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From Multi-valued Logics to Fuzzy Logic

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Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 349))

Abstract

When, in February 2008, I visited the European Centre for Soft Computing (ECSC) in Mieres Asturias, Spain, for the first time to give a talk, I stayed for about a week. It was Enric Trillas who extended the invitation, and it was Claudio Moraga who made my first weekend in Asturias—this then unknown and foreign landscape—enjoyable. One of the first places of interest he showed me was the old church of San Julián de los Prados, or Santullano (built between the years 812 and 842 AD) in Oviedo’s suburb Pumarí, close to the A-6 motorway.

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Notes

  1. 1.

    There are other properties of classical logics that we don’t consider here: Monotonicity and Idempotency of entailment, Commutativity of conjunction, De Morgan duality, double negative elimination, and the principle of explosion (ex falso (sequitur) quodlibet, “from falsehood, anything (follows).”

  2. 2.

    Aristotle said in On Interpretation [1] that of two contradictory propositions one must be true, and the other false—(Contradictory propositions are propositions where one proposition is the negation of the other.) Russell and Whitehead stated this principle as a theorem of propositional logic in the Principia Mathematica.

  3. 3.

    Ich kann ohne Widerspruch annehmen, dass meine Anwesenheit in Warschau in einem bestimmten Zeitmoment des nächsten Jahres, z.B. mittags den 21. Dezember, heutzutage weder im positiven noch im negativen Sinne entschieden ist. Es ist somit möglich, aber nicht notwendig, dass ich zur angegebenen Zeit in Warschau anwesend sein werde. Unter diesen Voraussetzungen kann die Aussage: “Ich werde mittags den 21. Dezember nächsten Jahres in Warschau anwesend sein”, heutzutage weder wahr noch falsch sein. Denn ware sie heutzutage wahr, so müsste meine zukünftige Anwesenheit in Warschau notwendig sein, was der Voraussetzung widerspricht; und wäre sie heutzutage falsch, so müsste meine zukünftige Anwesenheit in Warschau unmöglich sein, was ebenfalls der Voraussetzung widerspricht. Der betrachtete Satz ist daher heutzutage weder wahr noch falsch und muss einen dritten, von “0” oder dem Falschen und von “1” oder dem Wahren verschiedenen Wert haben. Diesen Wert können wir mit “1 / 2” bezeichnen; es ist eben ‘das Mögliche’, das als dritter Wert neben “das Falsche” und “das Wahre” an die Seite tritt. Diesem Gedankengang verdankt das dreiwertige System des Aussagenkalküls seine Entstehung. [20, p. 165].

  4. 4.

    “Es war mir von vornherein klar, dass unter allen mehrwertigen Systemen nur zwei eine philosophische Bedeutung beanspruchen können: das dreiwertige und das unendlichwertige System. Denn werden die von “0” und “1” verschiedenen Werte als “das Mögliche” gedeutet, so konnen aus guten Gründen nur zwei Fälle unterschieden werden: entweder nimmt man an, dass das Mögliche keine Gradunterschiede aufweist, und dann erhält man das dreiwertige System; oder man setzt das Gegenteil voraus, und dann ist es am natürlichsten ebenso wie in der Wahrscheinlichkeitsrechnung anzunehmen, dass unendlich viele Gradunterschiede des Möglichen bestehen, was zum unendlichwertigen Aussagenkalkul führt. Ich glaube, dass gerade dieses letztere System vor allen anderen den Vorzug verdient. Leider ist dieses System noch nicht genau untersucht; insbesondere ist auch das Verhältnis des unendlichwertigen Systems zur Wahrscheinlichkeitsrechnung noch nicht geklärt.” [22, p. 173].

  5. 5.

    For a detailed presentation of the history of Fuzzy Set Theory see [33].

  6. 6.

    Here we need the definition of the “orthocomplement” \(x^\perp \) of the element x: This element satisfies (1) the complement law \(x^\perp \vee x = 1\) and \(x^\perp \wedge x = 0\); (2) the involution law \(x^{\perp \perp } = x\); (3) the order reserving law if \(x\le y\) then \(y^\perp \le x\).

  7. 7.

    The lecture notes of this course Physical Information Theory (1959–60) were published under the title “Algebra of Observations” but not until 1966 unfortunately [38].

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Acknowledgments

I would like to thank Mark Winstanley for proofreading and suggestions for improvement.

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  1. Geschichte der Nat, “Ernst-Haeckel-Haus”, Friedrich-Schiller-Universität Geschichte der Nat, Jena, Germany

    Rudolf Seising

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  1. Escuela de IngenieríaInformática, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile

    Rudolf Seising

  2. Geschichte der Nat, "Ernst-Haeckel-Haus", Deutsches Museum, Munich, Germany

    Héctor Allende-Cid

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Seising, R. (2017). From Multi-valued Logics to Fuzzy Logic. In: Seising, R., Allende-Cid, H. (eds) Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing. Studies in Fuzziness and Soft Computing, vol 349. Springer, Cham. https://doi.org/10.1007/978-3-319-48317-7_1

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