Use of abstraction and logic in mathematics

“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. Cover 1 Title Page 5 Copyright 6 DECLARATION 7 ABOUT THE EDITOR 9 TABLE OF CONTENTS 11 List of Contributors 17 List of Abbreviations 21 Preface 23 Chapter 1 Classical Logic and Quantum Logic with Multiple and Common Lattice Models 25 Abstract 25 Introduction: Is Logic Empirical? 26 Kinds of Logic 28 Lattices 31 Soundness and Completeness 37 Discussion 47 Acknowledgments 49 References 50 Chapter 2 A Novel Categorical Approach to Semantics of Relational First-Order Logic 55 Abstract 55 Introduction 56 A Relational First-Order Logic 59 Category Theory 66 A Categorical Semantics 73 An Implementation of the Categorical Semantics 83 Conclusions 89 Author Contributions 89 Funding 90 References 91 Chapter 3 Infinitary Classical Logic: Recursive Equations and Interactive Semantics 95 Introduction 96 Preliminaries: Positions and Labeled Trees 99 Infinitary Classical Logic 101 Interactive Seantics 112 References 121 Chapter 4 Formalization of Linear Space Theory in the Higher-Order Logic Proving System 123 Abstract 123 Introduction 124 Preliminaries in Hol 124 Formalization of Linear Space Theory in Hol4 125 Conclusion 132 Acknowledgment 132 References 133 Chapter 5 Language and Proofs for Higher-Order SMT (Work in Progress) 135 Introduction 136 A Syntax Extension for the Smt-Lib Language 137 An Extension for the Verit Proof Format 139 Conclusion and Future Work 143 Acknowledgment 144 References 145 Chapter 6 Alternation Is Strict For Higher-Order Modal Fixpoint Logic 147 Introduction 147 Alternating Parity Krivine Automata 150 APKA and HFL 158 The Alternation Hierarchy for Alternating Parity Krivine Automata 161 Discussion 168 Acknowledgements 169 References 170 Chapter 7 Bisimulation in Inquisitive Modal Logic 173 Introduction 174 Inquisitive Modal Logic 176 Inquisitive Bisimulation 181 An Ehrenfeucht–Fra ̈Isse Theorem 182 Relational Inquisitive Models 184 The ~-Invariant Fragment of FO 187 Conclusion 191 References 192 Chapter 8 Graphical Sequent Calculi for Modal Logics 195 Introduction 196 The Syntax of Modal Graphs 197 The Graphical Calculi Kg 199 Extensions 203 Graphical and Sequent Calculi 205 Conclusion 209 Acknowledgements 210 References 211 Chapter 9 Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems 215 Abstract 215 Introduction 216 Basic Framework 219 Meet-Combinations 221 Soundness 224 c-Completeness 227 Conservativeness and Consistency 230 Examples from Classical Propositional Logic 231 References 234 Chapter 10 Fuzzy Logic versus Classical Logic: An Example in Multiplicative Ideal Theory 237 Abstract 237 Introduction 237 Preliminaries and Notations 238 Fuzzy Logic versus Classical Logic: An Example 241 References 245 Chapter 11 Link Prediction Using A Probabilistic Description Logic 247 Abstract 247 Introduction 248 Background 249 Link Prediction with CR ALC 255 Experiments 257 Conclusion 266 Acknowledgments 267 References 268 Chapter 12 Reasoning about Social Choice and Games in Monadic Fixed-Point Logic 271 Introduction 272 The Improvement Graph Structure 275 Monadic Fixed-Point Logic With Counting 280 Model Checking Algorithm 285 Discussion 289 Acknowledgements 290 References 291 Chapter 13 Formal Analysis of 2D Image Processing Filters using Higher-order Logic Theorem Proving 295 Abstract 295 Introduction 296 Contributions of the Paper 297 Preliminaries 298 Methods 301 Results 301 Discussions 314 Conclusions 315 Acknowledgements 316 References 317 Chapter 14 GRAN3SAT: Creating Flexible Higher-Order Logic Satisfiability in the Discrete Hopfield Neural Network 319 Abstract 319 Introduction 320 G-Type Random K Satisfiability 324 Gran3sat in the Discrete Hopfield Neural Network 326 Experimental Setup 329 Results and Discussion 335 Conclusions 354 Author Contributions 354 Acknowledgments 354 References 355 Chapter 15 Design of a Computable Approximate Reasoning Logic System for AI 359 Abstract 359 Introduction 360 Mathematical Logic System Based on Precise Reasoning 362 Irrationality of a Fuzzy Logic System 363 Preliminary Knowledge of a Regression Logic Route in Fuzziness Research 366 Redundancy Theory: Computable Logic, Approximate Reasoning Logic 372 Generalized Dynamic Logic System Characterized By Machine Learning 378 Conclusions 382 Author Contributions 382 References 383 Chapter 16 On the Possibility of Correct Concept Learning in Description Logics 387 Abstract 387 Introduction 388 Notation and Semantics of Description Logics 393 Concept Normalization 396 A Concept Learning Algorithm 406 C-Learnability in Description Logics 410 On Concept Learning Using Queries 411 Concluding Remarks 413 Acknowledgements 414 References 415 Index 419 Back Cover 424