Source
Wenmackers (2011), LIT-163, chapters 3 and 4, read in full in NOTE-139.
What was actually shown
The problem. The threshold Lockean thesis (believe x iff P(x) ≥ θ < 1) violates the conjunction principle even for two beliefs (eq. 3.3, proved). In a large fair lottery one believes of each ticket that it loses, and not that all lose.
The diagnosis (§4.2). The infinite lottery’s failure of countable additivity and the Lottery Paradox’s failure of conjunction are cast as one “adding problem”: each term carries a negligible rounding error, and many times small is not small.
The model. In Hrbacek’s relative analysis, with levels as a context parameter and ≃_V meaning ultraclose at level V:
- Stratified Belief (Def. 1, eq. 3.4). B(x) iff P(x) ≃_V 1.
- Stratified Conjunction Principle (Def. 2, eq. 3.5). If both conjuncts are ultraclose to 1 at V, so is their conjunction, by a short proof, extended in §3.6.3.2 to any standard number of conjuncts.
- Its limit. For an ultralarge number N of conjuncts it fails (1 − N/N = 0).
- A threshold variant, SB_θ (Def. 3). P(x) ≳_V θ keeps conjunction only for an arbitrary event plus a singleton. It is offered as a model of epistemicist vagueness.
Rational belief about a fair infinite lottery then takes two roundings: first to the standard part, then relative-infinitesimal rounding (§4.4).
What could have come out otherwise. The conjunction rule could have failed inside the framework even for two conjuncts. It does not. That is a result about the model, not about believers.
What this does not say
- That belief has levels. Relative analysis’s levels have no explicit boundary by design. Any explicit threshold collapses the model to the standard one (§3.5.5, §3.7.2.1), so it is not operational. The author calls it “a crude model” (§3.6.4).
- That it is psychologically real. The support is an analogy with Dehaene’s mental number line (C10, weak).
- That the Lottery Paradox is a vagueness problem. That is argued informally (C9, weak).
- Anything about knowledge. The knowledge version is explicitly set aside (fn. 1, fn. 22). The bank case and voting-paradox applications are sketches (§3.7).
- That belief is graded. Belief is binary here. One context level is allowed per agent at a time.
Note, 2026-10-21
The conditional result in this title, that conjunction holds for any standard number of believed propositions and can fail for an ultralarge number, is now THEORY-829, Active, with the re-derivation written out. The title’s “fails for an ultralarge number” is stronger than what is proved: conjunction can fail for an ultralarge number (the fair lottery), not for every ultralarge collection. What stays open here is the thesis: that belief has context levels and that the Lottery Paradox is an accumulation of negligible rounding errors.
Connections
-
It extends THEORY-829, the conditional result inside relative analysis.
-
It applies the rounding diagnosis of the thesis’s lottery construction (THEORY-401) to belief.
-
Pettigrew’s accuracy-first credence (LIT-171) is a neighbour on the relation of credence to outright belief. It is not engaged, and the thesis predates it.