0)(g,w') → dawn=t1 ∧ ∂(t1<now) ∧ ∃t'[t'⊆t1 ∧ ride(w',t',f)]]"
          ]
        },
        {
          "sentence": "Frodo wants to ride at dawn.",
          "tree": "[.S [.DP Frodo ] [.VP [.V wants ] [.CP [.TenseP [.TenseP [.Tense PAST ] [.AspP [.Asp PFV ] [.vP [.DP Frodo ] [.VP rides ] ] ] ] [.PP [.P at ] [.DP dawn ] ] ] ] ] ]",
          "targets": [
            "∀w'[boul(w0)(f,w') → dawn=t1 ∧ ∂(t1<now) ∧ ∃t'[t'⊆t1 ∧ ride(w',t',f)]]"
          ]
        }
      ]
    }
  ]
}