<i>The Laws of Belief: Ranking Theory &amp; its Philosophical Applications</i>, by Wolfgang Spohn.

Probabilists model partial belief by assigning it a degree between 0 and 1; the higher the degree, the more credible the proposition. In contrast, ranking theorists assign numbers or ranks to propositions so that the higher the number, the more the proposition is disbelieved. In The Laws of Belief, Wolfgang Spohn develops ranking theory in detail and applies it to a host of philosophical problems. The book has an extraordinary sweep, taking in many areas of metaphysics, epistemology, and philosophy of science and even the philosophy of language. Long in the writing, it synthesises thirty years’ work on ranking theory. A ranking function κ is a function from possible worlds to the natural numbers 0, 1, 2, 3, … and ∞ (infinity). Let W be a set of possible worlds and A an algebra of propositions over W, a proposition being a subset of W. The ranking function κ is extended from worlds to propositions by setting κ(A)= min{κ(w): w is in A}, from which it follows that κ(A1 ∨ A1 ) = min{κ(A1), κ(A2)}. This gives a grading of disbelief: κ(A) = 0 means that the proposition A is not disbelieved at all (A may be believed or neither believed nor disbelieved), κ(A) = 1 means that A is disbelieved to the least degree, κ(A) = 2 that A is disbelieved to the second least degree, …, and κ(A) = ∞ that A is utterly disbelieved. Contradictions have rank (i.e. κ-value) ∞ and are therefore utterly disbelieved, whereas tautologies have rank 0. The subject believes A if and only if κ(¬A) > 0; equivalently, if and only if she does not disbelieve A and also disbelieves ¬A to some extent.