Abstract The conditional probabilistic representation of propositions based on the uniform, or structural, distribution over the constituents of a propositional language was introduced in Kuipers (2025, J. Logic Comput., 35, exaf016). A main finding was that the corresponding ‘city-block’ similarity between two propositions based on the structural distribution is itself a conditional probability. The present paper shows, in the first place, that this can be generalized to any underlying probability distribution. The meaningfulness of the latter is illustrated by a botanic example. It is also argued that conditional probabilities between two propositions, based on the uniform or any non-uniform distribution, can well be seen as explication of the idea of a degree of entailment of one proposition by the other, or the degree of validity of the argument that the second proposition entails the first one. The general version is again illustrated by the botanic example. The notion of similarity between two propositions is of particular interest in the context of truthlikeness, the idea that one theory can be closer, or more similar, to the truth than another. The paper presents some consequences of the above findings and proposals for truthlikeness and illustrates them with the electric circuit example presented in Kuipers (2025, J. Logic Comput., 35, exaf016).

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.