Probability's Evolution Mirrors Rationality: Bayesian vs Fuzzy vs Deep Learning

The paper, authored by eight researchers including Jean-Louis Le Mouël and Vincent Courtillot, spans 44 pages and offers a sweeping historical-epistemological analysis. It begins with the origins of probability in Pascal and Fermat's work on games of chance, then moves through Bayes and Laplace's inductive logic, Poisson's statistics, and Kolmogorov's axiomatic formalization. This arc culminates in modern Bayesian inference, particularly Tarantola's view of probability as a logic of information—where prior knowledge and data are coherently combined. However, the authors identify a fundamental limit: probability quantifies uncertainty about well-defined propositions but cannot formalize the vagueness of the concepts used to describe propositions.

To address this gap, the paper introduces fuzzy logic as a rigorous language for graded meaning and qualitative judgment, and analyzes deep learning as a distinct mode of prediction based on geometric interpolation and optimization—not explicit inference. By situating all three frameworks—probability, fuzzy logic, and deep learning—in a common historical and epistemological perspective, the authors argue that contemporary scientific rationality cannot be reduced to data-driven performance alone. Instead, it requires the explicit articulation of uncertainty, vagueness, and inference. The paper serves as a timely reminder that AI tools based solely on deep learning lack the probabilistic and logical foundations essential for truly rational decision-making.