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Graded Decagon of Opposition with Fuzzy Quantifier-Based Concept-Forming Operators

We introduce twelve operators called fuzzy quantifier-based operators. They are proposed as a new tool to help to deepen the analysis of data in fuzzy formal concept analysis. Moreover, we employ them to construct a graded extension of Aristotle’s square, namely the graded decagon of opposition.


Introduction
Formal Concept Analysis (FCA) is a mathematical theory applied to the analysis of data (see [6]). The input of FCA is a triple called formal context that consists of a set of objects, a set of attributes, and a binary relation between objects and attributes. FCA techniques extract a collection of formal concepts from every formal context.
Formal concepts are special clusters that correspond to concepts such as "numbers divisible by 5", or "white roses in the garden". Fuzzy Formal Concept Analysis (FFCA) generalizes formal concept analysis to include also vague information. The input of FFCA is an L-context (X, Y, I) where L is a support of an algebra of truth values, X is a set of objects, Y a set of attributes, and I is a fuzzy relation I : A fuzzy concept is a pair (A, B) where A, B are fuzzy sets A : X −→ L, B : Y −→ L. A is called extent and it is a fuzzy set of all objects x ∈ X that have all attributes of B, and B is called intent and it is a fuzzy set of all attributes y ∈ Y being satisfied by all objects of A. Namely, A(x) is the degree to which "x has all attributes of B", and B(y) is the degree to which "the attribute y is satisfied by all objects of A".
In this article, we define twelve special operators as a tool to deepen the analysis of data. To explain their function, let us consider the following situation.
Let (X, Y, I) be an L-context, where X is a set of students, Y are their skills, I(x, y) is the degree to which "a student x has the skill y". Thus, given a fuzzy concept (A, B), we know that A is the fuzzy set representing all students with all skills in B.
Let us now ask, how many students share "almost all skills in B" ("most skills in B", or "many skills in B"). Additionally, we may need to classify students with respect to how many skills of B they do not have and exactly, to consider the following fuzzy sets of X: students sharing "few skills in B", students who do not have "most skills in B", or students "do not have many skills in B". Similarly, we can also consider a fuzzy set of Y formed of all skills shared by "almost all" ("most", "many", or "few") students of A, and the fuzzy set of Y made of all skills that are not shared by "most" (or "many") students in A. Each of the previous sets is generated by a fuzzy quantifier-based concept-forming operator, that allows us to introduce an extended notion of fuzzy concept.
Fuzzy quantifier-based operators are defined taking into account expressions of natural language extremely big, very big, and not small that are formalized within the theory of evaluative linguistic expressions [8]. Finally, starting from the Lukasiewicz MV-algebra, we employ the fuzzy quantifier-based operators to represent a graded decagon of opposition, which is a graded extension of Aristotle's square (see Fig. 1 The article is organized as follows. Section 2 reviews some basic notions and results regarding MV-algebras, fuzzy formal concept analysis, and the graded square of opposition. Section 3 introduces the fuzzy quantifier-based operators and the corresponding new notions of fuzzy concepts. In Sect. 4, we construct a graded decagon of opposition using the former. Finally, in the last section we discuss further possible development of our results.

Preliminaries
This section describes some fundamental notions and results regarding MValgebras, fuzzy formal concept analysis, and the graded square of opposition.

Definition 3 ([3,11]
). An MV-algebra is a residuated lattice We will also work with the following additional operations on L:

Example 1. A special MV-algebra is the standard Lukasiewicz MV-algebra
In the following lemma, we list some properties of complete MV-algebras 1 that will be used below.

Fuzzy Formal Concept Analysis
In this subsection, we recall the definition of two pairs of fuzzy concept-forming operators ( ↑ , ↓ ), and ( ∩ , ∪ ) existing in literature. Given a complete residuated lattice L, by a fuzzy set of the universe X we mean a function A : for all x ∈ X and y ∈ Y .
The A ↑ (y) and B ↓ (x) correspond to the truth degrees of the statements "an attribute y is shared by all objects of A" and "an object x has all attributes of B", respectively.
The operators ∩ and ∪ are borrowed from the rough set theory. Namely, A ∩ (y) and B ∪ (x) correspond to the truth degrees of the statements "an attribute y is shared by at least one object of A" and "an object x has no attributes outside B", respectively. Each

Theorem 2. The pair of mappings
Definition 6. Given a set X and a complete residuated lattice L, by a fuzzy preposet we mean a pair (X, R) where R is a fuzzy relation on X that is reflexive, for each x, y, z ∈ X.

Graded Square of Opposition and Fuzzy Concept-Forming Operators
In this subsection, we define graded square of opposition referring to [5], and we enunciate a theorem that shows how this square can be obtained using the fuzzy concept-forming operators introduced in Subsect. 2.2.

Definition 7.
Let P A and P B be properties represented by A, B ⊂ ∼ X, then we say that Definition 8. In a graded square of opposition the vertices A, E, I, and O are fuzzy sets representing the propositions P A , P E , P I , and P O such that the following conditions hold: 1. P A and P E are contraries; 2. P I and P O are sub-contraries; 3. P A and P I are sub-alterns, as well as P E and P O ; 4. P A and P O are contradictories, as well as P E and P I .
From now, given the L-contexts (X, Y, I), we suppose that L is the Lukasiewicz MV-algebra, because we will need the double negation law, i.e. ¬¬a = a for each a ∈ L. Moreover, we put (¬I)(x, y) = ¬I(x, y). In the standard Lukasiewicz algebra, ¬I(x, y) = 1 − I(x, y), for all x ∈ X and y ∈ Y .
This lemma follows from the results found in [5].

Lemma 2.
Let A ⊂ ∼ X be a normal fuzzy set 3 , then are the vertices of a graded square of opposition, and they represent proprieties that are in relation of contrary, sub-contrary, sub-altern, and contradictory as shown in Fig. 2.
Observe that we obtain the graded square of opposition defined in [4] when fixing y ∈ Y .

Fuzzy Quantifier-Based Operators
In this section, we introduce the fuzzy quantifier based-operators extending the notion of fuzzy concept. Our theory is based on the theory of intermediate quantifiers presented in [7,9] and elsewhere. The theory is based on the concept of evaluative linguistic expression. These are expressions of natural language such as "small, very big, rather medium", etc. In this paper we confine only to "not small", "very big" and "extremely big" and use a simplified model in which we consider only extensions in the (linguistic) context 0, 0.5, 1 4 that are fuzzy sets BiEx, BiVe, ¬Smν depicted in Fig. 4. For justification of this model, see [8,10].

Remark 1. It is clear that
The cardinality of A ⊂ ∼ X is defined by |A| = x∈X A(x). Furthermore, given A, B ⊂ ∼ X, we consider the following measure that expresses how large the size of A is w.r.t. the size of B (see [9]) For our further reasoning, we need a special operation called cut of a fuzzy set. It is motivated by the need to form a new fuzzy set from a given one by extracting several elements together with their membership degrees and putting the other membership degrees equal to 0.

Definition 9 ([7]). Let
The cut of A with respect to B is the fuzzy set Now, we give the definition of positive and negative fuzzy quantifier basedoperators that are based on the relation I, and on the functions ¬Smv, BiVe and BiEx. Our aim is to capture positive, or negative information in (X, Y, I).

(i) Positive fuzzy quantifier based-operators
and

(ii) Negative fuzzy quantifier based-operators
and Informal explanation of the formulas in Definition 10 is the following:  Since BiEx, BiVe and ¬Smv lay behind the definition of the intermediate quantifiers almost, most and many (cf. [9]), formulas A ↑ I,Ev (y), A ↑ ¬I,Ev (y), B ↓ I,Ev (x) and B ↓ ¬I,Ev (x) can be understood as interpretation of the linguistic expressions summarized in Table 1.  In this section, we introduce the definition of graded decagon of opposition, which is a generalization of the graded square of opposition given in Definition 8. Moreover, we construct a graded decagon of opposition using some fuzzy quantifier-based operators. The graded decagon of opposition is depicted in Fig. 5.
The following theorem shows that we can obtain a decagon of oppositions starting from our operators.  Fig. 6.
Proof. The proof follows by Theorem 3, Lemma 3, Lemma 4, Lemma 5, and Lemma 6. Then we obtain the graded decagon of opposition depicted in Fig. 7.

Future Directions
In this article, a graded decagon of opposition is introduced as a graded generalization of Aristotle's square, and it is constructed using some fuzzy quantifierbased operators. As future work, we intend to analyze more deeply the role that the fuzzy quantifier-based operators could have in fuzzy formal concept analysis. Moreover, fixed an evaluative linguistic expression Ev 1 , we will find another evaluative linguistic expression Ev 2 such that the pair of operators ↑ I,Ev1 and ↓ I,Ev2 forms a Galois connection. Finally, we would like to propose our operators as fuzzy generalizations of the scaling quantifiers used in Relational concept analysis [2].