Conditional Random Quantities and Compounds of Conditionals

In this paper we consider finite conditional random quantities and conditional previsions assessments in the setting of coherence. We use a suitable representation for conditional random quantities; in particular the indicator of a conditional event $E|H$ is looked at as a three-valued quantity with values 1, or 0, or $p$, where $p$ is the probability of $E|H$. We introduce a notion of iterated conditional random quantity of the form $(X|H)|K$ defined as a suitable conditional random quantity, which coincides with $X|HK$ when $H \subseteq K$. Based on a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of coherence. We give a representation of the conjoined conditional and we show that this new object is a conditional random quantity. We examine some cases of logical dependencies, by also showing that the conjunction may be a conditional event; moreover, we introduce the negation of the conjunction and by De Morgan's Law the operation of disjunction. Finally, we give the lower and upper bounds for the conjunction and the disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold

Frontiers in Psychology / Imprecise Uncertain Reasoning : A Distributional Approach

The contribution proposes to model imprecise and uncertain reasoning by a mental probability logic that is based on probability distributions. It shows how distributions are combined with logical operators and how distributions propagate in inference rules. It discusses a series of examples like the Linda task, the suppression task, Doherty's pseudodiagnosticity task, and some of the deductive reasoning tasks of Rips. It demonstrates how to update distributions by soft evidence and how to represent correlated risks. The probabilities inferred from different logical inference forms may be so similar that it will be impossible to distinguish them empirically in a psychological study. Second-order distributions allow to obtain the probability distribution of being coherent. The maximum probability of being coherent is a second-order criterion of rationality. Technically the contribution relies on beta distributions, copulas, vines, and stochastic simulation.(VLID)311645

Exchangeability in Probability Logic

The paper investigates exchangeability in the context of probability logic. We study generalizations of basic inference rules and inferences involving cardinalities. We compare the results with those obtained in the case in which only identical probabilities are assumed