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" right-downward. "Some" right-upward.

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

Symbols.

Symbol	Unicode	Meaning
Q(A,B)	—	generalized quantifier
⟨1,1⟩	—	type of determiner
CONS	—	conservativity
MON↑	—	upward monotone

Metatheory. Set relations. Universal constraints. Monotonicity. Type theory.

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

Limitations. Context-dependence. Exceptional quantifiers. Dynamic aspects. Plural.