Nomic truthlikeness in the light of a probabilistic representation of propositions

Abstract Assuming a propositional language, there appears to be a very natural way to represent propositions as a special kind of probability distributions over the propositional constituents. This enables to define a plausible (normalized metric) distance function between propositions, and hence a similarity function between them. A particularly interesting application of the latter is using it to define the degree of nomic truthlikeness of a proposition as its degree of similarity with the nomic truth, the proposition characterizing the set of nomic possibilities. The ‘probabilistic distance’ between two propositions is a typical ‘vertical’ function by its being based on the differences between the probabilities assigned to each constituent, in contrast to the usual ‘horizontal’ definitions, based on a distance function between the constituents. This leads, for example, to a so-called ‘content’ definition of truthlikeness, as opposed to the usual ‘likeness’, or ‘similarity’, definitions. The ‘probabilistic distance’ appears to be strongly related to the so-called fractional distance between quantities. Moreover, it has one obvious competitor, the well-known symmetric difference distance between propositions, which can also be seen as a kind of vertical measure. In comparison, there are good reasons to prefer the probabilistic one, not in the least because of its ‘micro-foundation’ in the sense that it is a plausible application of a very general approach to the (normalized) distance between two probability distributions over the constituents. The two nomic truthlikeness measures, and a third one, related to the probabilistic one, are illustrated by a simple electric circuit, about which the nomic truth is easy to determine. The three distance functions are applied to three theories about the circuit. Finally, some issues for further elaboration are indicated.