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 $\mathcal{F}\mathcal{P}\mathcal{M}$.

Investigating the additive probability of repeated language production decisions

This paper introduces an experimental paradigm based on probabilistic evidence of the interaction between construction decisions in a parsed corpus. The approach is demonstrated using ICE-GB, a one million-word corpus of English. It finds an interaction between attributive adjective phrases in noun phrases with a noun head, such that the probability of adding adjective phrases falls successively. The same pattern is much weaker in adverbs preceding a verb phrase, implying this decline is not a universal phenomenon. Noun phrase postmodifying clauses exhibit a similar initial fall in the probability of successive clauses modifying the same NP head, and embedding clauses modifying new NP heads. Successive postmodification shows a secondary phenomenon of an increase in additive probability in longer sequences, apparently due to ‘templating’ effects. The author argues that these results can only be explained as cognitive and communicative natural phenomena acting on and within recursive grammar rules.

A foundation for probabilistic beliefs with or without atoms

We propose two novel axioms for qualitative probability spaces: (i) unlikely atoms, which requires that there is an event containing no atoms that is at least as likely as its complement; and (ii) third‐order atom‐swarming, which requires that for each atom, there is a countable pairwise‐disjoint collection of less‐likely events that can be partitioned into three groups, each with union at least as likely as the given atom. We prove that under monotone continuity, each of these axioms is sufficient to guarantee a unique countably‐additive probability measure representation, generalizing work by Villegas to allow atoms. Unlike previous contributions that allow atoms, we impose no cancellation or solvability axiom.

On non-additive probability measures

The additivity of considered measures or integrals resp. can be omitted in some problems of mathematical analysis and its applications. In the paper it is shown that similar situations are possible also in the probability theory. As an example is proved a generalized version of the central limit theorem about the convergence of arithmetical means of independent random variables to the Gauss distribution.

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.

NONCONGLOMERABILITY FOR COUNTABLY ADDITIVE MEASURES THAT ARE NOT κ-ADDITIVE

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.