Generalized Quantifier Theory

Origin. Mostowski (1957), Lindström, Barwise & Cooper (1981). Quantifiers as relations. NP denotations. Conservativity. Foundation of quantifier semantics.

Models. Quantifiers as relations between sets. Determiners as type ⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩. Universal constraints. Natural language quantification.

Formalism.

Generalized quantifier: Q: ℘(E) × ℘(E) → {0,1}. Relation between two sets. Q(A,B): "Q A's are B's." Determiner meaning.

Examples: every(A,B) iff A ⊆ B. some(A,B) iff A ∩ B ≠ ∅. no(A,B) iff A ∩ B = ∅. most(A,B) iff |A ∩ B| > |A - B|.

Type: Det: ⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩. NP: ⟨⟨e,t⟩,t⟩. Det combines with N, yields NP. "every student": ⟨⟨e,t⟩,t⟩.

Conservativity: Q(A,B) iff Q(A, A∩B). Only the A's matter. Universal constraint on natural language. "Every student smokes" = "Every student is a student who smokes."

Extension (quantity): Q depends only on |A∩B|, |A-B|, |B-A|, |E-(A∪B)|. Cardinality-based. Most determiners satisfy.

Monotonicity: Right upward: Q(A,B) and B ⊆ C implies Q(A,C). Right downward: Q(A,B) and C ⊆ B implies Q(A,C). "Every" and "some" are both right-upward: from A ⊆ B and B ⊆ C one gets A ⊆ C. "No" is right-downward. Monotonicity in the left argument is the other coordinate, and that is where "every" differs: A' ⊆ A with A ⊆ B gives A' ⊆ B, so "every" is left-downward while "some" is left-upward. The familiar description of "every" as downward-entailing concerns its first argument, the restrictor — which is why "every student who has ever been to Paris" licenses the negative polarity item and "every student has ever been to Paris" does not.

Polyadic quantifiers: "More... than..." Branching quantifiers. Resumptive quantifiers. Beyond type ⟨1,1⟩.

Symbols.

Metatheory. Mostowski (1957) defined a quantifier as an isomorphism-closed class of structures and Lindström (1966) generalized this to arbitrary types, which is what makes abstract model theory possible; Barwise and Cooper (1981) applied the apparatus to determiners and proposed conservativity as a semantic universal. Van Benthem (1984) proved the representation result behind the theory's predictive content: a type ⟨1,1⟩ quantifier satisfying CONS, EXT, and ISOM is determined by the two cardinals |A−B| and |A∩B| alone, so such quantifiers can be displayed on the number tree. His semantic-automata theorem (1986) grades expressive strength by machine class: the quantifiers recognized by acyclic finite automata are exactly the first-order definable ones — unrestricted finite automata add modular-counting quantifiers such as an even number of — while most requires a push-down automaton — the automata-theoretic form of the fact that most is not first-order definable on finite models. Keenan and Stavi (1986) showed that on a finite universe every conservative, extensional, isomorphism-closed quantifier is Boolean-definable from the English determiners. Monotonicity does linguistic work: Ladusaw (1979) tied negative polarity licensing to downward entailment, weakened by von Fintel (1999) to Strawson downward entailment.

Applies to. Determiner semantics. Quantifier scope. Inference. Typology.

Limitations. The universals have counterexamples. Only is not conservative — "Only students smoke" is not "Only students are students who smoke" — so either it is not a determiner or the universal is a tendency; Westerståhl's many and Partee's (1989) proportional readings violate ISOM and EXT, since what counts as many depends on a background frequency the quantifier is not supposed to see. Conservativity is close to vacuous as a constraint, being derivable from any syntax that makes the restrictor an argument. The framework is static and extensional, so donkey anaphora, intensional determiners, and collective readings all need machinery from outside it. Keenan (1992) showed that branching and reciprocal constructions do not reduce to iterated ⟨1,1⟩ quantifiers, so the type that carries the universals does not cover the data. And the theory fixes truth conditions without fixing scope.

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