THEORY-829: In Hrbacek's relative analysis, if belief at a context level is credence ultraclose to 1 at that level, the conjunction of any standard number of believed propositions is believed, while for an ultralarge number it can fail: in a fair lottery with an ultralarge number of tickets each 'this ticket loses' is believed and their conjunction has probability 0

Source

Wenmackers (2011), LIT-163, chapter 3, read in full in NOTE-139. Stratified Belief is Def. 1 (eq. 3.4), the Stratified Conjunction Principle is Def. 2 (eq. 3.5), and its extension to any standard number of conjuncts is §3.6.3.2. The level axioms are Hrbacek’s (§3.5.3.1). NOTE-139 rates the result strong within relative analysis (C8).

What was actually shown

Work at a context level V. A real x is V-ultrasmall if x ≠ 0 and |x| < r for every V-standard r > 0, and x ≃_V y means x − y is V-ultrasmall or 0. Stratified Belief: E is believed iff P(E) ≃_V 1.

Re-derivation, step by step. Let E₁, …, Eₙ be believed, with n V-standard, and put eᵢ = 1 − P(Eᵢ). Each eᵢ is 0 or V-ultrasmall, and eᵢ ≥ 0.

1. Finite subadditivity, an ordinary theorem for every finite n: 1 − P(E₁ ∧ … ∧ Eₙ) = P(¬E₁ ∨ … ∨ ¬Eₙ) ≤ e₁ + … + eₙ.
2. Fix any V-standard r > 0. Since r and n are V-standard, r/n is V-standard, by the Closure principle: anything defined from V-standard parameters is V-standard.
3. Each eᵢ is below every V-standard positive number, so eᵢ < r/n for each i, and e₁ + … + eₙ < n·(r/n) = r.
4. Since r was arbitrary, the sum is 0 or V-ultrasmall. As 1 ≥ P(E₁ ∧ … ∧ Eₙ) ≥ 1 − (e₁ + … + eₙ), the conjunction is ultraclose to 1, so it is believed.
5. For an ultralarge number, take a fair lottery with N tickets, N V-ultralarge, so 1/N is V-ultrasmall. “Ticket i loses” has probability 1 − 1/N ≃_V 1 and is believed for every i. The conjunction of all N has probability 0 (1 − N/N = 0), so it is not believed. Step 2 is what fails: r/N is not V-standard.

The argument uses Closure, not induction on n. That matters, because induction fails across levels: a level is a predicate, not a set (NOTE-139, glossary).

What could have come out otherwise. Conjunction could have failed inside the framework even for two conjuncts, as it does for any fixed threshold below 1 (eq. 3.3). It does not.

What this does not say

• That belief works this way. That is Wenmackers’s thesis, which THEORY-647 carries and which stays Proposed. Relative analysis’s levels have no explicit boundary by design, and any explicit threshold collapses the model to the Lockean one (§3.5.5).
• That conjunction fails for every ultralarge collection. It can fail; N copies of a single believed proposition never do.
• That the Lottery Paradox is dissolved. Inside the model the paradox’s conjunction is not believed. Whether that is the right diagnosis is the argument THEORY-647 makes.
• Anything for the threshold variant SB_θ. Its conjunction rule holds only for an arbitrary event plus a singleton (Def. 3); it is not covered here.

Connections

• THEORY-647 extends this result into a thesis about outright belief.
• THEORY-401 holds the thesis’s fair-lottery construction, whose rounding diagnosis this result applies to belief.