On a Certain Type of Unary Operators J´ ozsef Dombi University of Szeged Hungary Email: dombi@inf.u-szeged.hu Abstract—In our study we give a general form for modifiers that includes negation, different types of hedge and the sharpness operators. We will show that the four operators have a common form in the Pliant system and they will be called modifier operators. By changing the parameter value of a modifier we get the modalities, negation and the sharpness operators. Index Terms—modalities, sharpness operator, negation, Pliant system I. I NTRODUCTION The concept of hedge and modifiers appears at the very beginning of fuzzy set theory. They are related to an attempt to model meanings like ”very”, ”more or less”, ”somewhat”, ”rather” and ”quite”. A hedge modifies the shape of the fuzzy set, inducing a change in the membership function. Thus a hedge transforms one fuzzy set into another fuzzy set. Here we will deal with strictly monotonously increasing and decreasing membership function. A. Historical background In the early 1970s, Zadeh [1] introduced a class of powering modifiers. He proposed computing with words as an extension of fuzzy sets and logic theory (Zadeh [2]). As pointed out by Zadeh [3]–[5], linguistic variables and terms are closer to human thinking (which emphasise importance more than certainty) and are used in everyday life. For this reason, words and linguistic terms can be used to model human thinking systems. Zadeh [1] said that a proposition such as ”The sea is very rough” can be interpreted as ”It is very true that the sea is rough.” A number of studies [6], [7] have been conducted that discuss fuzzy logic and fuzzy reasoning with linguistic truth values. Basic notions of linguistic variables were formalized in different works by Zadeh in the mid 1970s [3]–[5]. These papers sought to provide a mathematical model for linguistic variables. II. THE PLIANT OPERATOR SYSTEM Here, we will be concerned with strict operators (strict t- norms and t-conorms). Using the general representation theorem, we have for the strict t-norm (conjunctive operator) and the strict t-conorm (disjunctive operator). (, )= −1 ( ()+ ()),(, )= −1 ( ()+ ()). (1) Here () : [0, 1] → [0, ∞] and ( () : [0, 1] → [0, ∞]) are continuous and strictly decreasing (increasing) monotone functions and they are the generator functions of the strict t-norms and strict t-conorms. Those familiar with fuzzy logic theory will find that the terminology used here is slightly different from that used in standard texts [8]–[12]. In the Pliant system, we look for a class of operators with infinitely many negation operators. Definition 1: If () ()=1, ∈ [0, 1] (2) then we call the generated connectives a Pliant system. Theorem 2: (, ) and (, ) build a DeMorgan system for ∗ () where ∗ ( ∗ )= ∗ for all ∗ ∈ (0, 1) if and only if () ()=1. (3) Proof: See [13]. The general form of the multiplicative Pliant system is (, )= −1 ( ( ()+ ()) 1/ ) (4) where () is the generator function of the strict t-norm operator and : [0, 1] → [0, ∞] is a continuous and strictly decreasing function. If > 0, than (, ) conjunctive operator (t-norm). If < 0, than (, ) disjunctive operator (t-conorm). (5) The corresponding negation operator is ()= −1 ( ( 0 ) ( ) () ) or (6) ∗ ()= −1 ( 2 ( ∗ ) () ) , (7) where ∗ is the fix point of the negation, i.e. ( ∗ )= ∗ and when we fix a certain 0 threshold then takes this value, i.e. ( )= 0 . A characterization of this operator class can be found in [13]. We can introduce the aggregative operator (uninorm) con- sistent with the conjunctive and disjunctive operators and 2038 U.S. Government work not protected by U.S. copyright WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia FUZZ IEEE