{
  "compose": 1,
  "id": "ch13-modals",
  "title": "Modals: Necessity & Possibility (§13.4.4)",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w"
    },
    "variables": {
      "e": "x y z",
      "s": "w0 w w' w''",
      "st": "p q",
      "et": "P Q",
      "set": "R S"
    }
  },
  "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(w0,x)"
    },
    {
      "words": [
        "sings",
        "sing"
      ],
      "denotation": "Lx.sing(w0,x)"
    },
    {
      "words": [
        "rests",
        "rest"
      ],
      "denotation": "Lx.rest(w0,x)"
    },
    {
      "words": [
        "reads",
        "read"
      ],
      "denotation": "Ly.Lx.read(w0,x,y)"
    },
    {
      "words": [
        "finds",
        "find"
      ],
      "denotation": "Ly.Lx.find(w0,x,y)"
    },
    {
      "words": [
        "sees",
        "see"
      ],
      "denotation": "Ly.Lx.see(w0,x,y)"
    },
    {
      "words": [
        "guards",
        "guard"
      ],
      "denotation": "Ly.Lx.guard(w0,x,y)"
    },
    {
      "words": [
        "ring"
      ],
      "denotation": "Lx.ring(w0,x)"
    },
    {
      "words": [
        "map"
      ],
      "denotation": "Lx.map(w0,x)"
    },
    {
      "words": [
        "gate"
      ],
      "denotation": "Lx.gate(w0,x)"
    },
    {
      "words": [
        "dragon"
      ],
      "denotation": "Lx.dragon(w0,x)"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(w0,x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(w0,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": [
        "necessarily"
      ],
      "denotation": "Lp.Aw'[acc(w0)(w') -> p(w')]"
    },
    {
      "words": [
        "possibly"
      ],
      "denotation": "Lp.Ew'[acc(w0)(w') & p(w')]"
    },
    {
      "words": [
        "necessarily_deon"
      ],
      "denotation": "Lp.Aw'[deon_acc(w0)(w') -> p(w')]"
    },
    {
      "words": [
        "possibly_deon"
      ],
      "denotation": "Lp.Ew'[deon_acc(w0)(w') & p(w')]"
    },
    {
      "words": [
        "necessarily_meta"
      ],
      "denotation": "Lp.Aw'[meta_acc(w0)(w') -> p(w')]"
    },
    {
      "words": [
        "possibly_epist"
      ],
      "denotation": "Lp.Ew'[epist_acc(w0)(w') & p(w')]"
    },
    {
      "words": [
        "possibly_meta"
      ],
      "denotation": "Lp.Ew'[meta_acc(w0)(w') & p(w')]"
    },
    {
      "words": [
        "necessarily_epist"
      ],
      "denotation": "Lp.Aw'[epist_acc(w0)(w') -> p(w')]"
    },
    {
      "words": [
        "seeks",
        "seek"
      ],
      "denotation": "LR.Lx.seek(w0,x,R)"
    },
    {
      "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,
      "predicateModification": true,
      "predicateAbstraction": true
    }
  },
  "exercises": [
    {
      "title": "A. Necessarily (IFA)",
      "instructions": "necessarily ↝ λp⟨s,t⟩.∀w′[acc(w0)(w′) → p(w′)] has type ⟨⟨s,t⟩,t⟩. Its sister has type t (with w0 free). IFA wraps the sister as λw0.φ before applying necessarily — this is the intension of the embedded clause, a proposition (function from worlds to truth values).",
      "items": [
        {
          "sentence": "Necessarily, Frodo rides.",
          "tree": "[.IP [.Adv necessarily ] [.S [.DP Frodo ] [.VP rides ] ] ]",
          "targets": [
            "∀w'[acc(w0)(w') → ride(w',f)]"
          ]
        },
        {
          "sentence": "Necessarily, Gandalf sings.",
          "tree": "[.IP [.Adv necessarily ] [.S [.DP Gandalf ] [.VP sings ] ] ]",
          "targets": [
            "∀w'[acc(w0)(w') → sing(w',g)]"
          ]
        },
        {
          "sentence": "Necessarily, Frodo finds the ring.",
          "tree": "[.IP [.Adv necessarily ] [.S [.DP Frodo ] [.VP [.V finds ] [.DP [.D the ] [.NP ring ] ] ] ] ]",
          "targets": [
            "∀w'[acc(w0)(w') → find(w',f,ιz[ring(w',z)])]"
          ]
        }
      ]
    },
    {
      "title": "B. Possibly (IFA)",
      "instructions": "possibly ↝ λp⟨s,t⟩.∃w′[acc(w0)(w′) ∧ p(w′)] is the existential dual of necessarily. Apply IFA in the same way — the two derivations are identical in structure, only the truth conditions differ: ∃ instead of ∀. Note how possibly does NOT entail the truth of its argument in the actual world.",
      "items": [
        {
          "sentence": "Possibly, Aragorn rests.",
          "tree": "[.IP [.Adv possibly ] [.S [.DP Aragorn ] [.VP rests ] ] ]",
          "targets": [
            "∃w'[acc(w0)(w') ∧ rest(w',a)]"
          ]
        },
        {
          "sentence": "Possibly, Gandalf finds the ring.",
          "tree": "[.IP [.Adv possibly ] [.S [.DP Gandalf ] [.VP [.V finds ] [.DP [.D the ] [.NP ring ] ] ] ] ]",
          "targets": [
            "∃w'[acc(w0)(w') ∧ find(w',g,ιz[ring(w',z)])]"
          ]
        }
      ]
    },
    {
      "title": "C. Modal flavor",
      "instructions": "Different accessibility relations capture different modal flavors. Deontic accessibility (deon_acc) ranges over worlds compatible with the prevailing rules; metaphysical (meta_acc) ranges over worlds metaphysically possible from here; epistemic (epist_acc) ranges over worlds compatible with what is known. The derivation is identical across flavors; only the accessibility constant changes.",
      "items": [
        {
          "sentence": "It is obligatory that Frodo guards the gate. (deontic)",
          "tree": "[.IP [.Adv necessarily_deon ] [.S [.DP Frodo ] [.VP [.V guards ] [.DP [.D the ] [.NP gate ] ] ] ] ]",
          "targets": [
            "∀w'[deon_acc(w0)(w') → guard(w',f,ιz[gate(w',z)])]"
          ]
        },
        {
          "sentence": "It is metaphysically possible that Sam sings. (metaphysical)",
          "tree": "[.IP [.Adv possibly_meta ] [.S [.DP Sam ] [.VP sings ] ] ]",
          "targets": [
            "∃w'[meta_acc(w0)(w') ∧ sing(w',s)]"
          ]
        },
        {
          "sentence": "It is epistemically possible that an elf finds the ring.",
          "tree": "[.IP [.Adv possibly_epist ] [.S [.DP [.D the ] [.NP elf ] ] [.VP [.V finds ] [.DP [.D the ] [.NP ring ] ] ] ] ]",
          "targets": [
            "∃w'[epist_acc(w0)(w') ∧ find(w',ιx[elf(w',x)],ιz[ring(w',z)])]"
          ]
        }
      ]
    }
  ]
}