Bertrand’s paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace’s principle of insufficient reason. In this article we show that Bertrand’s paradox contains two different problems: an «easy» problem and a «hard» problem. The easy problem can be solved by formulating Bertrand’s question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace’s principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible «ways of selecting» an interaction between the entity under investigation and that producing the randomization.

We present a new type of spin market model, populated by hierarchical agents, represented as configurations of sites and arcs in an evolving network. We describe two analytic techniques for investigating the asymptotic behavior of this model: one based on the spectral theory of Markov chains and another exploiting contingent submartingales to construct a deterministic cellular automaton that approximates the stochastic dynamics. Our study of this system documents a phase transition between a sub-critical and a super-critical regime based on the values of a coupling constant that modulates the tradeoff between local majority and global minority forces. In conclusion, we offer a speculative socioeconomic interpretation of the resulting distributional properties of the system.

This essay discusses the advantages of a probabilistic agent-based approach to questions in theoretical economics, from the nature of economic agents, to the nature of the equilibria supported by their interactions. One idea we propose is that «agents» are meta-individual, hierarchically structured objects, that include as irreducible components groupings of different dimensions. We also explore the effects of non-ergodicity, by constructing a simple stochastic model for the contingent nature of economic interactions.

A new computational method is provided to implement the system of deductive logic presented in Aristotle’s Prior Analytics. Each Aristotelian problem is interpreted as a parametric probability network in which the premises give constraints on probabilities relating the problem’s categorical terms (major, minor, and middle). The problem’s figure (schema) describes which specific probabilities are constrained, relative to those that are queried. Using numerical optimization methods, the minimum and maximum feasible values of certain queried probabilities are computed. These computed values determine whether a syllogism is present and if so, which precise conclusion has been deduced from the premises. This method of analysis prevents existential fallacies, and reveals new complementary patterns of syllogism that were previously unappreciated.

We count the number and patterns of pairs and tuples of independent events in a simple random experiment: first a fair coin is flipped and then a fair die is tossed. The first number, equal to 888,888, suggest that there are some open questions about the structure of independence even in a finite sample space. We discuss briefly these questions and possible approaches to answer them.

This paper is devoted to Poincar\’e’s work in probability. Though the subject does not represent a large part of the mathematician’s achievements, it provides significant insight into the evolution of Poincar\’e’s thought on several important matters such as the changes in physics implied by statistical mechanics and molecular theories. After having drawn the general historical context of this evolution, I focus on several important steps in Poincar\’e’s texts dealing with probability theory, and eventually consider how his legacy was developed by the next generation.

We count the number and patterns of pairs and tuples of independent events in a simple random experiment: first a fair coin is flipped and then a fair die is tossed. The first number, equal to 888,888, suggest that there are some open questions about the structure of independence even in a finite sample space. We discuss briefly these questions and possible approaches to answer them.