## An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation SystemsProfessor Merrie Bergmann presents an accessible introduction to the subject of many-valued and fuzzy logic designed for use on undergraduate and graduate courses in non-classical logic. Bergmann discusses the philosophical issues that give rise to fuzzy logic - problems arising from vague language - and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three-valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical systems - Lukasiewicz, Gödel, and product logics - are then presented as generalisations of three-valued systems that successfully address the problems of vagueness. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, that ask students to continue proofs begun in the text, and that engage students in the comparison of logical systems. |

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### Contents

1 | |

12 | |

3 Review of Classical FirstOrder Logic | 39 |

4 Alternative Semantics for TruthValues | 57 |

Semantics | 71 |

6 Derivation Systems for ThreeValued Propositional Logic | 100 |

Semantics | 130 |

9 Alternative Semantics for ThreeValued Logic | 161 |

10 The Principle of Charity Reconsidered and a New | 174 |

Semantics | 176 |

12 Fuzzy Algebras | 212 |

13 Derivation Systems for Fuzzy Propositional Logic | 223 |

Semantics | 262 |

if we really wish to assert that well any person | 269 |

15 Derivation Systems for Fuzzy FirstOrder Logic | 287 |

8 Derivation Systems for ThreeValued FirstOrder Logic | 146 |

aboutaoranythingelseandadoesnotoccurineitherPorQAndthecondition | 149 |

16 Extensions of Fuzziness | 300 |

17 Fuzzy Membership Functions | 309 |

### Common terms and phrases

antecedent atomic formulas axiom schema axiomatic system BL-algebra Bochvar’s bold conjunction bold meet Boolean algebra Chapter Charity premise classical logic classical propositional logic classical tautology clause commutation conditional conjunction and disjunction conjunctive normal form connectives Deduction Theorem deﬁnable deﬁned deﬁnition degree of membership degree-valid derivation system derived axiom dissatisﬁed Double Negation downward distance entailment equivalent example Excluded Middle exercise external assertion F T F false ﬁnite ﬁrst first-order logic FLPA fuzzy logic fuzzy set FuzzyL G¨odel GCON graded formula graded value height inﬁnite Iv(P Iv(Tsi L3PA Law of Excluded least the value Lukasiewicz MV-algebra normal form operations Principle of Charity Proof Prove quasi-tautology Result satisﬁed schemata Section semantic set of formulas Sorites argument Sorites paradox speciﬁc t-norm three-valued system true truth-conditions truth-function truth-tables truth-value assignment Tx a Eyx unit interval valid value F variable assignment Vx)P weak conjunction x-variant zero

### Popular passages

Page 14 - All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life, but only to an imagined celestial existence.

Page 14 - It is one of the paper's main contentions that with the provision of an adequate symbolism the need is removed for regarding vagueness as a defect of language. The ideal standard of precision which those have in mind who use vagueness as a term of reproach, when it is not a shifting standard of a relatively less vague symbol, is the standard of scientific precision. But the indeterminacy which is characteristic of vagueness is present also in all scientific measurement. "There is no...

Page 14 - laws" of logic or mathematics prescribe modes of existence to which intelligible discourse must necessarily conform. It will be argued, on the contrary, that deviations from the logical or mathematical standards of precision are all pervasive in symbolism; that to label them as subjective aberrations sets an impassable gulf between formal laws and experience and leaves the usefulness of the formal sciences an insoluble mystery.

Page 14 - But the indeterminacy which is characteristic of vagueness is present also in all scientific measurement. "There is no experimental method of assigning numerals in a manner which is free from error. If we limit ourselves strictly to experimental facts we recognize that there is no such thing as true measurement, and therefore no such thing as an error involved in a departure from it."6 Vagueness is...

Page 7 - Aguzzoli S., Ciabattoni A. [2000], Finiteness in infinite-valued Lukasiewicz logic, Journal of Logic Language and Information, 9, 5-29.