Source
Wenmackers (2011), LIT-163, chapter 5 with its appendices A and B, read in full in NOTE-139. NOTE-139 corrects the thesis’s summary of this chapter, and this document follows the correction.
What was actually shown
The model. M atomic sentences give 2^M worlds; a theory is a set of worlds, coded as a bit string, and the empty set is the inconsistent theory. Each of N agents independently draws a uniformly random consistent theory (an impartial culture). In one simultaneous update, each agent averages each bit over the peers within Hamming distance D and rounds; a tie keeps the old bit. A “zero-update” is an update to the inconsistent theory. There is no truth, evidence, trust or network structure, and only one step is analysed.
The counts (eq. 5.1, Table 5.3). F_AG is the fraction of agents that zero-update; F_OP the fraction of opinion profiles in which at least one does. Both were computed exactly for N ≤ 21 (M = 2) and N ≤ 4 (M = 3), and by sampling 10^6 profiles per point up to N = 200 (M = 2) and N = 2500 (M = 3).
- No zero-update occurs for M = 1, D = 1 or N = 2. The minimal case, M = 2, N = 3, D = 2, gives F_OP = F_AG = 24/3375 = 0.711%, checked by hand.
- F_AG is at most 1.8112% for M = 2 (at D = 4, N = 5) and at most 0.3215% for M = 3.
- F_OP reaches 6.4298% for M = 2 and 16.867% for M = 3 (D = 6, N = 1780).
- Odd and even N alternate in an unexplained wobble.
What could have come out otherwise. Averaging could have produced inconsistency often, or never. It produces it rarely per agent, and the group-level risk rises from M = 2 to M = 3.
What this does not say
- That the risk falls with more sentences. The abstract’s “arbitrarily small” by raising M is extrapolated from two values, and Appendix B says the data are “insufficient to predict the shape of D-curves for higher values of M”. For F_OP the trend from M = 2 to M = 3 runs the other way (NOTE-139, Corrections).
- That the bound is proved. It is computed for M = 2 and 3 only, after a single update, in an impartial culture.
- That real groups should avoid even sizes. The chapter’s recommendation (§5.5) conflicts with its own finding that odd N gives the higher per-agent fraction; the even-favouring wobble is in F_OP. The Delphi recommendations are not validated.
- Anything about truth-tracking. Agents average attitudes, not evidence, and no theory is true.
Connections
- It is a theory-valued counterpart to the continuous-opinion models in the record, such as LIT-070, and sits beside the discursive dilemma (List 2005, which the chapter cites), where proposition-wise aggregation of consistent judgements also yields inconsistency. The record’s account of that impossibility is THEORY-127.
- The network-epistemology results of THEORY-422 concern pooling estimates and learning by trying; none of them asks whether the pooled state is logically consistent.