Generalized Quantifiers: Scope Dominance and Branching

Abstract

Let P and Q be proper upward monotone unary generalized quantifiers on a fixed domain D. We study the validity, over all structures M with domain D, of M ⊨ Px Qy R(x,y) → Qy Px R(x,y), called scope dominance. The iteration of two quantifiers lies between their branching product, which is generated by rectangles, and the dual of the branching product of their duals. We give criteria for equality with these bounds, obtain a new proof of the countable characterization of scope dominance, and characterize the case in which the inner quantifier is "at least κ many." Under the Generalized Continuum Hypothesis, Goldberg's theorem shows that, for ultrafilters, scope dominance is equivalent to equality between iteration and the branching product, whereas an elementary counterexample on a domain of size ℵ₂ shows that this equivalence fails for arbitrary upward monotone quantifiers. For the case where both quantifiers are filters, the question becomes set-theoretic: the existence of a counterexample is equiconsistent with the existence of a measurable cardinal.

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BibTeX RIS

BibTeX

@misc{engstrom2026generalized,
  title = {{Generalized Quantifiers: Scope Dominance and Branching}},
  author = {Fredrik Engström},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31243},
  eprint = {2609.31243},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31243},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - Generalized Quantifiers: Scope Dominance and Branching
AU  - Fredrik Engström
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31243
UR  - https://arxiv.org/abs/2609.31243
AB  - Let P and Q be proper upward monotone unary generalized quantifiers on a fixed domain D. We study the validity, over all structures M with domain D, of M ⊨ Px Qy R(x,y) → Qy Px R(x,y), called scope dominance. The iteration of two quantifiers lies between their branching product, which is generated by rectangles, and the dual of the branching product of their duals. We give criteria for equality with these bounds, obtain a new proof of the countable characterization of scope dominance, and characterize the case in which the inner quantifier is "at least κ many." Under the Generalized Continuum Hypothesis, Goldberg's theorem shows that, for ultrafilters, scope dominance is equivalent to equality between iteration and the branching product, whereas an elementary counterexample on a domain of size ℵ_2 shows that this equivalence fails for arbitrary upward monotone quantifiers. For the case where both quantifiers are filters, the question becomes set-theoretic: the existence of a counterexample is equiconsistent with the existence of a measurable cardinal.
ER  -

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