“The Use of Abstraction and Logic in Mathematics” is an edited book consisting of 16 contemporaneous open-access articles that are essentially devoted to mathematical logic research, from classical to non-classical logical systems, from algebraic logic to fuzzy logic. The book addresses the following mathematical logic topics: first-order and higher-order logic; as well as infinitary, description, modal; fixed-point, algebraic and fuzzy logic. This book also includes examples of practical applications of logical systems in link prediction and image processing tasks, as well as in the training of neural networks and artificial intelligence. The intended audience of this book is undergraduate and graduate students, as well as junior researchers. Familiarity with first-order and higher-order logics, as well as set theory, and algebra is essential to grasp the concepts and methods described in this book.
Author(s): Olga Moreira
Publisher: Arcler Press
Year: 2022
Language: English
Pages: 422
City: Boston
Cover
Title Page
Copyright
DECLARATION
ABOUT THE EDITOR
TABLE OF CONTENTS
List of Contributors
List of Abbreviations
Preface
Chapter 1 Classical Logic and Quantum Logic with Multiple and Common Lattice Models
Abstract
Introduction: Is Logic Empirical?
Kinds of Logic
Lattices
Soundness and Completeness
Discussion
Acknowledgments
References
Chapter 2 A Novel Categorical Approach to Semantics of Relational First-Order Logic
Abstract
Introduction
A Relational First-Order Logic
Category Theory
A Categorical Semantics
An Implementation of the Categorical Semantics
Conclusions
Author Contributions
Funding
References
Chapter 3 Infinitary Classical Logic: Recursive Equations and Interactive Semantics
Introduction
Preliminaries: Positions and Labeled Trees
Infinitary Classical Logic
Interactive Seantics
References
Chapter 4 Formalization of Linear Space Theory in the Higher-Order Logic Proving System
Abstract
Introduction
Preliminaries in Hol
Formalization of Linear Space Theory in Hol4
Conclusion
Acknowledgment
References
Chapter 5 Language and Proofs for Higher-Order SMT (Work in Progress)
Introduction
A Syntax Extension for the Smt-Lib Language
An Extension for the Verit Proof Format
Conclusion and Future Work
Acknowledgment
References
Chapter 6 Alternation Is Strict For Higher-Order Modal Fixpoint Logic
Introduction
Alternating Parity Krivine Automata
APKA and HFL
The Alternation Hierarchy for Alternating Parity Krivine Automata
Discussion
Acknowledgements
References
Chapter 7 Bisimulation in Inquisitive Modal Logic
Introduction
Inquisitive Modal Logic
Inquisitive Bisimulation
An Ehrenfeucht–Fra¨Isse Theorem
Relational Inquisitive Models
The ~-Invariant Fragment of FO
Conclusion
References
Chapter 8 Graphical Sequent Calculi for Modal Logics
Introduction
The Syntax of Modal Graphs
The Graphical Calculi Kg
Extensions
Graphical and Sequent Calculi
Conclusion
Acknowledgements
References
Chapter 9 Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems
Abstract
Introduction
Basic Framework
Meet-Combinations
Soundness
c-Completeness
Conservativeness and Consistency
Examples from Classical Propositional Logic
References
Chapter 10 Fuzzy Logic versus Classical Logic: An Example in Multiplicative Ideal Theory
Abstract
Introduction
Preliminaries and Notations
Fuzzy Logic versus Classical Logic: An Example
References
Chapter 11 Link Prediction Using A Probabilistic Description Logic
Abstract
Introduction
Background
Link Prediction with CR ALC
Experiments
Conclusion
Acknowledgments
References
Chapter 12 Reasoning about Social Choice and Games in Monadic Fixed-Point Logic
Introduction
The Improvement Graph Structure
Monadic Fixed-Point Logic With Counting
Model Checking Algorithm
Discussion
Acknowledgements
References
Chapter 13 Formal Analysis of 2D Image Processing Filters using Higher-order Logic Theorem Proving
Abstract
Introduction
Contributions of the Paper
Preliminaries
Methods
Results
Discussions
Conclusions
Acknowledgements
References
Chapter 14 GRAN3SAT: Creating Flexible Higher-Order Logic Satisfiability in the Discrete Hopfield Neural Network
Abstract
Introduction
G-Type Random K Satisfiability
Gran3sat in the Discrete Hopfield Neural Network
Experimental Setup
Results and Discussion
Conclusions
Author Contributions
Acknowledgments
References
Chapter 15 Design of a Computable Approximate Reasoning Logic System for AI
Abstract
Introduction
Mathematical Logic System Based on Precise Reasoning
Irrationality of a Fuzzy Logic System
Preliminary Knowledge of a Regression Logic Route in Fuzziness Research
Redundancy Theory: Computable Logic, Approximate Reasoning Logic
Generalized Dynamic Logic System Characterized By Machine Learning
Conclusions
Author Contributions
References
Chapter 16 On the Possibility of Correct Concept Learning in Description Logics
Abstract
Introduction
Notation and Semantics of Description Logics
Concept Normalization
A Concept Learning Algorithm
C-Learnability in Description Logics
On Concept Learning Using Queries
Concluding Remarks
Acknowledgements
References
Index
Back Cover
