Probabilistic Geometry Based on the Fuzzy Playfair Axiom

The study of the interplay between belief and probability can be posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. Probability approximations of belief functions form two homogeneous groups, which we call "affine" and "epistemic" families. In this paper we focus on relative plausibility, belief, and uncertainty of probabilities of singletons, the "epistemic" family. They form a coherent collection of probability transformations in terms of their behavior with respect to Dempster's rule of combination. We investigate here their geometry in both the space of all pseudo belief functions and the probability simplex, and compare it with that of the affine family. We provide sufficient conditions under which probabilities of both families coincide.

International Journal of Statistics and Probability, 2016

In the domain of the logic of certainty we examine the objective notions of the subjective probability with the clear aim of identifying their fundamental characteristics before the assignment, by the individual, of the probabilistic evaluation. Probability is an additional and subjective notion that one applies within the range of possibility, thus giving rise to those gradations, more or less probable, that are meaningless in the logic of certainty. Each criterion for evaluations under conditions of uncertainty is a device or instrument for obtaining a measurement; it furnishes an operational definition of probability or prevision P and together with the corresponding conditions of coherence can be taken as a foundation for the entire theory of probability. When we examine these criteria and their corresponding conditions of coherence we show the inevitable dichotomy between the subjective or psychological or empirical aspect of probability and the objective or logical or geometri...

Theoretical Computer Science, 2011

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) and continuous logic stand in a similar close

International Conference on Scientific Computing, 2010

By accepting the uncertainty principle in an Euclidean plane, the position of point, line, and every geometric shape can be identi…ed by the statistical parameters of mean and standard deviation. A heuristic model of Euclidean Statistical Geometry will be established. An axiomatic approach for unde-…ned terms and their use in geometric objects is presented. The consistency of geometric interpretations of the mean, variance, and correlation coe¢ cient for random vectorial events in the Euclidean plane are discussed.

Classical and Quantum Gravity, 2001

We develop a kind of pregeometry consisting of a web of overlapping fuzzy lumps which interact with each other. The individual lumps are understood as certain closely entangled subgraphs (cliques) in a dynamically evolving network which, in a certain approximation, can be visualized as a time-dependent random graph. This strand of ideas is merged with another one, deriving from ideas, developed some time ago by Menger et al, that is, the concept of probabilistic-or random metric spaces, representing a natural extension of the metrical continuum into a more microscopic regime. It is our general goal to find a better adapted geometric environment for the description of microphysics. In this sense one may it also view as a dynamical randomisation of the causal-set framework developed by e.g. Sorkin et al. In doing this we incorporate, as a perhaps new aspect, various concepts from fuzzy set theory.

Physica a, 2007

Following our discussion [E. Canessa, Physica A 375 (2007) 123] to associate an analogous probabilistic description with spacetime geometry in the Schwarzschild metric from the macro- to the micro-domain, we argue that there is a possible connection among normalized probabilities P, spacetime geometry (in the form of Schwarzschild radii rs) and quantum mechanics (in the form of complex wave functions ψ), namely √{Pθ,φ,t(n)}≈Rs(n)/rs=|ψn(n)(X)|2/|ψn(x)|2. We show how this association along different (n)-nested surfaces-representing curve space due to an inhomogeneous density of matter-preserves the postulates of quantum mechanics at different geometrical scales.

Journal of Applied Non-Classical Logics, 2008

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Image Analysis & Stereology, 2012

The paper provides an account of the history of geometric probability and stereology from the time of Newton to the early 20th century. It depicts the development of two parallel ways: on one hand, the theory of geometric probability was formed with minor ...

International Journal of Cosmology, Astronomy and Astrophysics, 2021

This article describes a new theory formulated to improve our understanding of cosmological phenomena. First, an enumeration is offered of shortcomings of current cosmological theories. Second, the components of the new theory are delineated. Third, the theory's explanations of poorly understood cosmological phenomena are presented. Finally, numerous testable predictions are described that differentiate the theory from existing theories.

Kolmogorov’s Heritage in Mathematics