Remarks on continuously distributed sequences

In the first part of the paper we define the notion of the density as certain type of finitely additive probability measure and the distribution function of sequences with respect to the density. Then we derive some simple criterions providing the continuity of the distribution function of given sequence. These criterions we apply to the van der Corput's sequences. The Weyl's type criterions of continuity of the distribution function are proven.

Probability logic: A model-theoretic perspective

This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

Epistemic adjectives: Likely and probable

This chapter investigates the (near-)synonymous relative adjectives likely and probable, starting with the hypothesis that they live on an upper- and lower-bounded ratio scale. If it is correct, then the scale in question is provably equivalent to a representation in terms of finitely additive probability. This would explain the puzzle around disjunction noted in chapter 3, and it is supported by the acceptability of ratio modifiers such as three times as likely and item-by-item consideration of ratio scale axioms (with a caveat involving connectedness). The second part of the chapter turns to a theoretical puzzle: in Kennedy’s (2007) theory, likely and probable could not be relative adjectives if their scale is bounded. However, this theory is falsified on independent grounds: among other empirical problems, relative adjectives routinely occur on bounded scales. Likely and probable provide two more counter-examples to the claim that relative adjectives are restricted to open scales.

Finitely Additive Probability and Benford's Law

This chapter demonstrates the robustness of Benford's law even with regard to basic underlying mathematical hypotheses. It records, without proof, several of the main theorems pertinent to a theory of Benford's law in the context of finitely additive probability, and points the interested reader to further references on that theory in the literature.

Translation invariance and finite additivity in a probability measure on the natural numbers

Inspired by the “two envelopes exchange paradox,” a finitely additive probability measuremon the natural numbers is introduced. The measure is uniform in the sense thatm({i})=m({j})for alli,j∈ℕ. The measure is shown to be translation invariant and has such desirable properties asm({i∈ℕ|i≡0(mod2)})=1/2. For anyr∈[0,1], a setAis constructed such thatm(A)=r; however,mis not defined on the power set ofℕ. Finally, a resolution to the two envelopes exchange paradox is presented in terms ofm.