Zwart and Franssen’s impossibility theorem holds for possible-world-accounts but not for consequence-accounts to verisimilitude

Abstract

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).

Matrix- based logic for avoiding paradoxes and its paraconsistent alternative

“…But RC could also be formulated by using just 'wff' instead of 'atomic wff ' (or propositional variable). This has been proved in Schurz, Weingartner (1987).…”

“…The criteria originate in a paper devoted to give a solution of the problem of verisimilitude in the sense of rehabilitating Popper's original idea by restricting the logical consequence class to relevant consequentelements only (cf. Schurz, Weingartner (1987). Further developments are in Schurz (1991) and Weingartner (2000) and Schurz, Weingartner (2009)).…”

“…Schurz, Weingartner (1987). Further developments are in Schurz (1991) and Weingartner (2000) and Schurz, Weingartner (2009)). …”

“…. For difficulties in the domain of Quantum Physics see Mittelstaedt (1978Mittelstaedt ( , 2004, Weingartner (2004Weingartner ( , 2009Weingartner ( , 2010. The criteria originate in a paper devoted to give a solution of the problem of verisimilitude in the sense of rehabilitating Popper's original idea by restricting the logical consequence class to relevant consequentelements only (cf.…”

Introduction and Overview

“…The set of relevant elements of a theory T is written as T r and consists of all minimal but still relevant 'conjunctive parts' of T. In propositional languages, these relevant elements are given as clauses, i.e., disjunctions of b-claims (or literals) satisfying the additional 'relevance' condition that no proper subdisjunction of them is entailed by T. It can be proved that the set T r preserves the logical content of T. This is the difference with the view of conjunctive parts as b-claims, which preserve a theory's content only if it is a c-theory in the sense of Cevolani et al Based on earlier work, Schurz goes on to introduce his comparative definition of verisimilitude, according to which T 1 C V T 2 iff (T 1 ) tr k-(T 2 ) tr and (T 2 ) fr k-(T 1 ) fr , where (T) tr and (T) fr denote the set of T's true and T's false relevant elements, respectively (''k-'' for ''logical consequence''). He extends this comparative notion to a quantitative concept of verisimilitude defined over relevant elements that has been introduced in Schurz and Weingartner (2010).…”