Quantifiers in Natural Language

How quantifiers work

The simplest quantified sentence has three parts: a determiner, a noun (the restrictor), and a predicate (the nuclear scope).

[Every]   [student]      [passed].
 ───────   ─────────      ────────
 determiner  restrictor    nuclear scope

Generalized Quantifier Theory (GQT) treats the determiner as a function that takes two sets — the restrictor's denotation and the scope's denotation — and returns a truth value. Every takes the set of students S and the set of pass-ers P and returns true iff S ⊆ P. Some returns true iff S ∩ P ≠ ∅. Most returns true iff |S ∩ P| > |S − P|. No returns true iff S ∩ P = ∅.

This uniform treatment is the move. First-order logic could only translate every and some; most required jumping to second-order or to a generalized quantifier. The unification reveals that natural language treats them all the same way grammatically — they all combine determiner-with-noun-with-predicate — and the semantic differences are purely in which set-relation each determiner picks out.

Two universal-ish properties

Conservativity. A determiner Q is conservative iff Q(A, B) ↔ Q(A, A ∩ B). In words: only the A's matter. "Every dog barks" depends only on what dogs do; non-dogs are irrelevant. Every, some, no, most, three, few, many — all conservative. Keenan and Stavi (1986) argue every extensional natural-language determiner is conservative, a striking universal because logically there's no reason it should be.

Monotonicity. A determiner is upward-monotone in its second argument if Q(A, B) and B ⊆ B′ imply Q(A, B′). Downward-monotone if Q(A, B) and B′ ⊆ B imply Q(A, B′). The same applies to the first argument. The monotonicity profile predicts Negative Polarity Item (NPI) licensing: any, ever, lift a finger are grammatical only in downward-entailing environments.

Every  [+student] [+walked]   →  ↓ in arg1, ↑ in arg2
Some   [+student] [+walked]   →  ↑ in arg1, ↑ in arg2
No     [+student] [+walked]   →  ↓ in arg1, ↓ in arg2
Most   [+student] [+walked]   →  non-monotone in arg1, ↑ in arg2

Test: NPI any requires a downward-entailing context.
"Nobody who has any money complained" ✓ (no = ↓ in arg1)
"Somebody who has any money complained" ✗ (some = ↑ in arg1)

Restricted vs unrestricted quantifiers

	Restricted (NL)	Unrestricted (FOL)
Domain	Provided by the noun (restrictor)	Whole universe
Form	[Det N] VP — "every student passed"	∀x. (φ(x) → ψ(x))
Type	⟨⟨e,t⟩, ⟨⟨e,t⟩, t⟩⟩	⟨⟨e,t⟩, t⟩
Conservativity	Built in (only restrictor matters)	N/A — no restrictor
Most expressible	Yes (Det as set relation)	No (requires second-order)
Vague Det (many, few)	Yes	No
Cross-linguistic	Universal pattern (Det N + scope)	Logician's normal form
Quantifier raising	Generates inverse-scope readings	Scope fixed by formula

The disconnect is one reason logicians and linguists historically talked past each other. Frege wrote ∀x. (STUDENT(x) → PASSED(x)), folding the restriction into a conditional. Linguists pointed out that most resists this trick: ∀x. (STUDENT(x) → PASSED(x)) means every student passed, but you cannot rewrite "most students passed" as a single first-order formula.

A typology of natural-language Det's

• Logical. every, all, no, some, a/an — definable in first-order logic, isomorphism-invariant, cross-linguistically near-universal.
• Cardinal. three, seven, more than five, fewer than ten, exactly two — count-based, definable with cardinality predicates.
• Proportional. most, half, two-thirds, ten percent, the majority — compare cardinalities of A∩B and A−B.
• Vague / context-dependent. many, few, several, a lot, hardly any — sensitive to context-set, expectations, and comparison classes.
• Definite / specific. the, this, that, these, those — analyzed as quantifiers in some frameworks (Russell 1905, Strawson 1950, Heim 1982).
• Possessive. John's, my, every student's — quantify over a possession relation.
• Generic. Bare plurals and the singular generic — "dogs bark," "a dog barks" — quantify over normal individuals or kinds (Carlson 1977, Krifka et al. 1995).

Cross-linguistic patterns

• English. Det-N order; rich proportional system (most, majority, the bulk of); strong Quantifier Raising for inverse scope.
• Mandarin. No definite article; classifier-based quantification (三本书 sān běn shū "three CL book"); strong surface-scope preference. Mandarin lacks a direct lexicalization of most; speakers use 大多数 dàduōshù ("great-majority") or 多半 duōbàn ("more-than-half").
• Japanese. Floating quantifiers (学生が三人来た gakusei-ga san-nin kita "students three-CL came") — the quantifier appears separated from the noun. Strong surface-scope; Japanese inverse readings are rare and pragmatically marked (Kuroda 1971).
• German. Quantifiers carry case agreement; scope is fairly fixed by surface order (more rigid than English).
• Pirahã. Famously claimed (Everett 2005) to lack proportional quantifiers entirely — the only quantifier-like terms are something like "a small amount" and "a large amount." The claim is contested but a useful boundary case for the typology.
• Hungarian. Strict left-to-right scope determined by syntactic position; no Quantifier Raising. Hungarian is the textbook case for surface-scope languages.

Worked example: scope ambiguity

Consider:

Every student read a book.

Two readings:

• Surface scope (∀ > ∃): ∀x. STUDENT(x) → ∃y. BOOK(y) ∧ READ(x, y). Each student read some book or other; possibly different books.
• Inverse scope (∃ > ∀): ∃y. BOOK(y) ∧ ∀x. STUDENT(x) → READ(x, y). There is one specific book that every student read.

In Generalized Quantifier Theory the two readings correspond to two different orders of function application — the indefinite scoping over the universal, or vice versa. May (1985) introduced Quantifier Raising (QR), an LF-level movement operation that adjoins a quantifier phrase to a sentential node, deriving the inverse reading from the same surface structure.

Languages differ in QR availability:

English  ∀>∃ and ∃>∀  both available, ambiguous
Mandarin ∀>∃ strongly preferred (Aoun & Li 1989)
Japanese ∀>∃ strongly preferred
German   roughly surface-only
Hungarian surface-only — no QR

Other quantificational phenomena

• Donkey anaphora. "Every farmer who owns a donkey beats it" — the indefinite a donkey behaves like a universal in scope, motivating Discourse Representation Theory (Kamp 1981) and File Change Semantics (Heim 1982).
• Cumulative readings. "Three professors taught five courses" can mean a total of 3 professors and 5 courses with arbitrary pairings — neither pure ∀>∃ nor ∃>∀.
• Polyadic quantifiers. "Different students read different books" — quantifiers that don't decompose into iterated monadic ones (van Benthem 1989, Keenan 1992).
• Choice functions. Reinhart (1997) — wide-scope existentials over indefinites are analyzed as choice functions, sidestepping standard scope mechanisms.
• Q-particles. Some languages mark question-quantifier scope morphologically — Japanese -ka, Sinhala -da, Tlingit -sá.
• Numeral classifiers. Mandarin, Japanese, Korean, Vietnamese require a classifier between numeral and noun (三本书 sān běn shū). Treated semantically as part of the quantification structure (Chierchia 1998).

Common pitfalls

• Treating most as "more than half." Closer than "many," but not identical — "most" usually implies a clear majority, not a 51% one. Hackl (2009) gives experimental evidence the lexical entry is > half; pragmatic strengthening adds the "clear" flavor.
• Confusing any with every. NPI any needs a downward-entailing context; free-choice any in modal contexts is universal-flavored. Two distinct lexical items, often conflated.
• Assuming all languages have proportional quantifiers. Not all do — Pirahã is the famous edge case; many others lack a direct lexicalization of most.
• Reading scope off surface order. English allows inverse scope freely; Mandarin and Japanese typically don't. Cross-linguistic predictions matter.
• Forgetting conservativity. If you propose a new determiner, check whether only the restrictor matters — non-conservative entries are nearly impossible to find in natural language.
• Conflating all and every. "All students passed" allows collective readings; "every student passed" forces distributive (each one). The difference shows up in "all students gathered" ✓ vs "every student gathered" ✗.