Belief and Desire

The sum and product rules are consequences of associating a single number with a conditioned proposition: the Boolean algebra of the propositions induces an algebra for the numbers, which with a few additional assumptions gives the sum and product rules. We call the associated number probability, and can interpret it as representing strength of belief that the proposition be true. The extra assumptions are empirical and cannot be derived from logical argument alone; consequently, probabilistic logic has a semi-empirical basis, and is not a pure consequence of consistency requirements. In addition to the capacity for belief we also have the capacity for desire: but numerical measure of the strength of desire that a conditioned proposition be true does not obey the same laws, because the additional assumptions which led for probability to the sum and product rules do not hold for desirability. The equations, whose solution gives a calculus of desirabilities, are derived. The notion of desirability clarifies the relative status of probabilistic inference and decision theory, in which probability and desirability are combined when the desirability that a proposition be true is conditioned on an action of our choosing. Desirability is then known — among other names — as loss function; however. it remains a valid concept when there is no choice of action. It is a valuable clarifying notion.