## A Treatise on Many-valued LogicsA growing interest in many-valued logic has developed which to a large extent is based on applications, intended as well as already realised ones. These applications range from the field of computer science, e.g. in the areas of automated theorem proving, approximate reasoning, multi-agent systems, switching theory, and program verification, through the field of pure mathematics, e.g. in independence of consistency proofs, in generalized set theories, or in the theory of particular algebraic structures, into the fields of humanities, linguistics and philosophy. |

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### Contents

General Background | 3 |

The Formalized Language and its Interpretations | 15 |

Logical Validity and Entailment | 29 |

Copyright | |

23 other sections not shown

### Common terms and phrases

according additionally adequate axiomatization algebraic structures axiom schemata basic BL-algebra characterized classical logic condition conjunction consider Corollary corresponding defined definition denoted derivation designated truth degree disjunction elements equivalence relation exists extended filter finitely many-valued first-order first-order logic functionally complete Furthermore fuzzy logic fuzzy relation fuzzy sets G-algebra Godel systems hence Heyting algebra holds true immediately implication connective inference rules infinitely isomorphic language lattice ordering left continuous t-norm Lindenbaum algebra logical calculus logically valid Lukasiewicz systems many-valued logic many-valued systems means modal monoidal MV-algebras n-ary negation function notion operations ordinal sum Post algebras predicate symbol product algebra Proof propositional variables prove quantifiers residuated lattice result S-sequent satisfies schema sentences sequence signed formulas subset suitable Suppose systems of many-valued t-norm based truth degree constants truth degree functions truth degree set truth value unary universe of discourse valuation wff H