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AbstractSo far, the most prominent measure for actual truthlikeness, i.e. the likeness of a theory to the actual truth, is Ilkka Niiniluoto's minsum definition, which is purely based on distances. A competing definition is the average distance measure proposed by Pavel Tichy and Graham Oddie. We will define three related, distance and size based, measures for actual truthlikeness and compare them with the two well-known options. However, we will start, Sect. 2, from a trio of such measures for nomic truthlikeness. The nomic truth, or the true theory, here refers to what is nomically, e.g. physically, possible. In a nomic (and factual) context there are two basic kinds of theories, viz. either based on an exclusion claim or on an inclusion claim. Two-sided theories combine these claims, with the maximal claim as extreme special case. We will base truthlikeness measures for exclusion, inclusion, two-sided, and hence maximal, nomic theories on two similarity measures, one in terms of distances between conceptual possibilities and the other in terms of sizes of sets of such possibilities. In Sect. 3 we will treat actual truthlikeness as extreme special case of nomic truthlikeness, viz. assuming that there is just one nomic possibility, the actual one. Next we will compare the resulting measures mutually and with the above mentioned measures of Niiniluoto and Tichy & Oddie. Finally, in Sect. 4, we will sum up the results and explore five questions for further research.

AbstractTheories of truth approximation in terms of truthlikeness (or verisimilitude) almost always deal with (non-probabilistically) approaching deterministic truths, either actual or nomic. This paper deals first with approaching a probabilistic nomic truth, viz. a true probability distribution. It assumes a multinomial probabilistic context, hence with a lawlike true, but usually unknown, probability distribution. We will first show that this true multinomial distribution can be approached by Carnapian inductive probabilities. Next we will deal with the corresponding deterministic nomic truth, that is, the set of conceptually possible outcomes with a positive true probability. We will introduce Hintikkian inductive probabilities, based on a prior distribution over the relevant deterministic nomic theories and on conditional Carnapian inductive probabilities, and first show that they enable again probabilistic approximation of the true distribution. Finally, we will show, in terms of a kind of success theorem, based on Niiniluoto's estimated distance from the truth, in what sense Hintikkian inductive probabilities enable the probabilistic approximation of the relevant deterministic nomic truth. In sum, the (realist) truth approximation perspective on Carnapian and Hintikkian inductive probabilities leads to the unification of the inductive probability field and the field of truth approximation.