Zwart and Franssen's impossibility theorem reveals a conflict between the possible-world-based content-definition and the possible-world-based likeness-definition of verisimilitude. In Sect. 2 we show that the possible-world-based content-definition violates four basic intuitions of Popper's consequence-based content-account to verisimilitude, and therefore cannot be said to be in the spirit of Popper's account, although this is the opinion of some prominent authors. In Sect. 3 we argue that in consequence-accounts, content-aspects and likeness-aspects of verisimilitude are not in conflict with each other, but in agreement. We explain this fact by pointing towards the deep difference between possible-world-and the consequence-accounts, which does not lie in the difference between syntactic (object-language) versus semantic (meta-language) formulations, but in the difference between 'disjunction-of-possibleworlds' versus 'conjunction-of-parts' representations of theories. Drawing on earlier work, we explain in Sect. 4 how the shortcomings of Popper's original definition can be repaired by what we call the relevant element approach. We propose a quantitative likeness-definition of verisimilitude based on relevant elements which provably agrees with the qualitative relevant content-definition of verisimilitude on all pairs of comparable theories. We conclude the paper with a plea for consequence-accounts and a brief analysis of the problem of language-dependence (Sect. 6).

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The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense “irrelevant” from those which are “relevant” or “nonredundant” in the sense of allowing only the most informative consequences in the derivations. The latter derivations are strictly valid inRMQ, whereas the former are only materially valid.RMQis a decidable matrix calculus which possesses a semantics and has the finite model property. It is shown in the paper howRMQby its strictly valid derivations can avoid the difficulties with commensurability, distributivity, and Bell's inequalities when it is applied to quantum physics.

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