On the logic of high probability (Q1083444)

Let 'H(A,B)' symbolize the value proposition that P(A,B) is high, where A and B are elements of a boolean algebra and P(A,B) is the conditional probability of A given B, P being a probability function for the algebra. Letting I(A,B) be the improbability function 1-P(A,B), it is proved that a necessary and sufficient condition for the inequality \[ I(A_ 1,B_ 1)+...+I(A_ n,B_ n)\leq I(C_ 1,D_ 1)\times...\times I(C_ m,D_ m) \] to hold for all improbability functions is that \(H(C_ 1,D_ 1)\vee...\vee H(C_ m,D_ m)\) should be derivable from \(H(A_ 1,B_ 1)\&...\&H(A_ n,B_ n)\) according to simple rules of inference, and it is also shown that if it is not derivable by these rules then for all positive \(\epsilon\) there exists an improbability function I such that \(I(A_ i,B_ i)\leq \epsilon\) for \(i=1,...,n\), while \(I(C_ j,D_ j)\geq 1/m\), for \(j=1,...,m\). A simple decision procedure is described for determining derivability according to the rules, and a number of metatheoretic consequences are derived from its properties. Connections with one of D. Lewis's systems of counterfactual logic are noted.