Daniele Mundici

Algebraic Probabilistic Consistency:
Boole, Łukasiewicz, de Finetti, Kolmogorov

DETAILS
EXPLANATIONS (Book Summary)

This book investigates the foundations of probability theory and logic, intertwining historical insights with modern interpretations. It explores the evolution of probability theory from Boole’s seminal question on the very object of probability, through de Finetti’s finitely additive probability and his consistency notion, also known as “non-Dutchbookability”, to the intricate relationship between logic independence and stochastic independence.

Using the recent characterization of Łukasiewicz logic as the only logic generated by a continuous $[0,1]$-valued operation having the two minimal properties of what is commonly understood as an implication, the author extends the results of the first part of the book from yes-no events to continuous real-valued events. The book culminates with a detailed examination of the symbiosis between de Finetti’s finitely additive and Kolmogorov’s countably additive probability on compact spaces. By providing a rigorous and cohesive narrative, this book serves as an essential resource for scholars and students in mathematical logic eager to grasp the profound connections between logic, probability, and algebraic structures.

TABLE OF CONTENTS

  1. Geometry of finite boolean algebras and their states

  2. De Finetti's “Fundamental Theorem of Probability”

  3. De Finetti's Consistency Theorem

  4. Boolean independence, consistency, and the product law

  5. Interlude: de Finetti's exchangeability theorem

  6. The logic $L_{\infty}$ of continuous $[0, 1]$ valued events

  7. MV algebraic probabilistic consistency

  8. The product law for continuous $[0, 1]$ events

  9. Finite/countable additivity


    Dov M. Gabbay (King's College London), John Woods (University of British Columbia)

    A Logical Tour of Modal Logic: Multi-Modal Systems, Axioms, and Completeness

    :
    DETAILS
    EXPLANATIONS (Book Summary)

    This book provides a comprehensive and accessible guide to the landscapes of modern modal logic, focusing specifically on multi-modal systems, their foundational axioms, and completeness theorems. The authors bridge classic propositional modal logic with advanced structural proof theory, offering new insights into how multiple modalities (such as time, knowledge, and belief) interact within unified logical frameworks.

    Designed for advanced students and researchers in mathematical logic, computer science, and philosophy, the text systematically deconstructs complex relational semantics (Kripke structures) and guides the reader through step-by-step canonical model constructions. By unifying disparate techniques in modal proof theory, the book serves as both a thorough textbook and a definitive reference for understanding completeness in complex multi-modal environments.

    TABLE OF CONTENTS (Chapters)

    1. Introduction to Multi-Modal Frameworks

    2. Relational Semantics and Kripke Frames

    3. Basic Normal Modal Logics and Their Axiomatization

    4. Interactions Between Modalities: Epistemic and Doxastic Mergers

    5. Temporal Modalities and Non-Linear Time Structures

    6. The Technique of Canonical Models

    7. Completeness Theorems for Multi-Modal Systems

    8. Decidability and the Finite Model Property

    9. Advanced Structural Proof Theory in Modal Logic




      Karl G. Jöreskog, Ulf H. Olsson, Fan Y. Wallentin

      Multivariate Analysis with LISREL (Second Edition)

      DETAILS

      • Publisher: Springer Nature Switzerland / Springer (Springer Series in Statistics)

      • Publication Date: June 2026 (予定)

      • Language: English

      • Format: Hardcover / eBook

      • ISBN-13: 978-3-03224-305-8

      • ISBN-10: 303224305X

      EXPLANATIONS (Book Summary)

      This second edition traces the theory and methodology of multivariate statistical analysis and shows how it can be conducted in practice using the LISREL computer program. It presents not only the typical uses of LISREL, such as confirmatory factor analysis (CFA) and structural equation models (SEM), but also several other multivariate analysis topics, including regression (univariate, multivariate, censored, logistic, and probit), generalized linear models, multilevel analysis, and principal components.

      The book is uniquely topic-oriented, describing the method and its basic statistical theory first, before providing numerous examples mostly using real data to demonstrate how to perform the analysis with LISREL. It discusses and interprets the results, illustrated with sections of output from the LISREL program in the context of the example. This textbook is intended for master's and PhD students, as well as researchers in the social, behavioral, economic, and other sciences who require a solid understanding of multivariate statistical theory and practical analysis methods.

      TABLE OF CONTENTS (Chapters)

      1. Getting Started

      2. Regression Models (Univariate, Multivariate, Censored, Logistic, and Probit)

      3. Generalized Linear Models

      4. Multilevel Analysis

      5. Principal Components (PCA)

      6. Exploratory Factor Analysis (EFA)

      7. Confirmatory Factor Analysis (CFA)

      8. Structural Equation Models (SEM) with Latent Variables

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        Krishna Suryanarayan

        Formal Proofs in Elementary Set Theory:
        Rules of Logic for Formal Proofs in Zermelo Set Theory

        DETAILS

        • Publisher: Springer Nature Switzerland / Springer (SpringerBriefs in Computational Intelligence / SpringerBriefs in Applied Sciences and Technology)

        • Publication Date: June 2026 (予定)

        • Language: English

        • Format: Paperback / eBook

        • Pages: Approx. 135 pages

        • ISBN-13: 978-3-03222-238-1

        • ISBN-10: 3032222389

        EXPLANATIONS (Book Summary)

        This book gives a comprehensive summary of the rules of logic required to write formal mathematical proofs. It focuses on providing explicit, step-by-step formal proofs for foundational theorems in elementary set theory, strictly based on the Zermelo axioms.

        A key highlight of the text includes a rigorous proof of the existence of a Peano system, which serves as the structural foundation for modern number theory. Designed to help students and researchers transition into writing airtight formal proofs, the book functions as both a handy reference for mathematical logic courses and a theoretical blueprint for developing software meant to write and computationally verify formal mathematical arguments.

        TABLE OF CONTENTS (Expected Outline)

        1. Summary of Rules of Logic for Formal Proofs

        2. The Zermelo Axioms of Set Theory

        3. Formal Proofs of Elementary Theorems in Set Theory

        4. The Peano System and Foundations of Number Theory

        5. Structural Frameworks for Computational Proof Verification


          Stephen S.-T. Yau (清華大学), Bingyi Chen, Xiankui Meng, Huaiqing Zuo (清華大学)

          • Complex Singularity Theory and Cauchy–Riemann Geometry

            s:
          DETAILS

          • Publisher: Springer Nature Switzerland / Springer (Springer Monographs in Mathematics)

          • Publication Date: June 2026 (予定)

          • Language: English

          • Format: Hardcover / eBook

          • ISBN-13: 978-3-03224-659-2

          • ISBN-10: 3032246598

          EXPLANATIONS (Book Summary)

          This advanced monograph provides a deep and cohesive exploration of the intersections between complex singularity theory and Cauchy-Riemann (CR) geometry. The text brings together sophisticated techniques from differential geometry, several complex variables, and algebraic singularity theory within a unified mathematical framework.

          A central focus of the work is the application of these geometric methods to structural problems, including the explicit construction of infinite-dimensional moduli spaces for certain CR manifolds. Designed for researchers, faculty, and advanced graduate students specializing in complex geometry and mathematical analysis, this book serves as both a comprehensive guide to the current state-of-the-art research and a definitive reference for modern analytical and algebraic techniques in CR structures.

          TABLE OF CONTENTS (Expected Chapters)

          1. Foundations of Cauchy-Riemann (CR) Geometry

          2. Complex Singularity Theory and Isolated Singularities

          3. Relational Semantics between Singularities and CR Structures

          4. Moduli Spaces of CR Manifolds

          5. Infinite-Dimensional Constructive Methods (Yau's Multipliers and Moduli)

          6. Deformation Theory of Complex and CR Structures

          7. Differential Geometric Invariants in Complex Manifolds

          8. Advanced Topics in CR Embeddability and Singularities

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            Rick Durrett (Richard Durrett / デューク大学名誉教授)

            Dynamics on Graphs

            Authors:

            DETAILS

            • Publisher: Springer Nature Switzerland / Springer

            • Publication Date: 2026年8月 (予定)

            • Language: English

            • Format: Hardcover / eBook

            • Pages: Approx. 376 pages

            • ISBN-13: 978-3-03221-907-7

            • ISBN-10: 3032219076

            EXPLANATIONS (Book Summary)

            This book is an extensive revision of the author's 2007 landmark volume, Random Graph Dynamics. In contrast to the previous edition which focused heavily on the varied geometry of random graphs, this new version simplifies the graph models considered—concentrating primarily on the classical Erdős–Rényi model, the configuration model, and inhomogeneous random graphs—in order to deeply investigate a wide variety of stochastic dynamics that run on them.

            The text provides up-to-date results on the convergence to equilibrium for random walks on random graphs. It also explores major topics that have emerged as mature research areas over the last fifteen years, including the contact process, voter models, coalescing random walks, and the mathematical modeling of epidemics (such as the SIR model). Relying primarily on probabilistic methods rather than purely combinatorial ones, this book serves as both a comprehensive guide and a definitive textbook for graduate students and researchers in probability, interacting particle systems, and network science.

            TABLE OF CONTENTS (Chapters 1–4 Outline)

            1. Erdős–Rényi Random Graphs

              • Branching Processes

              • Cluster growth as a branching process / random walk

              • Threshold for connectivity / Long paths

              • Central Limit Theorem for the size of the giant component

              • Critical regime and critical exponents

            2. General Degree Distributions

              • Configuration model / Limiting degree distribution approach

              • Subcritical cluster sizes

              • Distance between two randomly chosen vertices

              • First passage percolation / Percolation / Critical regime

            3. Inhomogeneous Random Graphs

              • Finitely many types

              • Motivating examples

              • Computing the survival probability

              • Component sizes in the subcritical case

            4. SIR Epidemics

              • On the complete graph

              • Fixed and general infection times

              • Miller–Volz equations and their rigorous derivations

              • Household and dorm models

              • Epidemics on $\mathbb{Z}^2$