Approaching probabilistic laws

Abstract

In the general problem of verisimilitude, we try to define the distance of a statement from a target, which is an informative truth about some domain of investigation. For example, the target can be a state description, a structure description, or a constituent of a first-order language (Sect. 1). In the problem of legisimilitude, the target is a deterministic or universal law, which can be expressed by a nomic constituent or a quantitative function involving the operators of physical necessity and possibility (Sect. 2). The special case of legisimilitude, where the target is a probabilistic law (Sect. 3), has been discussed by Roger Rosenkrantz (Synthese, 1980) and Ilkka Niiniluoto (Truthlikeness, 1987, Ch. 11.5). Their basic proposal is to measure the distance between two probabilistic laws by the Kullback–Leibler notion of divergence, which is a semimetric on the space of probability measures. This idea can be applied to probabilistic laws of coexistence and laws of succession, and the examples may involve discrete or continuous state spaces (Sect. 3). In this paper, these earlier studies are elaborated in four directions (Sect. 4). First, even though deterministic laws are limiting cases of probabilistic laws, the target-sensitivity of truthlikeness measures implies that the legisimilitude of probabilistic laws is not easily reducible to the deterministic case. Secondly, the Jensen-Shannon divergence is applied to mixed probabilistic laws which entail some universal laws. Thirdly, a new class of distance measures between probability distributions is proposed, so that their horizontal differences are taken into account in addition to vertical ones (Sect. 5). Fourthly, a solution is given for the epistemic problem of estimating degrees of probabilistic legisimilitude on the basis of empirical evidence (Sect. 6).