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Modalities and Hedges in Unary Operators: A Theoretical Overview

Explore various representation theorems, modalities induced by negations, distributive properties, modifiers, and examples of modal operators applied on variables with a focus on modalities and hedges in a comprehensive presentation.

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Modalities and Hedges in Unary Operators: A Theoretical Overview

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  1. On a CertainType of Unary Operators József Dombi Brisbane, Australia 2012

  2. Outline of thepresentation

  3. Negation • Tworepresentationtheorems, Trillas • Representationtheorems, Dombi

  4. Negation and itsparameters

  5. Modalities • Necessity operator Dubois

  6. Possibility operator Dubois

  7. Modalitiesinducedbytwonegations

  8. General form of modalities

  9. Particularcase of negations and modalities

  10. Modalitiesbasedondistributiveproperties • Distributivityproperty of themodalities • The solution of theabovefunctionalequations is:

  11. Modalitiesbasedonconnectives

  12. Simultaneousdistributivity • Theorem:

  13. The sharpness operator • Aggregative operator (representableuninorm)

  14. The threetypes of modifiers

  15. General form of modifiers • Theorem: Negation, hedgeand sharpness of thegeneralformofthemodifiers

  16. Modalities and hedges • If we have a logical expression andbefore the logical expression there are modal operators, we can apply these modaloperators directly on the variables.WeinterpretthisactiononthemembershipfunctionasHedge.In this case, ”very true” is the same as ”necessarily true”, or ”who is very young”isthe same as ”he/she is in our opinion necessarily young”.

  17. Modalities and hedges: An example

  18. Charachterizationby a differentequation • Theorem:

  19. Kappafunctioninthe dombi operator case

  20. Kappafunctioninthe dombi operator case

  21. Pliant operator system

  22. THANK YOU FOR YOUR ATTENTION! dombi@inf.u-szeged.hu http://www.inf.u-szeged.hu/~dombi/

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