Group in Logic and the Methodology of Science

The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation $\succsim$ on events that does not preclude a probabilistic interpretation, in the sense that $\succsim$ has extensions that are probabilistically representable, we characterize the extension $\succsim^+$ of $\succsim$ that is exactly the intersection of all probabilistically representable extensions of $\succsim$. This extension $\succsim^+$ gives us all the additional comparisons that we are entitled to infer from $\succsim$, based on the assumption that there is some probability measure of which $\succsim$ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation, this problem has a more general interpretation involving measurement of any additive quantity: e.g., given comparisons between the weights of individual objects, what comparisons between the weights of groups of objects can we infer?