## Classical and Fuzzy Concepts in Mathematical Logic and Applications, Professional VersionClassical and Fuzzy Concepts in Mathematical Logic and Applications provides a broad, thorough coverage of the fundamentals of two-valued logic, multivalued logic, and fuzzy logic. Exploring the parallels between classical and fuzzy mathematical logic, the book examines the use of logic in computer science, addresses questions in automatic deduction, and describes efficient computer implementation of proof techniques. Specific issues discussed include: The authors consider that the teaching of logic for computer science is biased by the absence of motivations, comments, relevant and convincing examples, graphic aids, and the use of color to distinguish language and metalanguage. Classical and Fuzzy Concepts in Mathematical Logic and Applications discusses how the presence of these facts trigger a stirring, decisive insight into the understanding process. This view shapes this work, reflecting the authors' subjective balance between the scientific and pedagogic components of the textbook. Usually, problems in logic lack relevance, creating a gap between classroom learning and applications to real-life problems. The book includes a variety of application-oriented problems at the end of almost every section, including programming problems in PROLOG III. With the possibility of carrying out proofs with PROLOG III and other software packages, readers will gain a first-hand experience and thus a deeper understanding of the idea of formal proof. |

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### Contents

Introduction to Naive Mathematical Logic | 1 |

PROPOSITIONAL LOGIC | 15 |

The Formal Language of Propositional Logic | 23 |

The Truth Structure on 0 in Semantic Version | 37 |

The Truth Structure on 0 in The Syntactic Version | 69 |

Connections Between the Truth Structures on 0 | 97 |

Other Syntactic Versions of the Truth Structure on 0 | 107 |

Elements of Fuzzy Propositional Logic | 131 |

The Formal Language of Predicate Logic | 197 |

The Semantic Truth Structure on the Language С | 205 |

The Syntactic Truth Structure on the Language | 233 |

Elements of Fuzzy Predicate Logic | 259 |

Further Applications of Logic in Computer Science | 273 |

Exercises Part II | 315 |

A Boolean Algebras | 321 |

B MVAlgebras | 339 |

Applications of Propositional Logic in Computer Science | 153 |

Exercises Part I | 178 |

Introductory Considerations | 187 |

General Considerations about Fuzzy Sets | 351 |

359 | |

### Common terms and phrases

alphabet arbitrary formulas assertion axiom bijective Boolean algebra Boolean functions Boolean interpretation called closed formula consequence Consider the following contains correspondence deduction from H defined DEFINITION disjunction elements equivalent F V F F V G finite sequence finitely consistent formal construction formal language formal word formula F functor fuzzy logic fuzzy propositional logic fuzzy set fuzzy tautology hence hL F Horn clauses hypothesis inconsistent individual variable inductive assumption inference rule Karnaugh map Let F Let us consider Lindenbaum algebra literal symbols mathematical mathematical induction means METATHEOREM min{l MP-rule MV-algebra notation obtain operations predicate logic Prolog PROOF Let propositional variable prove quantifiers REMARK resolution method result s-consistent sentences set H subset suppose surjective switching network syntactic version system of clauses Taking into account true truth structure truth table truth values unique valid formula write