## Metamathematics of Fuzzy LogicThis book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named `fuzzy inference' can be naturally understood as logical deduction. There are two main groups of intended readers. First, logicians: they can see that fuzzy logic is indeed a branch of logic and may find several very interesting open problems. Second, equally important, researchers involved in fuzzy logic applications and soft computing. As a matter of fact, most of these are not professional logicians so that it can easily happen that an application, clever and successful as it may be, is presented in a way which is logically not entirely correct or may appear simple-minded. (Standard presentations of the logical aspects of fuzzy controllers are the most typical example.) This fact would not be very important if only the bon ton of logicians were harmed; but it is the opinion of the author (who is a mathematical logician) that a better understanding of the strictly logical basis of fuzzy logic (in the usual broad sense) is very useful for fuzzy logic appliers since if they know better what they are doing, they may hope to do it better. In addition, a better mutual understanding between (classical) logicians and researchers in fuzzy logic, promises to lead to deeper cooperation and new results. |

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### Contents

III | 1 |

IV | 6 |

V | 10 |

VI | 15 |

VII | 20 |

VIII | 22 |

IX | 27 |

X | 35 |

XXVI | 154 |

XXVII | 161 |

XXVIII | 167 |

XXIX | 168 |

XXX | 177 |

XXXI | 189 |

XXXII | 195 |

XXXIII | 205 |

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### Common terms and phrases

1-tautologies algebra arithmetic assume axiomatization axioms binary BL h BL-algebra BLV h BLV proves Boolean logic C-algebra classical closed formula completeness theorem completes the proof conjunction connectives conservative extension containing continuous t-norm Conversely Corollary corresponding countable crisp deduction rules deduction theorem defined Definition domain element equivalent evaluation example finite models function symbols fuzzy control fuzzy logic Godel hence implication induction interpretation isomorphic Kripke model L-model Lemma linearly ordered MV-algebra Lukasiewicz logic many-valued mapping modal logic modus ponens MV-algebra natural numbers non-empty notion o-group object variables Pavelka predicate calculus predicate language predicate logic probabilistic product logic propositional calculus propositional logic propositional variables provable proves the following quantifiers rational recursive Remark residuated lattices RPLV satisfies semantics semigroup Similarly structure subsets TAUT tautology theory truth constants truth degree truth functions truth value unary predicates

### Popular passages

Page 285 - Prade, H. Fuzzy sets in approximate reasoning, Part 1 : Inference with possibility distributions. Fuzzy Sets and Systems, 40(1): 143-202, March 5, 1991. [11] Dubois, D., Lang, J., and Prade, H. Fuzzy sets in approximate reasoning, Part 2: Logical approaches.

Page 283 - Alsina, C., Trillas, E. and Valverde, L., "On some logical connectives for fuzzy set theory," Journal of Mathematical Analysis and Application 93, 15-26, 1983.