Quantitative Truthlikeness and Truth Approximation
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In Subsection 10.2.2. we have elaborated our scepsis about the idea of quantitative truth approximation, mainly because it presupposes real-valued distances between structures. Since such distances are not presupposed by quantitative confirmation we are even more skeptical about quantitative approaches to truth approximation than to confirmation. However, we will nevertheless present some main lines of such approaches. We will first deal, in Section 12.1., with quantitative actual truthlikeness and truth approximation, which is relatively unproblematic when there is a plausible distance function on Mp. For the basic nomic case, to be dealt with in Subsection 12.2.1. and part of Subsection 12.3.1., the situation is also relatively clear, but the refined nomic case faces us with essentially two types of problems. First, for the idea of quantitative truthlikeness, discussed in Subsection 12.2.2., we have to find a plausible distance function between two theories, and hence between two subsets of Mp, based on a distance function on Mp. Even guided by some plausible principles, our best refined proposal is not completely satisfactory. Second, we have to choose between two essentially different ways of measuring the quantitative success of theories: a non-probabilistic and a probabilistic way. In the first case, presented in Subsection 12.3.1., the success, or better the failure, of a theory in meeting the available evidence is quantitatively expressed in a similar way as the chosen distance function between two theories. The second case, presented in Subsection 12.3.2., arises from an adaptation of Niiniluoto’s plausible idea of estimating the distance of a theory from the complete, hence actual, truth on the basis of the available evidence. It will be shown that there is, in the case that probabilities as well as distances are plausible, a very sophisticated way of doing so, where ideas of Carnap, Hintikka and Festa are combined, leading to ‘double’ truth approximation. However, it will also be argued that this primarily concerns improper cases of nomic truth approximation, that is, it amounts to refined actual truth approximation in two respects.