The theorem has two critical conditions: individual reliability greater than chance, and independence of errors among participants. The first condition is what makes universal education a mathematical necessity rather than merely a social good — the reliability of democratic decisions depends on the reliability of the individual judgments composing them.
The independence condition is what makes diversity a mathematical requirement rather than merely a political value. A 2024 study applying the theorem to ensembles of large language models found that majority voting across multiple LLMs produced only marginal improvements — the models, despite apparent diversity, had been trained on overlapping data and shared architectures, so their errors correlated. When errors correlate, the theorem's guarantee collapses. The fishbowl becomes the governing failure mode.
Researchers have explicitly deployed the theorem in neural network ensembles for medical diagnosis — combining outputs of multiple deep learning models trained on radiograph images, using majority voting to achieve diagnostic accuracy exceeding any individual model. The theorem provides the mathematical guarantee, and the guarantee depends on precisely the conditions Condorcet specified.
The theorem's application to AI governance is direct. Decisions about AI development are currently made by a narrow group whose information, assumptions, and professional networks substantially overlap. Under the theorem's conditions, their collective reliability is constrained by the correlation of their errors. Broadening participation — genuinely diverse participation, not demographic tokenism — would, under the theorem, produce more reliable collective decisions, provided the new participants are adequately informed.
The theorem appeared in the Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (1785), a 500-page treatise applying probability calculus to collective judgment — a founding document of social choice theory, decision theory, and the statistical analysis of testimony.
The result was rediscovered and formalized by Duncan Black in the 1950s, generalized in machine learning by Robert Schapire's 1990 paper 'The Strength of Weak Learnability,' and is now standard material in computer science curricula — usually without mention of the eighteenth-century mathematician whose proof it is.
Individual reliability matters. Below chance, larger groups converge on error; above chance, they converge on truth.
Independence is essential. Correlated errors nullify the theorem's guarantee, regardless of group size.
Diversity is mathematical, not decorative. A homogeneous group produces a chorus, not a jury.
The theorem runs inside AI. Random forests, boosting, and ensemble voting are literal implementations of Condorcet's proof.