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Generalized Quantifiers: Scope Dominance and Branching — arXiv:2609.31243

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.31243
License (as archived): CC0-1.0
Archived: 2026-10-08T03:11:47.334Z
SHA-256 of the archived text: 9a60aabbeca2693f3cfb33ddc30d214bdcf9e6b93257f88a8120651214730ad9
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How to cite

Generalized Quantifiers: Scope Dominance and Branching — arXiv:2609.31243 — arXiv (en), https://arxiv.org/abs/2609.31243 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bc32e2d32297 (Metraipolis archive dc_bc32e2d32297, sha256 9a60aabbeca2693f3cfb33ddc30d214bdcf9e6b93257f88a8120651214730ad9)

BibTeX

@misc{dcbc32e2d32297,
  title = {Generalized Quantifiers: Scope Dominance and Branching — arXiv:2609.31243},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.31243},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bc32e2d32297; sha256 9a60aabbeca2693f3cfb33ddc30d214bdcf9e6b93257f88a8120651214730ad9},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.31243. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 \[ M \models Px\,Qy\,R(x,y) \rightarrow Qy\,Px\,R(x,y), \] 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 $\kappa$ 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 $\aleph_2$ 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-28 in math.LO).
Categorie arXiv: Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2609.31243 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T02:47:39.403Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-28?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).