When Rational Disagreement is Impossible

It is natural to assume that a group sharing the same information and respecting each other's opinions may reasonably disagree. I shall prove, on the contrary, that if the members of such a group search for truth and accept the mathematical implications of their state, then they must converge toward consensus. Disagreement is demonstrably irrational.' The proof I shall offer will demonstrate that a group of people who give some weight to the probability assignments of others will converge toward a consensual probability assignment. The probability assignment, p?, of an individual i at stage O may, for my purposes, be thought of either as determining reasonable acceptance2 or as constituting degrees of belief.3 The principal thesis of my paper is best sustained, however, by interpreting the probability assignment of an individual as his estimate of the chances a proposition has of being true. My first assumption is that each member of the group has at least some small degree of positive respect for the probability assignments of other members of the group. This degree of respect is formulated as a weight which he gives to the probability assignment of others in the group. Thus, wij is the weight that person i gives to the probability assignment of person j. A person assigns a positive weight to each probability assignment, including his own, and the set of weights, wil, Wi2, and so forth to win, which a person i assigns to the n probability functions, sum to 1. I call this the respect assumption. It amounts to each person giving some positive weight to the opinion of others. Actually, a much weaker