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There is a plurality of formal constraints for aggregating probabilities of a group of individuals. Different constraints characterise different families of aggregation rules. In this paper, we focus on the families of linear and geometric opinion pooling rules which consist in linear, respectively, geometric weighted averaging of the individuals’ probabilities. For these families, it is debated which weights exactly are to be chosen. By applying the results of the theory of meta-induction, we want to provide a general rationale, namely, optimality, for choosing the weights in a success-based way by scoring rules. A major argument put forward against weighting by scoring is that these weights heavily depend on the chosen scoring rule. However, as we will show, the main condition for the optimality of meta-inductive weights is so general that it holds under most standard scoring rules, more precisely under all scoring rules that are based on a convex loss function. Therefore, whereas the exact choice of a scoring rule for weighted probability aggregation might depend on the respective purpose of such an aggregation, the epistemic rationale behind such a choice is generally valid.

In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, intuitionistic, paraconsistent and quantum logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages.

In this paper Reichenbach's best alternative account (BAA) to induction is examined. In the first section, three versions of the BAA are distinguished that have been discussed in the literature. The major objections against all three versions are presented. In the second section it is shown by a text analysis that Reichenbach (The theory of probability, University of California Press, California, 1949) argues for all three versions of the BAA and does not sufficiently distinguish between them. In the third section it is explained how Reichenbach's third version of the BAA can be transformed into a provable optimality theorem within the account of meta-induction.

Tacking by conjunction is a well-known problem for Bayesian confirmation theory. In the first section, disadvantages of existing Bayesian solution proposals to this problem are pointed out and an alternative solution proposal is presented: that of genuine confirmation (GC). In the second section, the notion of GC is briefly recapitulated and three versions of GC are distinguished: full (qualitative) GC, partial (qualitative) GC and quantitative GC. In the third section, the application of partial GC to pure post-facto speculations is explained. In the fourth section it is demonstrated that full GC is a necessary condition for Bayesian convergence to certainty based on the accumulation of conditionally independent pieces of evidence. It is found that whenever a hypothesis is equivalent to a disjunction of more fine-grained hypotheses conveying different probabilities to the evidence, then conditional independence of the evidence fails. This failure occurs typically for unspecific negations of hypotheses. A refined version of the convergence to certainty theorem that overcomes this difficulty is developed in the final section.

The universal optimality theorem for metainduction works for epistemic agents faced with a choice among finitely many prediction methods. Eckhart Arnold and Tom Sterkenburg objected that it breaks down for infinite or unboundedly growing sets of methods. In this article the metainductive approach is defended against this challenge by extending the optimality theorem (i) to unboundedly growing sets of methods whose number grows less than exponentially in time, (ii) to sequences of methods with an application to Goodman's problem, and (iii) to infinite sets of methods whose number of predictive equivalence classes grows less than linearly in time.

The paper starts with the distinction between conjunction-of-parts accounts and disjunction-of-possibilities accounts to truthlikeness (Sects. 1, 2). In Sect. 3, three distinctions between kinds of truthlikeness measures (t-measures) are introduced: (i) comparative versus numeric t-measures, (ii) t-measures for qualitative versus quantitative theories, and (iii) t-measures for deterministic versus probabilistic truth. These three kinds of truthlikeness are explicated and developed within a version of conjunctive part accounts based on content elements (Sects. 4, 5). The focus lies on measures of probabilistic truthlikeness, that are divided into t-measures for statistical probabilities and single case probabilities (Sect. 4). The logical notion of probabilistic truthlikeness (evaluated relative to true probabilistic laws) can be treated as a subcase of deterministic truthlikeness for quantitative theories (Sects. 4–6). In contrast, the epistemic notion of probabilistic truthlikeness (evaluated relative to given empirical evidence) creates genuinely new problems, especially for hypotheses about single case probabilities that are evaluated not by comparison to observed frequencies (as statistical probabilities), but by comparison to the truth values of single event statements (Sect. 6). By the method of meta-induction, competing theories about single case probabilities can be aggregated into a combined theory with optimal predictive success and epistemic truthlikeness (Sect. 7).