LIT-145: Ultrafilters as Propositional Theories

Jason Turner (2025), Philosophy Compass 20(7), e70047 — DOI-10.1111/phc3.70047

Key takeaways

• Filters and ultrafilters, and with them Łoś’s theorem and compactness, become accessible to philosophers once read as propositional theories: sets of possible-worlds propositions that are consistent and closed under conjunction and implication, and, for an ultrafilter, negation-complete.

Seeded from the abstract and a skim, before the reading in NOTE-177. What follows is what the work says about itself.

Turner gives a philosophically friendly explanation of filters and ultrafilters from mathematics and model theory. A filter is treated as a propositional theory: a consistent set of propositions closed under finite conjunction and under implication. An ultrafilter is a filter that is also negation-complete. The central ultrafilter theorem is shown to be a propositional analogue of Lindenbaum’s lemma. Products and ultraproducts are introduced and Łoś’s theorem is proved, with compactness as an application.

Standing in the record

Filed from the 2026-09-26 sweep of the papers-feed tracker for philosophy: 105 seconds of active reading over 4 sessions. It was filed Deferred, unread. NOTE-177 is the close reading of 2026-09-26, and it placed the work: Active — worth reading as the most accessible philosopher’s route into ultrafilters and Łoś’s theorem, provided the reader supplies the fixes to a few slips in the proofs (listed under Limitations).

When seeded, before the reading in NOTE-177, the entry gave a priority for a deeper reading: medium — the owner returned to it four times (t=105s), which is the strongest reading signal in the batch. But it is expository (“none of the results are novel”) and the skim captures its structure, so a deeper reading means working the proofs rather than extracting claims.

What the seeding asked a deeper reading to check (NOTE-177 is that reading):

• A teaching piece rather than an argument, so it is philosophy only in the sense of philosophical logic. The closed vocabulary has no logic topic. mathematics is the only honest tag and none of the nine philosophy topics fits, so the vocabulary wants logic if the record keeps it. Otherwise it is a candidate to drop as out of scope.
• It is useful as a technical primer if the record takes up ultraproduct or compactness arguments in metaphysics or philosophy of science.
• Check whether the published version (2025) differs from the 2021 preprint read here.

Access when seeded: The Wiley page and PDF and PhilArchive (TURUAP-2) all returned Cloudflare challenge pages. I read the author’s own preprint “Ultrafilters-Web.pdf” (22 pp., dated November 20, 2021) from jasonturner.arizona.edu by text extraction. The published version may differ. The online date 2025-07-26 is from Crossref, and volume/issue/article number 20(7) e70047 are also from Crossref. Keywords are my own; the preprint lists none.