Montague (1973): The Proper Treatment of Quantification in Ordinary English #
[Mon73], with the category-to-type correspondence in the form adopted by [dowty-wall-peters-1981] (Ch. 7, Bennett's simplification).
PTQ's two central moves, exercised over the shared Semantics.Montague toy
fragment:
- Compositional truth conditions. Each predication receives its truth
value from the
ToyLexicondenotations alone (§ Predication). - Term phrases denote generalized quantifiers (PTQ's category
T), and a single English string can be assigned more than one meaning by quantifying a term in at one position or another (PTQ's rule S14). This is the source of quantifier scope ambiguity — handled in the grammar, not by a post-hoc scope-assignment rule (§ Category-to-type correspondence,§ Scope ambiguity).
Grounding, not restipulation #
The category-to-type function f and the named PTQ categories come from the
canonical Intensional.CategoryType (PtqCat/catToTy), so the IL types here
are the paper's — verbs and common nouns take individual concepts ⟨s,e⟩,
not bare entities. The two scope readings are the shared GQ-level operators
Quantification.Polyadic.surfaceScope/inverseScope over the propositional
denotations every_sem/some_sem, and the ambiguous string is a
Semantics.Scope.ScopeDerivation. This makes PTQ's quantifying-in directly
comparable to the QR (HeimKratzer1998), Glue (Asudeh2022), and RSA
(ScontrasPearl2021) treatments of the same ambiguity, which
target the same substrate.
PTQ's distinctive intensional payload — individual concepts doing semantic work
in the temperature/price puzzle ("the temperature is ninety but it is rising")
and the de dicto/de re ambiguity of "John seeks a unicorn" — needs a model with
a nontrivial index set and is left to a dedicated intensional study; the toy
fragment here is extensional (W = Unit), so the "sees" example exercises only
the quantificational, not the intensional, side of PTQ.
Predication #
Compositional truth conditions by direct function application over the toy fragment.
Predication discriminates individuals in vs. out of the extension.
Transitive predication discriminates ordered pairs.
Category-to-type correspondence #
PTQ assigns every syntactic category a unique IL type via f (catToTy in
Intensional.CategoryType). The facts below anchor the categories relevant to
quantification; the general machinery lives in the substrate.
A term phrase (category T, e.g. "every student", "John") denotes a
generalized quantifier over property-intensions of individual concepts:
f(T) = ⟨⟨s, ⟨⟨s,e⟩, t⟩⟩, t⟩. Treating "every student" as a function on
(intensions of) VP-meanings rather than as an entity is exactly what lets
scope be assigned by the grammar.
A determiner (category T/CN, e.g. "every", "a") combines with a
common-noun intension to yield a term-phrase meaning.
Common nouns and intransitive verbs share the IL type of sets of
individual concepts ⟨⟨s,e⟩, t⟩ — PTQ's device for the
temperature/price puzzle, where a verb predicates of a concept whose value
varies, not of a fixed individual.
Scope ambiguity via quantifying-in (S14) #
"Every student sees a person" has two readings, generated by quantifying the
two term phrases in at different points (PTQ's S14). They are the two linear
Polyadic readings of an every/some pair:
- surface (∀∃): every student
xsees some person — possibly a different one for eachx; - inverse (∃∀): there is one person whom every student sees.
The paper's own non-intensional illustration is "a woman loves every man" (§4), introduced precisely to show "that ambiguity can arise even when there is no element of intensionality, simply because quantifying terms may be introduced in more than one order."
The toy "sees" relation as a subject-first binary predicate x sees y,
read off the (object-first) fragment denotation sees_sem. abbrev so the
Decidable (sees_sem _ _) instance below transports to seesRel.
Equations
Instances For
Each concrete sees_sem a b reduces to True/False, mirroring the
fragment's own DecidablePred pattern.
Equations
- One or more equations did not get rendered due to their size.
Surface reading is true on the fragment: every student sees some person (John sees Mary, Mary sees John).
Inverse reading is false on the fragment: no single person is seen by every student (John sees only Mary; Mary sees only John).
The string is genuinely ambiguous: true on its surface reading, false on its inverse reading.
∃∀ ⊨ ∀∃. For an every/some pair, the inverse (wide-existential)
reading entails the surface (wide-universal) reading, over any domain,
restrictors, and relation. PTQ's S14 generates both readings; this is the
one-directional entailment between them, and it is why a context can never
make the inverse reading true without the surface reading being true too.
"Every student sees a person" as a single syntactic form carrying both
scope-indexed meanings, in the shared ScopeDerivation representation —
the same type targeted by HeimKratzer1998 (QR), Asudeh2022 (Glue), and
ScontrasPearl2021 (RSA).
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Instances For
Truth of the surface (∀∃) reading over the single extensional world. The
Prop is written explicitly (rather than as everyStudentSeesAPerson.meaningAt .surface) so Decidable synthesises over ToyEntity; surfaceTruth_iff
certifies it is the same reading.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Truth of the inverse (∃∀) reading; inverseTruth_iff ties it to the
ScopeDerivation.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Bool surface truth function reflects the ScopeDerivation's surface
meaning — certifying that the explicitly-written Prop in surfaceTruth is
everyStudentSeesAPerson.meaningAt .surface, not merely asserted to be.
The Bool inverse truth function reflects the ScopeDerivation's inverse
meaning.
In the shared ScopeEntailment vocabulary (cf.
ScontrasPearl2021.every_not_scope_entailment, which gets
surfaceEntailsInverse for "every…not"), the every/some pair
classifies as inverseEntailsSurface: inverse ⊨ surface (held across the
checked world), surface ⊭ inverse (the toy world makes surface true and
inverse false). The general direction is every_some_inverse_entails_surface.