## Mathematics Behind Fuzzy LogicMany results in fuzzy logic depend on the mathematical structure the truth value set obeys. In this textbook the algebraic foundations of many-valued and fuzzy reasoning are introduced. The book is self-contained, thus no previous knowledge in algebra or in logic is required. It contains 134 exercises with complete answers, and can therefore be used as teaching material at universities for both undergraduated and post-graduated courses. "Chapter 1" starts from such basic concepts as order, lattice, equivalence and residuated lattice. It contains a full section on BL-algebras. "Chapter 2" concerns MV-algebra and its basic properties. "Chapter 3" applies these mathematical results on Lukasiewicz-Pavelka style fuzzy logic, which is studied in details; besides semantics, syntax and completeness of this logic, a lot of examples are given. "Chapter 4" shows the connection between fuzzy relations, approximate reasoning and fuzzy IF-THEN rules to residuated lattices. |

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### Contents

Residuated Lattices | 1 |

MVAlgebras | 35 |

Fuzzy Propositional Logic | 75 |

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antitone Assume binary operation binary relation Boolean algebra complete lattice complete MV-algebra conclude congruence relation deductive system defined definition denoted Example Exercise 12 Exercises from Section extensional hull extensional w.r.t. following metaproof fuzzy equivalence relation fuzzy logic Fuzzy Propositional Logic fuzzy relation equation fuzzy rules fuzzy set fuzzy similarity fuzzy theory greatest element hence holds imp 0 imp imp q implies inner truth value isomorphic isotone least element locally finite BL-algebra Lukasiewicz structure Lukasiewicz valued fuzzy min{l Moreover MV-homomorphism natural number non-a non-logical axiom non-p prime ds proof is complete proper ds Proposition 60 residuated lattice respect Rgmp rules of inference satisfies second variable Similarly stand for Bob subset unit interval v(a imp valuation values of truth verify Vier wait IF wait Wajsberg algebra well-formed formulas whence x,y G L