{
  "compose": 1,
  "id": "montague-ptq",
  "title": "Montague's PTQ · A — The Extensional Core",
  "subtitle": "Montague 1973 — quantifier NPs, raised verbs, quantifying-in",
  "domain": {
    "multiLetterNames": true,
    "constants": {
      "e": "j m b"
    },
    "variables": {
      "e": "x y z",
      "et": "P Q X Y Z",
      "<<e,t>,t>": "T U"
    }
  },
  "lexicon": [
    {
      "words": [
        "John"
      ],
      "denotation": "j"
    },
    {
      "words": [
        "Mary"
      ],
      "denotation": "m"
    },
    {
      "words": [
        "Bill"
      ],
      "denotation": "b"
    },
    {
      "words": [
        "man"
      ],
      "denotation": "Lx.man(x)"
    },
    {
      "words": [
        "woman"
      ],
      "denotation": "Lx.woman(x)"
    },
    {
      "words": [
        "unicorn"
      ],
      "denotation": "Lx.unicorn(x)"
    },
    {
      "words": [
        "fish"
      ],
      "denotation": "Lx.fish(x)"
    },
    {
      "words": [
        "park"
      ],
      "denotation": "Lx.park(x)"
    },
    {
      "words": [
        "walks",
        "walk"
      ],
      "denotation": "Lx.walk(x)"
    },
    {
      "words": [
        "talks",
        "talk"
      ],
      "denotation": "Lx.talk(x)"
    },
    {
      "words": [
        "runs",
        "run"
      ],
      "denotation": "Lx.run(x)"
    },
    {
      "words": [
        "rises",
        "rise"
      ],
      "denotation": "Lx.rise(x)"
    },
    {
      "words": [
        "loves",
        "love"
      ],
      "denotation": "LT.Lx.T(Ly.love(x,y))"
    },
    {
      "words": [
        "finds",
        "find"
      ],
      "denotation": "LT.Lx.T(Ly.find(x,y))"
    },
    {
      "words": [
        "seeks",
        "seek"
      ],
      "denotation": "LT.Lx.T(Ly.seek(x,y))"
    },
    {
      "words": [
        "eats",
        "eat"
      ],
      "denotation": "LT.Lx.T(Ly.eat(x,y))"
    },
    {
      "words": [
        "is",
        "are",
        "be"
      ],
      "denotation": "LT.Lx.T(Ly.(x=y))"
    },
    {
      "words": [
        "every"
      ],
      "denotation": "LX.LY.Ax[X(x) -> Y(x)]"
    },
    {
      "words": [
        "a",
        "an"
      ],
      "denotation": "LX.LY.Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "no"
      ],
      "denotation": "LX.LY.~Ex[X(x) & Y(x)]"
    },
    {
      "words": [
        "the"
      ],
      "denotation": "LX.LY.Ex[X(x) & Ay[X(y) -> y=x] & Y(x)]"
    }
  ],
  "rules": {
    "composition": {
      "functionApplication": true,
      "predicateModification": false,
      "nonBranchingNodes": true,
      "predicateAbstraction": true
    },
    "typeShifts": [
      "lift",
      "lower"
    ],
    "quantifierRaising": true
  },
  "exercises": [
    {
      "title": "A. Every noun phrase is a quantifier",
      "instructions": "A determiner + noun is a quantifier ⟨⟨e,t⟩,t⟩ that takes the VP as its argument by FA. A name may either combine directly (e + ⟨e,t⟩) or, in Montague's uniform style, lift to a quantifier first — either way the result is the same.",
      "items": [
        {
          "id": "d1",
          "tree": "[.t [.T Bill ] [.IV walks ] ]",
          "sentence": "Bill walks.",
          "targets": [
            "walk(b)"
          ]
        },
        {
          "id": "d2",
          "tree": "[.t [.T [.Det a ] [.CN man ] ] [.IV walks ] ]",
          "sentence": "A man walks.",
          "targets": [
            "Ex[man(x) & walk(x)]"
          ]
        },
        {
          "id": "d3",
          "tree": "[.t [.T [.Det every ] [.CN man ] ] [.IV walks ] ]",
          "sentence": "Every man walks.",
          "targets": [
            "Ax[man(x) -> walk(x)]"
          ]
        }
      ]
    },
    {
      "title": "B. Transitive verbs take quantifier objects",
      "instructions": "A PTQ transitive verb is type ⟨⟨⟨e,t⟩,t⟩,⟨e,t⟩⟩ — it combines with a quantifier object. A name object must lift first; a determiner object is already a quantifier and takes narrow scope in situ.",
      "items": [
        {
          "sentence": "John loves Mary.",
          "tree": "[.t [.T John ] [.IV [.TV loves ] [.T Mary ] ] ]",
          "targets": [
            "love(j,m)"
          ],
          "reading": {
            "section": "2"
          }
        },
        {
          "sentence": "John loves every woman.",
          "tree": "[.t [.T John ] [.IV [.TV loves ] [.T [.Det every ] [.CN woman ] ] ] ]",
          "targets": [
            "Ay[woman(y) -> love(j,y)]"
          ],
          "reading": {
            "section": "2"
          }
        },
        {
          "sentence": "John finds a unicorn.",
          "tree": "[.t [.T John ] [.IV [.TV finds ] [.T [.Det a ] [.CN unicorn ] ] ] ]",
          "targets": [
            "Ey[unicorn(y) & find(j,y)]"
          ],
          "reading": {
            "section": "2"
          }
        }
      ]
    },
    {
      "title": "C. The verb be — identity and predication",
      "instructions": "One entry, be = λT.λx.T(λy.x=y), takes a quantifier object. With a (lifted) name it gives identity; with an indefinite it reduces, by the one-point law, to a predication.",
      "items": [
        {
          "sentence": "Bill is Mary.",
          "tree": "[.t [.T Bill ] [.IV [.TV is ] [.T Mary ] ] ]",
          "targets": [
            "b = m"
          ],
          "reading": {
            "section": "3"
          }
        },
        {
          "sentence": "Bill is a man.",
          "tree": "[.t [.T Bill ] [.IV [.TV is ] [.T [.Det a ] [.CN man ] ] ] ]",
          "targets": [
            "man(b)"
          ],
          "reading": {
            "section": "3"
          }
        }
      ]
    },
    {
      "title": "D. The (Russellian)",
      "instructions": "Montague's the carries existence and uniqueness: the man = λY.∃x[man(x) ∧ ∀y[man(y)→y=x] ∧ Y(x)].",
      "items": [
        {
          "sentence": "The man walks.",
          "tree": "[.t [.T [.Det the ] [.CN man ] ] [.IV walks ] ]",
          "targets": [
            "Ex[man(x) & Ay[man(y) -> y=x] & walk(x)]"
          ],
          "reading": {
            "section": "4"
          }
        }
      ]
    },
    {
      "title": "E. Quantifying-in and scope",
      "instructions": "An in-situ object quantifier takes narrow scope. For the inverse reading, raise the object quantifier over the clause (QR) and bind the trace — Montague's quantifying-in. \"Every man loves a woman\" is scope-ambiguous: derive both readings.",
      "items": [
        {
          "sentence": "Every man loves a woman.",
          "tree": "[.t [.T [.Det every ] [.CN man ] ] [.IV [.TV loves ] [.T [.Det a ] [.CN woman ] ] ] ]",
          "targets": [
            "Ax[man(x) -> Ey[woman(y) & love(x,y)]]",
            "Ey[woman(y) & Ax[man(x) -> love(x,y)]]"
          ],
          "reading": {
            "section": "5"
          }
        }
      ]
    }
  ],
  "reading": {
    "format": "latex",
    "markdown": "# Montague's PTQ · A — The Extensional Core\n\nPTQ opens: \"The aim of this paper is to present in a rigorous way the syntax\nand semantics of a certain fragment of a certain dialect of English.\" In the\nfragment every term phrase — a name, a quantifier phrase, a pronoun — is\ntranslated at one type, the generalized-quantifier type ⟨⟨e,t⟩,t⟩; the\nworksheet's lexical entries below are PTQ's own translations (rules T1–T2,\nquoted at the point of use, with PTQ's P{x} notation adapted to P(x) and\nintension/extension marks dropped as licensed by his meaning postulates —\nsee §6).\n\nThis reading develops the **extensional skeleton** of PTQ. The intensional half —\nthe de dicto reading of *seek*, the rising-temperature puzzle — needs functions\nfrom possible worlds and is noted in §6 but not derived here.\n\n## 1 Every noun phrase is a quantifier\n\nA name lifts to the set of its properties; PTQ's T1(e): \"heₙ translates into\nλP.P{xₙ}\" — the same shape, λP.P(j), for lifted names. For determiners,\nT2, quoted (notation adapted): \"If ζ ∈ P_CN and ζ translates into ζ′, then\nevery ζ translates into λP∀x[ζ′(x) → P{x}]\":\n\n\\begin{derivation}\nlift([[John]])    = lambda P.P(j)                       : <<e,t>,t>\n[[every]]         = lambda X.lambda Y.forall x[X(x) -> Y(x)] : <<e,t>,<<e,t>,t>>\n[[every man]]     = lambda Y.forall x[man(x) -> Y(x)]   : <<e,t>,t>\n\\end{derivation}\n\nA quantifier subject takes the VP as *its* argument (\\ref{npq}):\n\n\\begin{forest}\n[S{forall x[man(x) -> walk(x)]}\n  [DP{lambda Y.forall x[man(x) -> Y(x)]}\n    [D{lambda X.lambda Y.forall x[X(x) -> Y(x)]} every]\n    [NP{lambda x.man(x)} man]]\n  [VP{lambda x.walk(x)} walks]]\n\\end{forest}\n\n\\ex<npq> Bill walks.\n\\xe\n\n\\ex Every man walks.\n\\xe\n\n## 2 Transitive verbs take quantifier objects\n\nIn PTQ a transitive verb does not combine with an individual but with a *quantifier*.\nIts denotation feeds the object quantifier the relation's second slot:\n\n\\begin{derivation}\n[[loves]]            = lambda T.lambda x.T(lambda y.love(x,y)) : <<<e,t>,t>,<e,t>>\n[[loves every woman]] = lambda x.forall y[woman(y) -> love(x,y)] : <e,t>\n\\end{derivation}\n\nSo an in-situ object quantifier automatically takes **narrow scope** (\\ref{tv}). A name\nobject lifts first, and the truth conditions reduce to the simple relation:\n\n\\begin{derivation}\n[[loves Mary]]      = lambda x.love(x,m)     : <e,t>\n[[John loves Mary]] = love(j,m)              : t\n\\end{derivation}\n\n\\begin{forest}\n[S{forall y[woman(y) -> love(j,y)]}\n  [DP{j} John]\n  [VP{lambda x.forall y[woman(y) -> love(x,y)]}\n    [V{lambda T.lambda x.T(lambda y.love(x,y))} loves]\n    [DP{lambda Y.forall y[woman(y) -> Y(y)]} every woman]]]\n\\end{forest}\n\n\\ex<tv> John loves Mary.\n\\xe\n\n\\ex John loves every woman.\n\\xe\n\n\\ex John finds a unicorn.\n\\xe\n\n## 3 The verb *be*\n\nOne entry covers both identity and predication. PTQ's T1(b), quoted\n(notation adapted, extension marks dropped): \"be translates into\nλ𝒫λx.𝒫{ŷ[ˇx = ˇy]}\". Montague notes its extensionality need not be\nstipulated: \"The reason why the extensionality of be was not explicitly\nassumed is that it can be proved.\"\n\n\\begin{derivation}\n[[is]]          = lambda T.lambda x.T(lambda y.x=y)   : <<<e,t>,t>,<e,t>>\n[[is Bill]]     = lambda x.(x=b)                       : <e,t>     (Bill lifted)\n[[is a man]]    = lambda x.exists z[man(z) & x=z] = lambda x.man(x) : <e,t>\n\\end{derivation}\n\n*Bill is Mary* is the identity `j=b`; *Bill is a man* reduces — by the one-point\nlaw — to the predication `man(j)` (\\ref{beids}).\n\n\\ex<beids> Bill is Mary.\n\\xe\n\n\\ex Bill is a man.\n\\xe\n\n## 4 The (Russellian)\n\nT2 continues (notation adapted): \"the ζ translates into\nλP∃y[∀x[ζ′(x) ↔ x = y] ∧ P{y}]\" — Russell's existence-and-uniqueness:\n\n\\begin{derivation}\n[[the]]      = lambda X.lambda Y.exists x[X(x) & forall y[X(y) -> y=x] & Y(x)] : <<e,t>,<<e,t>,t>>\n[[the man walks]] = exists x[man(x) & forall y[man(y) -> y=x] & walk(x)]       : t\n\\end{derivation}\n\nThere is a man, he is the only man, and he walks (\\ref{theman}).\n\n\\ex<theman> The man walks.\n\\xe\n\n## 5 Quantifying-in and scope\n\nSurface scope falls out in situ (§2). The **inverse** reading, and de re readings\ngenerally, come from *quantifying-in*: build an open sentence with a pronoun —\nitself a lifted variable `λP.P(xₙ)` — then let a quantifier bind it. In COMPOSE this\nis Quantifier Raising: move the object to the front, leaving a trace.\n\n\\begin{derivation}\n[[every man loves a woman]]\n  surface (∀ > ∃) : forall x[man(x) -> exists y[woman(y) & love(x,y)]]\n  inverse (∃ > ∀) : exists y[woman(y) & forall x[man(x) -> love(x,y)]]\n\\end{derivation}\n\nThe inverse is derived by raising *a woman* over the clause and binding the trace\n(\\ref{scope}):\n\n\\begin{derivation}\n[[every man loves t1]] = forall x[man(x) -> love(x,x1)]            : t\nlambda-bind x1         = lambda x1.forall x[man(x) -> love(x,x1)]  : <e,t>\n[[a woman]](...)       = exists y[woman(y) & forall x[man(x) -> love(x,y)]] : t\n\\end{derivation}\n\n\\ex<scope> Every man loves a woman.\n\\xe\n\n\\ex John seeks a unicorn.\n\\xe\n\n## 6 A note on intensionality\n\nThis worksheet stays inside the extensional core: by Montague's own meaning\npostulates (MP1–MP4), names are rigid, ordinary common nouns and verbs are\nextensional, and the starred (reduced) translations used here are exactly\nwhat those postulates license. The paper's real subject — *seek*,\n*believe that*, *try to*, *necessarily*, and the temperature puzzle — lives\nin **Part B — The Intensional Fragment**, the companion worksheet on this\npage, which states explicitly how COMPOSE renders Montague's intensional\nlogic in Gallin's two-sorted TY2.\n\n## Credits\n\nMontague, R. (1973). The proper treatment of quantification in ordinary\nEnglish. In K. J. J. Hintikka et al. (eds.), *Approaches to Natural\nLanguage*, 221–242. Reidel. Translations quoted from T1–T2 with notation\nadapted as flagged above. Notes text: quotes as marked, plus flagged\nrendering notes; remaining prose is signposting."
  }
}