{
  "compose": 1,
  "id": "ch7-adjts",
  "title": "Adjective Type-Shifting: MOD (§7.2)",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "f s g n a b l m o w bi st pi tm el sr mr x1 x2"
    },
    "variables": {
      "e": "x y z",
      "et": "X Y Z P Q R F G H",
      "t": "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": [
        "is",
        "are"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.X"
    },
    {
      "words": [
        "not"
      ],
      "denotation": "LP.Lx.~P(x)"
    },
    {
      "words": [
        "hobbit"
      ],
      "denotation": "Lx.hobbit(x)"
    },
    {
      "words": [
        "wizard"
      ],
      "denotation": "Lx.wizard(x)"
    },
    {
      "words": [
        "elf"
      ],
      "denotation": "Lx.elf(x)"
    },
    {
      "words": [
        "creature"
      ],
      "denotation": "Lx.creature(x)"
    },
    {
      "words": [
        "ranger"
      ],
      "denotation": "Lx.ranger(x)"
    },
    {
      "words": [
        "human"
      ],
      "denotation": "Lx.human(x)"
    },
    {
      "words": [
        "king"
      ],
      "denotation": "Lx.king(x)"
    },
    {
      "words": [
        "brave"
      ],
      "denotation": "Lx.brave(x)"
    },
    {
      "words": [
        "mischievous"
      ],
      "denotation": "Lx.mischievous(x)"
    },
    {
      "words": [
        "mundane"
      ],
      "denotation": "Lx.mundane(x)"
    },
    {
      "words": [
        "magical"
      ],
      "denotation": "Lx.magical(x)"
    },
    {
      "words": [
        "traveling"
      ],
      "denotation": "Lx.traveling(x)"
    },
    {
      "words": [
        "wise"
      ],
      "denotation": "Lx.wise(x)"
    },
    {
      "words": [
        "alleged"
      ],
      "denotation": "LF.Lx.alleged(F)(x)"
    },
    {
      "words": [
        "former"
      ],
      "denotation": "LF.Lx.former(F)(x)"
    },
    {
      "words": [
        "sees",
        "see"
      ],
      "denotation": "Ly.Lx.see(x,y)"
    },
    {
      "words": [
        "every",
        "Every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "some",
        "Some"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "no",
        "No"
      ],
      "denotation": "LX.LY.~Ex[X(x) & Y(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "nonBranchingNodes": true,
      "predicateModification": true
    },
    "typeShifts": [
      "mod"
    ]
  },
  "exercises": [
    {
      "title": "A. MOD: predicative to attributive",
      "instructions": "An intersective adjective ⟨e,t⟩ can be shifted to a modifier ⟨⟨e,t⟩,⟨e,t⟩⟩ by MOD, then combined with the noun by Function Application. Apply MOD to the adjective, then use FA to combine with the noun.",
      "items": [
        {
          "sentence": "Pippin is a mischievous hobbit.",
          "tree": "[.S [.DP Pippin ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mischievous ] [.NP hobbit ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "mischievous(pi) ∧ hobbit(pi)"
          ],
          "id": "d1"
        },
        {
          "sentence": "Tom is not a mundane creature.",
          "tree": "[.S [.DP Tom ] [.VP [.Neg not ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mundane ] [.NP creature ] ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "¬(mundane(tm) ∧ creature(tm))"
          ],
          "id": "d2"
        },
        {
          "sentence": "Every traveling hobbit is a brave creature.",
          "tree": "[.S [.DP [.D every ] [.NP [.AP traveling ] [.NP hobbit ] ] ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP brave ] [.NP creature ] ] ] ] ]",
          "reading": {
            "section": "7.2"
          },
          "targets": [
            "∀x.traveling(x) ∧ hobbit(x) → brave(x) ∧ creature(x)"
          ],
          "id": "d3"
        }
      ],
      "id": "g1"
    },
    {
      "title": "B. Stacked MOD",
      "instructions": "Stack two adjective shifts: apply MOD to each adjective in turn, combine each with the noun by FA. The third adds negation.",
      "items": [
        {
          "sentence": "Tom is a mischievous magical creature.",
          "tree": "[.S [.DP Tom ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mischievous ] [.NP [.AP magical ] [.NP creature ] ] ] ] ] ]",
          "reading": {
            "section": "7.2.1"
          },
          "targets": [
            "mischievous(tm) ∧ magical(tm) ∧ creature(tm)"
          ],
          "id": "d1"
        },
        {
          "sentence": "Strider is not a human ranger.",
          "tree": "[.S [.DP Strider ] [.VP [.Neg not ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP human ] [.NP ranger ] ] ] ] ] ]",
          "reading": {
            "section": "7.2.1"
          },
          "targets": [
            "¬(human(st) ∧ ranger(st))"
          ],
          "id": "d2"
        },
        {
          "sentence": "No traveling hobbit is a mischievous brave creature.",
          "tree": "[.S [.DP [.D no ] [.NP [.AP traveling ] [.NP hobbit ] ] ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP mischievous ] [.NP [.AP brave ] [.NP creature ] ] ] ] ] ]",
          "reading": {
            "section": "7.2.1"
          },
          "targets": [
            "¬(∃x.traveling(x) ∧ hobbit(x) ∧ mischievous(x) ∧ brave(x) ∧ creature(x))"
          ],
          "id": "d3"
        }
      ],
      "id": "g2"
    },
    {
      "title": "C. Non-intersective adjectives via FA",
      "instructions": "Non-intersective adjectives are already modifiers ⟨⟨e,t⟩,⟨e,t⟩⟩ — no MOD shift needed. They take the nominal predicate directly by FA.",
      "items": [
        {
          "sentence": "Strider is an alleged king.",
          "tree": "[.S [.DP Strider ] [.VP [.V is ] [.NP [.D an ] [.NP [.AP alleged ] [.NP king ] ] ] ] ]",
          "reading": {
            "section": "7.2.2"
          },
          "targets": [
            "alleged(λx.king(x), st)"
          ],
          "id": "d1"
        },
        {
          "sentence": "Bilbo is a former traveling hobbit.",
          "tree": "[.S [.DP Bilbo ] [.VP [.V is ] [.NP [.D a ] [.NP [.AP former ] [.NP [.AP traveling ] [.NP hobbit ] ] ] ] ] ]",
          "reading": {
            "section": "7.2.2"
          },
          "targets": [
            "former(λx.traveling(x) ∧ hobbit(x), bi)"
          ],
          "id": "d2"
        },
        {
          "sentence": "Some alleged wise wizard sees Sauron.",
          "tree": "[.S [.DP [.D some ] [.NP [.AP alleged ] [.NP [.AP wise ] [.NP wizard ] ] ] ] [.VP [.V sees ] [.DP Sauron ] ] ]",
          "reading": {
            "section": "7.2.2"
          },
          "targets": [
            "∃x.alleged(λx.wise(x) ∧ wizard(x), x) ∧ see(x, n)"
          ],
          "id": "d3"
        }
      ],
      "id": "g3"
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Chapter 7 · Adjective Type-Shifting (MOD)\n\nThe same adjective–noun denotation can be reached by two routes. **Predicate Modification** treats both sisters as ⟨e,t⟩ and conjoins them directly. The **MOD type-shift** instead raises an intersective adjective to type ⟨⟨e,t⟩,⟨e,t⟩⟩, letting it combine with the noun by ordinary **Function Application**.\n\n## 7.2 The MOD shift\n\n**Type-Shifting Rule — MOD.** If α has type ⟨e,t⟩, it also has a derived translation of type ⟨⟨e,t⟩,⟨e,t⟩⟩:\n\n$$\\text{MOD}(\\alpha) = \\lambda P.\\lambda x.[\\alpha(x) \\wedge P(x)]$$\n\nThe shifted adjective takes the noun as its FA argument, returning the intersective conjunction. The result is identical to applying PM directly.\n\n\\ex<mod-ex> Pippin is a mischievous hobbit.\n\\xe\n\n\\ex Tom is not a mundane creature.\n\\xe\n\n\\ex Every traveling hobbit is a brave creature.\n\\xe\n\n\\begin{derivation}\n[[mischievous]]                    = lambda x.mischievous(x)                              : <e,t>\nMOD([[mischievous]])               = lambda P.lambda x.mischievous(x) /\\ P(x)             : <<e,t>,<e,t>>\n[[hobbit]]                         = lambda x.hobbit(x)                                   : <e,t>\n[[mischievous hobbit]]             = lambda x.mischievous(x) /\\ hobbit(x)                 : <e,t>\n[[Pippin is a mischievous hobbit]] = mischievous(pi) /\\ hobbit(pi)                        : t\n\\end{derivation}\n\nThe derivation produces exactly the same truth conditions as the PM route. MOD is a **silent operator** — it has no phonological form but a definite semantic effect, lifting the adjective's type so FA can apply.\n\n## 7.2.1 Stacking via MOD\n\nMultiple adjectives each undergo MOD in turn, applying to the growing nominal predicate one at a time.\n\n\\ex<stack-mod> Tom is a mischievous magical creature.\n\\xe\n\n\\ex Strider is not a human ranger.\n\\xe\n\n\\begin{derivation}\nMOD([[mischievous]])           = lambda P.lambda x.mischievous(x) /\\ P(x)         : <<e,t>,<e,t>>\nMOD([[magical]])               = lambda P.lambda x.magical(x) /\\ P(x)              : <<e,t>,<e,t>>\n[[creature]]                   = lambda x.creature(x)                              : <e,t>\n[[magical creature]]           = lambda x.magical(x) /\\ creature(x)               : <e,t>\n[[mischievous magical creature]] = lambda x.mischievous(x) /\\ magical(x) /\\ creature(x) : <e,t>\n\\end{derivation}\n\nThe innermost adjective applies first (its MOD-shifted form takes the bare noun), then the outer MOD-shifted adjective takes the result — each step is FA.\n\nWhen the subject is a quantifier, the MOD-derived predicate serves as the VP restriction exactly as in PM:\n\n\\begin{derivation}\n[[every]]                               = lambda X.lambda Y.forall x[X(x) -> Y(x)]              : <<e,t>,<<e,t>,t>>\n[[traveling hobbit]]                    = lambda x.traveling(x) /\\ hobbit(x)                    : <e,t>\n[[every traveling hobbit]]              = lambda Y.forall x[(traveling(x) /\\ hobbit(x)) -> Y(x)] : <<e,t>,t>\n[[brave creature]]                      = lambda x.brave(x) /\\ creature(x)                      : <e,t>\n[[Every traveling hobbit is a brave creature]] = forall x[(traveling(x) /\\ hobbit(x)) -> (brave(x) /\\ creature(x))] : t\n\\end{derivation}\n\n## 7.2.2 Non-intersective adjectives\n\nNon-intersective adjectives like *alleged* and *former* are **lexically** type ⟨⟨e,t⟩,⟨e,t⟩⟩ — they do not start as ⟨e,t⟩ and therefore do not undergo MOD. They take the nominal predicate directly by FA, exactly as in the PM set.\n\n\\ex<nonintersect-ts> Strider is an alleged king.\n\\xe\n\n\\ex Bilbo is a former traveling hobbit.\n\\xe\n\n\\ex Some alleged wise wizard sees Sauron.\n\\xe\n\n\\begin{derivation}\n[[alleged]]       = lambda F.lambda x.alleged(F)(x)    : <<e,t>,<e,t>>\n[[former]]        = lambda F.lambda x.former(F)(x)     : <<e,t>,<e,t>>\n[[alleged king]]  = lambda x.alleged(king)(x)          : <e,t>  (FA — no MOD)\n\\end{derivation}\n\nThe diagnostic: an adjective open to MOD is one that can appear *predicatively* (*Pippin is mischievous*); non-intersective adjectives resist purely predicative uses with their opaque reading.[^diag]\n\n[^diag]: *The king is former* is at best odd; *former* needs a nominal complement to make sense. This correlates with its lexical type: it is a modifier, not a predicate."
  }
}