Strict coherence, sigma coherence and the metaphysics of quantity

Abstract

In 1955 Abner Shimony introduced a strong notion of coherence which has become known as strict coherence. It is a variation on a theme by Ramsey and de Finetti: that of testing the coherence of your system of beliefs by seeing how much damage can be done by a cunning bettor with a finite number of bets, each of which you take to be fair. The bettor makes a Dutch Book against you if his bets taken together guarantee that you must suffer a loss; that is to say that no matter what the outcomes of the events upon which bets are placed you suffer a net loss. Let us say that a bettor makes a Weak Dutch Book against you if you can suffer a loss but cannot enjoy a gain: that is to say that there are possible outcomes of the events on which bets are made such that you suffer a net loss and no such outcomes which yield a net gain. If no Dutch book can be made against you, your beliefs are coherent; if- in addition no weak Dutch book can be made against you, your beliefs are strictly coherent. Betting quotients are coherent if and only if they are finitely additive probabilities. They satisfy the stronger requirement of strict coherence if and only if they are finitely additive probabilities which assign a positive probability to every possible event (Kemeny 1955). Coherence is a pragmatic consistency condition. Strict coherence adds to this a requirement of open-mindedness. The strictly coherent agent does not assign probability zero to anything unless it is must be judged impossible by virtue of the underlying Boolean logic. Strict coherence is epistemologically attractive and was easily implementable in the context of finite Carnapian inductive logic in which it was introduced. But