Graded dominance and related graded properties of fuzzy connectives

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Graded dominance and related graded properties of fuzzy connectives Libor Bˇehounek a,b , Ulrich Bodenhofer c , Petr Cintula a,∗ , Susanne Saminger-Platz d , Peter Sarkoci e a Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou vˇeží 2, 182 07 Prague, Czech Republic b NSC IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22,

701 03 Ostrava, Czech Republic c Institute of Bioinformatics, Johannes Kepler University Linz, Altenbergerstrasse 69, 4040 Linz, Austria d Dept. of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Altenbergerstrasse 69, 4040 Linz, Austria e Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia

Received 16 October 2013; received in revised form 24 February 2014; accepted 27 April 2014

Abstract Graded properties of binary and unary fuzzy connectives (valued in MTL -algebras) are studied, including graded monotony, a generalized Lipschitz property, commutativity, associativity, unit and null elements, and the dominance relation between fuzzy connectives. The apparatus of Fuzzy Class Theory (or higher-order fuzzy logic) is employed as a tool for easy derivation of graded theorems on the connectives. © 2014 Elsevier B.V. All rights reserved. Keywords: Fuzzy connective; Fuzzy relation; Dominance; Fuzzy Class Theory

1. Introduction Gradual properties, i.e., properties coming in degrees, pervade all areas of real-world applications, from medicine to engineering to information science. An efficient tool for manipulation with gradual properties is fuzzy set theory, which represents a gradual property by a fuzzy set, i.e., a function from some domain X to the real unit interval [0, 1] or another suitable algebra L of degrees. To aggregate two or more degrees of gradual properties, an n-ary function Ln → L, for n ≥ 2, is requisite; similarly, to modify a single degree of a gradual property, one uses a unary function L → L. Consequently, a theory of aggregation operators Ln → L (for n ≥ 1) that combine the degrees of gradual properties has been developed and extensively studied (see, e.g., [1,2]). Due to their resemblance to classical * Corresponding author.

E-mail addresses: [email protected] (L. Bˇehounek), [email protected] (U. Bodenhofer), [email protected] (P. Cintula), [email protected] (S. Saminger-Platz), [email protected] (P. Sarkoci). http://dx.doi.org/10.1016/j.fss.2014.04.025 0165-0114/© 2014 Elsevier B.V. All rights reserved.

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two-valued logical connectives (which combine bivalent rather than gradual properties), n-ary aggregation functions Ln → L are also known as fuzzy connectives. In this paper we restrict our attention to the most basic fuzzy connectives, namely unary and binary ones (L → L and L2 → L, respectively). Various properties (such as commutativity, associativity, monotonicity, etc.) and classes of fuzzy connectives (including, e.g., triangular norms and conorms, uninorms, copulas, etc.) have been studied in the literature (see, e.g., [3–5]). Being defined by formulae of ordinary—rather than fuzzy—mathematics, these properties of fuzzy connectives are themselves crisp: i.e., are either satisfied by a particular fuzzy connective, or not. For example, a binary fuzzy connective ∗: [0, 1]2 → [0, 1] is a t-norm if it is (i) commutative (i.e., x ∗ y = y ∗ x for all x, y ∈ [0, 1]), (ii) associative (i.e., (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z ∈ [0, 1]), (iii) monotone (i.e., x ∗ y ≤ x  ∗ y  for all x, x  , y, y  ∈ [0, 1] such that x ≤ x  and y ≤ y  ), and (iv) the value 1 acts as its neutral element (i.e., x ∗ 1 = 1 ∗ x = x for all x ∈ [0, 1]). Since the conditions (i)–(iv) are defined by crisp (in)equalities, the notion of a t-norm is crisp, rather than gradual: a function ∗: [0, 1]2 → [0, 1] either is or is not a t-norm. T-norms proved to be an important class of binary fuzzy connectives with a broad field of applications. The sundry theorems on t-norms known from the literature (see [3] or [6] for a comprehensive treatise), however, apply only to functions that satisfy the conditions (i)–(iv) unfalteringly, with absolute precision. If, on the other hand, a function ∗ violates any of the conditions (i)–(iv) by however small a margin, the theorems on t-norms do not say anything at all about ∗, as ∗ does not satisfy their premise that it be a t-norm. Thus, for instance, whenever a t-norm (e.g., the product t-norm x ∗ y = x · y) is just slightly distorted, e.g., by some random noise or rounding (e.g., is implemented in a computer with a fixed number of decimal places, however large the precision may be—e.g., 2−256 ), then strictly speaking the theorems on t-norms do not apply to the implemented function, since the function only approximates, rather than coincides with, a t-norm. It can be, though, reasonably expected that the properties of t-norms will be ‘almost satisfied’ by the function as well, as differs from a true t-norm only by a negligible variation. However, the standard theorems do not predict which properties, and to which extent, will be transmitted to the slightly perturbed function. The imperfect satisfaction of mathematical conditions and the transmission of provable properties to such functions comprises the study of defects of mathematical properties [7,8]. A specific approach to the defects of mathematical properties is one based on formal fuzzy logic. Indeed, graduality being a tenet of fuzzy set theory, the traditional crisp properties of fuzzy sets seem to contradict the very principles of the fuzzy approach: one would rather expect the properties of fuzzy sets to be defined by gradual conditions that can be satisfied to larger or lesser degrees. However, only a few works in fuzzy set theory have dealt with such graded properties of fuzzy structures: prominent early examples are Gottwald’s [9,10] and Bˇelohlávek’s [11] study of gradual properties of fuzzy relations and Ying’s study of a graded notion of fuzzy topological space [12]. A particularly suitable tool for the study of graded properties of fuzzy sets is formal fuzzy logic (see esp. [13,14]), as its formulae are inherently many-valued, and so the properties defined by means of formal fuzzy logic naturally come in degrees. Moreover, provable theorems of fuzzy logic can provide estimates of degrees of gradual properties inferred from imperfectly satisfied premises. Thus while traditional fuzzy set theory proves theorems of the following form: If the assumptions ϕ1 , . . . , ϕn are true (to degree 1) then the conclusion ψ is true (to degree 1), which is void if the assumptions ϕ1 , . . . , ϕn are only slightly violated, under the graded approach of formal fuzzy logic, theorems of the following form are proved: The larger the aggregated degree of the (graded) assumptions ϕ1 , . . . , ϕn , the larger the degree of the (graded) conclusion ψ. The former non-graded theorems of traditional fuzzy set theory are a special case (for the fully true assumptions) of the latter graded ones of formal fuzzy logic. The approach based on formal fuzzy logic thus automatically generalizes known traditional results. The approach based on formal fuzzy logic has already been successfully applied to graded properties of fuzzy relations in [15]. Here we propose an analogous approach to fuzzy connectives, which we consider equally important

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due to their role in aggregation of gradual data. Aggregating gradual properties by connectives characterized by gradual properties aims at preserving as much of the knowledge about the underlying structures as possible. E.g., it is well-known that in the classical setting, aggregating T -transitive relations with the aim to preserve the transitivity forces the applied aggregation function to dominate (in the classical sense) the t-norm involved [16,17]. However, we may face the situation that not all relations involved are of the same transitivity type; by allowing for gradual properties, we are not forced to focus on the weakest transitivity condition involved, but to treat weaker types of transitivity as gradual versions of stronger types of transitivity. Secondly, the demands of a particular application might force us to use a particular type of operation (e.g., some weighted mean) for carrying out the aggregation process. This operation might have some of the properties (like, e.g., associativity) just to a certain degree. Moving to a gradual framework for both—the properties of the relations involved and the connectives applied—allows us to still carry on with as much information as possible from the underlying structures to the newly constructed one. In order to have a more complete understanding of these impacts a detailed study of graded dominance has been carried out. In this paper we focus on gradual properties related to t-norms; nevertheless, the methods we demonstrate are clearly applicable as well to other properties and classes of fuzzy connectives. It will be seen that the logic-based approach is viable and yields a reasonable graded theory of fuzzy connectives and aggregation functions. (A brief sketch of the theory was published in the preliminary conference paper [18] by the present authors; the theory is further elaborated in [19].) Even though we do not have any particular application in mind, the above motivation suggests that the theory can be useful in many situations—namely whenever the perfect satisfaction of the defining conditions can be corrupted. Like in [15], we use formal fuzzy logic of a higher order (also known as Fuzzy Class Theory or FCT), as a tool for deriving general results on graded properties of fuzzy connectives. The readers interested only in the graded results will only need a cursory knowledge of the theory, at a level sufficient for ‘translation’ of its formulae into the more common semantic notation. The translation is briefly explained in Section 2; in more detail it can be found in [20] or [21]. Such readers can safely skip all formal proofs contained in this paper. The readers interested also in proofs and the formal aspects of the enterprise can find more details on the theory in [22], and on the underlying first-order fuzzy logic in [13,14,23]. General methodological remarks on the development of fuzzy mathematics based on formal fuzzy logic can be found in [24,25]. The paper is organized as follows: in Section 3 we deal with basic t-norm-related graded properties of fuzzy connectives—in particular, graded notions of monotony, commutativity, associativity, idempotence, and null and unit elements. Section 4 is devoted to a graded variant of the relation of dominance between fuzzy connectives (for the nongraded notion of dominance see [16,17,26]): after deriving general results in FCT, examples of particular estimates of dominance degrees are given. Finally, the notion of graded dominance is applied in the theory of fuzzy relations, generalizing the well-known result of [16] on the transmission of transitivity of fuzzy relations under dominance. 2. Preliminaries The general results of this paper are derived in the framework of higher-order fuzzy logic, also known as Fuzzy Class Theory (FCT). FCT is an axiomatic theory of Zadeh’s notions of fuzzy set [27] and fuzzy relation [28] in formal fuzzy logic (in the sense of [13,14]). For reference, a (slightly simplified) definition of FCT is given below; for more details see the original paper [22] or the freely available primer [20]. We shall use the variant of FCT over the logic MTL of all left-continuous t-norms [23], one of the weakest fuzzy logics suitable for this type of graded fuzzy mathematics. For details on the first-order logic MTL consult [14,23]; here we shall just briefly recall the standard semantics of its connectives and quantifiers over the real unit interval [0, 1]: & → ∧, ∨ ¬ ↔  ∀, ∃

... ... ... ... ... ... ...

any left-continuous t-norm ∗ the residuum ⇒∗ of ∗, defined as x ⇒∗ y =df sup{z | z ∗ x ≤ y} min, max ¬x =df x ⇒∗ 0 the bi-residuum ⇔∗ , defined as x ⇔∗ y =df (x ⇒∗ y) ∧ (y ⇒∗ x) x = 1 if x = 1; x = 0 otherwise inf, sup

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By means of this ‘dictionary’, the formal results formulated and proved in MTL can be translated into the more common semantic notation: for instance, the defining formula of Definition 3.1 on page 6 below,   Cng(u) =df (∀αβ) (α ↔ β) → (uα ↔ uβ) , expresses the following semantic definition of the degree of the fuzzy property Cng for a unary fuzzy connective u:     (α ⇔∗ β) ⇒∗ u(α) ⇔∗ u(β) , Cng(u) = α,β

for any given left-continuous t-norm ∗. In this manner, all formulae encountered in this paper can be understood as denoting the corresponding semantic facts about standard fuzzy sets and fuzzy relations. Recall further that since x ⇒∗ y equals 1 iff x ≤ y, theorems with implication as the principal connective express the comparison of degrees. Thus, e.g., in Claim (R9) of Theorem 5.6 on page 23 below, the formula Com(c) ⇒ Sym(R c R −1 ) expresses the semantic fact that the degree of Com(c) is less than or equal to the degree of Sym(R c R −1 ). (Notice that sometimes we use the sign ⇒ for → and ⇔ for ↔, by Convention 2.1 below.) Similarly theorems with ↔ as the principal connective express the identity of degrees; so for example Claim (C1) of Theorem 3.2 on page 7 below, MonCng(u) ⇔ Mon(u) ∧ Cng(u), expresses the fact that the degree of MonCng(u) equals the minimum of the degrees of Mon(u) and Cng(u). For more details on the meaning of formulae in MTL and FCT see [20,21]. Convention 2.1. For better readability of complex formulae, we shall alternatively use the comma (,) for &; the symbol ⇒ for →; and ⇔ for ↔. The symbols ⇒ and ⇔ can be chained, with ϕ1 ⇒ ϕ2 ⇒ . . . ⇒ ϕn representing the formula (ϕ1 → ϕ2 ) & (ϕ2 → ϕ3 ) & . . . & (ϕn−1 → ϕn ), and similarly for ⇔. The sign =df is employed to introduce a new abbreviation. By convention, the symbols ⇒ and ⇔ will have the lowest priority in formulae and the comma the second lowest priority. Of other symbols, → and ↔ will have lower priority than other binary connectives, and quantifiers and unary connectives will have the highest priority. We shall write ϕ n for ϕ & . . . & ϕ (n times). In atomic subformulae, the superscript can be attached directly to the predicate, e.g., Com2 (c) or c ≈3 d. Furthermore we shall employ the following defined connectives that express the ordering and equality of truth degrees: ϕ ≤ ψ =df (ϕ → ψ) ϕ = ψ =df (ϕ ↔ ψ) The priority of these connectives is the same as that of implication. Fuzzy class theory FCT, or Henkin-style higher-order fuzzy logic MTL , is an axiomatic theory over multi-sorted first-order logic MTL , with sorts of variables for: • • • •

Atomic elements, denoted by lowercase letters x, y, . . . Fuzzy classes1 of atomic elements, denoted by uppercase letters A, B, . . . Fuzzy classes of fuzzy classes of atomic elements, denoted by calligraphic letters A, B, . . . Etc.; in general for fuzzy classes of the n-th order, written as X (n) , Y (n) , . . .

If the order n of a variable is irrelevant, we shall use just lowercase letters x, y, . . . (the uppercase letters A, B, . . . then denote variables of the (n + 1)-st order, etc.). The primitive symbols of FCT are: • The identity predicate = on each sort2 1 For certain formal reasons, in FCT we use the term fuzzy class besides the more common fuzzy set (for details on this distinction, irrelevant in the present paper, see, e.g., [15, Section 2.1]). 2 Note that we use the same symbol for the identity predicate = of FCT and the equality connective = defined in Convention 2.1. This ambiguity is harmless, as the context always determines the meaning of this symbol: the connective occurs between formulae, while the predicate occurs between terms. Semantically, both denote equality (of individuals in the case of the predicate and of truth values in the case of the connective).

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• The membership predicate ∈ between successive sorts • The symbol for tuples x1 , . . . , xk  of individuals x1 , . . . , xk of any order, for all arities k ∈ N The formula x ∈ A and the term x1 , . . . , xk  may be abbreviated, respectively, as Ax and x1 . . . xk . FCT has the following axioms, for all formulae ϕ and variables of any order: • • • • •

The logical axioms of multi-sorted first-order logic MTL The identity axioms: x = x, and x = y → (ϕ(x) ↔ ϕ(y)) The tuple-identity axioms: x1 , . . . , xk  = y1 , . . . , yk  → xi = yi , for all k ∈ N and 1 ≤ i ≤ k The comprehension axioms: (∃A)(Ax = ϕ(x)) The extensionality axioms: (∀x)(Ax = Bx) → A = B

The axioms for identity entail that the identity predicate = on each sort is crisp (while the membership predicate ∈ can in general be fuzzy). Notice that due to the logical axioms, theorems of FCT need be proved by the rules of the logic MTL rather than classical logic. The models of FCT are systems of fuzzy sets and fuzzy relations of all finite arities and orders over a fixed crisp set X (the universe of discourse) that are closed under all FCT-definable operations and whose membership degrees take values in any fixed MTL -chain (standardly, the real unit interval [0, 1] equipped with a left-continuous t-norm). In particular, all theorems of FCT are valid for the system of all [0, 1]-valued fuzzy sets and fuzzy relations (of all finite arities and orders)—i.e., for the so-called Zadeh models of FCT [22]. Convention 2.2. By standard Łukasiewicz logic we mean Zadeh models of FCT valued in the standard MV-algebra (i.e., in the interval [0, 1] equipped with the Łukasiewicz t-norm x ∗Ł y = max(0, x + y − 1)). Semantic examples will usually be given in these models, i.e., in standard Łukasiewicz logic.3 Definition 2.3. In FCT, we introduce the following graded properties of and relations between fuzzy classes: A ⊆ B =df (∀x)(Ax → Bx)

inclusion

A  B =df (A ⊆ B)

crisp inclusion

A ≈ B =df (∀x)(Ax ↔ Bx)

weak bi-inclusion

A  B =df (A ⊆ B) & (B ⊆ A)

strong bi-inclusion

Hgt A =df (∃x)Ax

height

Plt A =df (∀x)Ax

plinth

Crisp A =df (∀x)(Ax ∨ ¬Ax)

crispness

Moreover, we define the following operations on fuzzy classes: Ak x1 . . . xk =df Ax1 & . . . & Axk R

−1

xy =df Ryx

Cartesian power converse relation

Finite fuzzy classes {α1 /a1 , . . . , αn /an } are defined as follows:     x ∈ {α1 /a1 , . . . , αn /an } =df (x = a1 ) & α1 ∨ . . . ∨ (x = an ) & αn . In this paper we deal with fuzzy connectives, i.e., algebraic operations on truth degrees. Even though truth degrees are not part of the primitive language of FCT, they can be represented in the theory by subclasses of a crisp singleton. The details of the representation (for which see [29, Section 3] and [19]) are not important for our present purpose; 3 Since the semantics of both the identity predicate and the equality connective is the metatheoretical identity (of objects or truth values), we do not distinguish between the object-level and meta-level equality symbols. Similarly we denote the truth value of a formula by the formula itself in the semantic examples.

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we shall thus simply assume that variables α, β, . . . for truth values are at our disposal in FCT, and that the ordering of truth values and the usual propositional connectives and the quantifiers ∀, ∃ are definable in FCT. The crisp class of the represented truth values will be denoted by L. By n-ary fuzzy connectives we understand n-ary operations on truth degrees i.e., crisp functions c: Ln → L. Being functions into L, they can equivalently be regarded as fuzzy relations on Ln ; i.e., c  Ln . Thus, e.g., fuzzy inclusion c ⊆ d of binary fuzzy connectives c, d will be understood as inclusion of fuzzy relations c, d  L2 , i.e., c ⊆ d should be understood as (∀αβ)(cαβ → dαβ), rather than as inclusion of crisp functions c, d: L2 → L. Similarly, the converse fuzzy connective is defined as c−1 αβ =df cβα (cf. Definition 2.3 of converse fuzzy relation). Convention 2.4. We shall always use lowercase Greek letters for truth values, the letters u, v, w for unary connectives, and the letters a, b, c, . . . for binary connectives. The infix notation α c β will usually be employed for binary connectives instead of the prefix notation cαβ. In formulae, infix binary connectives will by convention have the same priority as & : thus, e.g., ¬α c β → γ will mean ((¬α) c β) → γ . We will use the following notation for fuzzy connectives: • The composition of a unary connective u: L → L with an n-ary connective f: Ln → L will be denoted by uf: Ln → L, for n ≥ 0. • The identity on L will be denoted by id: L → L; i.e., id α = α for all α ∈ L. 3. Graded properties of unary and binary connectives Here we present the basics of the graded theory of fuzzy connectives. We focus on results needed for the main topic of this paper, the graded notion of dominance relation. For a further elaboration of the theory of graded properties of fuzzy connectives, including illustrative examples, see [19]. 3.1. Graded properties of unary connectives Graded properties of unary connectives can be obtained by ‘fuzzification’ (consisting in replacement of the crisp comparisons ≤, = by the graded operations →, ↔ in defining formulae) of usual crisp properties of functions: this method usually yields meaningful graded properties, and therefore will in this paper be employed systematically to obtain graded versions of well-known crisp properties of fuzzy connectives. Definition 3.1. In FCT, we define the following graded properties of a unary connective u  L:   Mon(u) =df (∀αβ) (α ≤ β) → (uα → uβ) Graded monotony   Graded congruence w.r.t. ↔ Cng(u) =df (∀αβ) (α ↔ β) → (uα ↔ uβ) Analogously to monotony we could define the property of antitony, defined as (∀αβ)((α ≤ β) → (uβ → uα)). We shall not study it in this paper due to its little relevance to the properties of dominance. For more details about this property see [19]. The graded property Cng(u) measures, roughly speaking, the degree to which u yields close values for close arguments, where closeness is evaluated in the sense of ↔. In particular, in models of FCT over standard Łukasiewicz logic, where ↔ corresponds to the Euclidean distance, the property  Cng(u) expresses the 1-Lipschitz property of the function u. The property will play an important role in many graded theorems on fuzzy connectives, as it denotes the largest guaranteed degree of intersubstitutivity of uα and uβ for close (in the sense of ↔) arguments α and β. In models of FCT, the crisp condition  Mon(u) satisfies    Mon(u) = (∀αβ) (α ≤ β) → (uα ≤ uβ) , and thus expresses the usual crisp property of monotony of u w.r.t. the ordering ≤ on L. Its graded version Mon(u) arises from replacing the second occurrence of ≤ by →. Notice that replacing both ≤’s in  Mon(u) by → would not yield a graded generalization of monotony, as the resulting notion   MonCng(u) =df (∀αβ) (α → β) → (uα → uβ)

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does not coincide with crisp monotony when fully true, but rather is stronger—cf. the following Theorem 3.2, according to which  MonCng(u) is not equivalent to  Mon(u), but rather to the conjunction of  Mon(u) and  Cng(u) (which is not equivalent to  Mon(u) by Example 3.3). However, Theorem 3.2 also shows that this notion is superfluous and can be characterized in terms of Mon and Cng: Theorem 3.2. FCT proves the following graded theorem: (C1) MonCng(u) ⇔ Mon(u) ∧ Cng(u) Proof. From left to right: First observe that trivially MonCng(u) → Mon(u), as (α → β) → (α → β). Second, from MonCng(u) we get both (α ↔ β) ⇒ (α → β) ⇒ (uα → uβ) and (α ↔ β) ⇒ (β → α) ⇒ (uβ → uα). Thus (α ↔ β) ⇒ (uα → uβ) ∧ (uβ → uα) ⇔ (uα ↔ uβ), and the rest is simple. For the converse direction we take the crisp cases α ≤ β and β ≤ α, which are exhaustive due to the prelinearity axiom of MTL. For α ≤ β we obtain (α → β) ⇒ (α → β) ⇒ (uα → uβ) by Mon(u). For β ≤ α we get (α → β) ⇒ (α ↔ β) ⇒ (uα → uβ) by Cng(u). Thus Mon(u) ∧ Cng(u) → MonCng(u). 2 Example 3.3. To demonstrate the difference between the three properties consider standard Łukasiewicz models of FCT and the connectives u1 (α) = min(α, 1 − α) and u2 (α) = min(2α, 1). Clearly Cng(u1 ) = Mon(u2 ) = 1 (as u1 is fully 1-Lipschitz and u2 is fully monotone), but an easy computation shows that Cng(u2 ) = Mon(u1 ) = MonCng(u1 ) = MonCng(u2 ) = 12 only. The following easy theorems show how the graded properties of unary connectives are transmitted to connectives which are close in the sense of ≈ or . The results of Theorem 3.4 actually follow from a general metatheorem [30, Theorem 3.5]. Since, however, the proof of the metatheorem is omitted in [30], and also in order for the present paper to be self-contained, we give direct proofs of the particular claims here. (The same remark applies also to further theorems on preservation under ≈ or  in this paper.) Theorem 3.4. FCT proves: (C2) Cng(u), u ≈2 v ⇒ Cng(v) (C3) Mon(u), u  v ⇒ Mon(v) Proof. (C2) (C3)

α ↔ β implies uα ↔ uβ by Cng(u), whence vα ↔ vβ by u ≈2 v (as uα ↔ vα by u ≈ v and uβ ↔ vβ by u ≈ v). α ≤ β implies uα → uβ by Mon(u); vα → uα by v ⊆ u; and uβ → vβ by u ⊆ v. The transitivity of implication then completes the proof. 2

Example 3.5. To illustrate the graded notions introduced in this paper we shall employ several unary fuzzy connectives representing some basic types of distortion of the identity function id: random noise, rounding, and a uniform shift. Variants of these connectives have already been employed in [19]; examples in the subsequent sections will make use of them to modify sample binary fuzzy connectives. For simplicity we use the fixed amplitude 0.1 for the noise, rounding, and shift; a parametrical generalization to arbitrary amplitudes is straightforward. Let us work in standard Łukasiewicz logic (see Convention 2.2) and define the following fuzzy connectives4 : ws (α) =df (α + 0.1) ∧ 1 10 · α + 0.5 wr (α) =df 10 4 Since we are working in a standard semantic model of FCT, the connectives need not be defined by formulae of FCT, but can refer to any operations on [0, 1].

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wn (α) =df

   rand(α) − 0.5 ∨0 ∧1 α+ 10

where rand(α) assigns a random number from [0, 1] to each α ∈ [0, 1]. Clearly, ws slightly shifts the identity function id upwards; wr rounds id to one decimal place; and wn adds to id a random noise with the amplitude 0.1. In standard Łukasiewicz logic, the values of the graded properties Mon and Cng for these functions can be computed rather easily: let us, for example, compute the value of Mon(wn ). By the semantic definitions of logical connectives and quantifiers in standard Łukasiewicz logic,        1 ∧ 1 − wn (α) + wn (β) = 1 ∧ 1 − wn (α) − wn (β) . Mon(wn ) = α≤β

α≤β

Now since the amplitude of the noise in wn is 0.1, the value of wn (α) − wn (β) can be at most 0.1 if α ≤ β. And since the  noise is random, for some α < β the value wn (α) − wn (β) will be arbitrarily close to 0.1 with probability 1. Thus α≤β (wn (α) − wn (β)) = 0.1 and Mon(wn ) = 1 ∧ (1 − 0.1) = 0.9 with probability 1. The values of further graded properties of id, ws , wr , and wn are computed similarly and are summarized in the following table5 : w

Mon(w)

Cng(w)

id ⊆ w

w ⊆ id

w ≈ id

w  id

id ws wr wn

1 1 1 0.9

1 1 0.9 0.9

1 1 0.95 0.95

1 0.9 0.95 0.95

1 0.9 0.95 0.95

1 0.9 0.9 0.9

We can compare the values with the estimates provided by Theorem 3.4. By (C2), Cng(ws ) ≥ Cng(id) ∗Ł (ws ≈ id)2 = 1 ∗Ł (0.9 ∗Ł 0.9) = 0.8. The estimate is not tight, as in fact Cng(ws ) = 1 (since the function ws is 1-Lipschitz). Similarly (C2) provides the estimates Cng(wr ) ≥ 0.9 and Cng(wn ) ≥ 0.9, which are tight. Analogously (C3) yields the estimates Mon(ws ) ≥ 0.9 (not tight), Mon(wr ) ≥ 0.9 (not tight), and Mon(wn ) ≥ 0.9 (tight). It can be seen that some of the estimates provided by Theorem 3.4 are rather loose. This is because theorems of this form give general lower bounds for all fuzzy connectives, including the ‘worst cases’. As shown by the tight estimates above, Theorem 3.4 cannot be strengthened by using just u ≈ v instead of (u ≈ v)2 in (C2) or instead of u  v in (C3). 3.2. Graded congruence and monotony of binary connectives We now turn to binary connectives. First we shall discuss the properties of congruence and monotony, which are analogous to the unary case. We define a separate variant for each argument, as well as for both arguments at once: Definition 3.6. In FCT, we define the properties of congruence and monotony for a binary fuzzy connective c  L2 as follows:   LCng(c) =df (∀αβγ ) (α ↔ β) → (α c γ ↔ β c γ )   RCng(c) =df (∀αβγ ) (α ↔ β) → (γ c α ↔ γ c β)   Cng(c) =df (∀αβγ δ) (α ↔ β) & (γ ↔ δ) → (α c γ ↔ β c δ)   LMon(c) =df (∀αβγ ) (α ≤ β) → (α c γ → β c γ )   RMon(c) =df (∀αβγ ) (α ≤ β) → (γ c α → γ c β)   Mon(c) =df (∀αβγ δ) (α ≤ β) & (γ ≤ δ) → (α c γ → β c δ) For convenience, we also define the abbreviation LRCng(c) =df LCng(c) & RCng(c), and similarly for monotony. 5 Throughout the paper, we always refer to what happens with probability 1 in the case of w . n

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Let us first study the interplay of the just defined variants of congruence and monotony: Theorem 3.7. FCT proves: (C4) Mon2 (c) ⇒ LRMon(c) ⇒ Mon(c) ⇒ LMon(c) ∧ RMon(c) (C5) Cng2 (c) ⇒ LRCng(c) ⇒ Cng(c) ⇒ LCng(c) ∧ RCng(c) Proof. We shall only prove (C4); the proof of (C5) is analogous. First we shall prove the second implication. By specification, LMon(c)

⇒

(α ≤ β) → (α c γ → β c γ )

RMon(c)

⇒

(γ ≤ δ) → (β c γ → β c δ).

Combining these formulae we obtain LRMon(c) ⇒ (α ≤ β) & (γ ≤ δ) → ((α c γ → β c γ ) & (β c γ → β c δ)). The transitivity of implication (applied to the consequent of the latter formula) then completes the proof. To prove the third implication just observe that setting γ = δ in the definition of Mon(c) yields: Mon(c)

⇒ (α ≤ β) & (γ ≤ γ ) → (α c γ → β c γ )

As clearly γ ≤ γ is a theorem of FCT, the proof is done. The first implication is a simple corollary of the third one.

2

The fact that none of the implications in Theorem 3.7 can in general be reversed is demonstrated in [19]. Observation 3.8. Let cγ (α) =df c(α, γ ) and cγ (α) =df c(γ , α) for all α ∈ L. Then FCT obviously proves: (C6) LCng(c) ⇔ (∀γ ) Cng(cγ ) (C7) RCng(c) ⇔ (∀γ ) Cng(cγ ) and analogously for LMon and RMon. Like in the unary case, the analogue of LCng featuring implication in place of equivalence, i.e.,   LMonCng(c) =df (∀αβγ ) (α → β) → (α c γ → β c γ ) , can again be reduced to min-conjunction of LCng and LMon: Theorem 3.9. FCT proves: (C8) LMonCng(c) ⇔ LMon(c) ∧ LCng(c) Proof. Let cγ (α) =df c(α, γ ) for all α ∈ L. Then the following chain of equivalences is provable in FCT:   (∀αβγ ) (α → β) → (α c γ → β c γ ) ⇔ (∀γ ) MonCng(cγ )   ⇔ (∀γ ) Mon(cγ ) ∧ Cng(cγ )

by Theorem 3.2

⇔ (∀γ ) Mon(cγ ) ∧ (∀γ ) Cng(cγ )

by quantifier distribution

⇔

by definition. 2

LMon(c) ∧ LCng(c)

by an analogue to Observation 3.8

The same claim can obviously be proved for the right-sided variants. However, the analogous reduction of the both-sided property   MonCng(c) =df (∀αβγ δ) (α → β) & (γ → δ) → (α c γ → β c δ)

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10

to the both-sided properties Mon(c) and Cng(c) does not hold; see [19] for a counterexample. The proofs of the following theorems are analogous to the proof of Theorem 3.4. Theorem 3.10. FCT proves: (C9) (C10) (C11) (C12)

LCng(c), c ≈2 d ⇒ LCng(d), and analogously for RCng LMon(c), c  d ⇒ LMon(d), and analogously for RMon Cng(c), c ≈2 d ⇒ Cng(d) Mon(c), c  d ⇒ Mon(d)

Example 3.11. In standard Łukasiewicz logic, consider the Łukasiewicz t-norm tŁ : [0, 1]2 → [0, 1], defined as αtŁ β = (α + β − 1) ∨ 0 for all α, β ∈ [0, 1]. Clearly Mon(tŁ ) = Cng(tŁ ) = 1, as tŁ is monotone and 1-Lipschitz in both arguments. Consider furthermore the composition of tŁ with the connectives ws , wr , and wn of Example 3.5. The resulting connectives ws tŁ , wr tŁ , wn tŁ : [0, 1]2 → [0, 1] distort tŁ , respectively, by the upward shift, rounding, and random noise with the amplitude 0.1 (see Fig. 1 for their graphs). The following table gives the values of congruence and monotony for these connectives in standard Łukasiewicz logic: c

c ≈ tŁ

c  tŁ

LMon(c)

LCng(c)

Mon(c)

Cng(c)

tŁ ws tŁ wr tŁ wn tŁ

1 0.9 0.95 0.95

1 0.9 0.9 0.9

1 1 1 0.9

1 1 0.9 0.9

1 1 1 0.9

1 1 0.9 0.9

Calculating the values in the table in standard Łukasiewicz logic is easy in all displayed cases. Let us, nevertheless, show how the theorems presented in this section help determine them, on the example of Mon(wn tŁ ). By Observation 3.8 we obtain LMon(wn tŁ ) = 0.9, since for every γ ∈ [0, 1], the cross-section (wn tŁ )γ clearly has Mon((wn tŁ )γ ) = 0.9 due to the amplitude 0.1 of the noise wn . By symmetry we then have RMon(wn tŁ ) = 0.9, too. Thus by (C4) we obtain Mon(wn tŁ ) ≥ LMon(wn tŁ ) ∗Ł RMon(wn tŁ ) = 0.9 ∗Ł 0.9 = 0.8. This estimate, however, is not tight, as (C12) yields Mon(wn tŁ ) ≥ Mon(tŁ ) ∗Ł (wn tŁ  tŁ ) = 1 ∗Ł 0.9 = 0.9. This is the exact value, since by (C4), Mon(wn tŁ ) ≤ LMon(wn tŁ ) = 0.9. Again, the tight estimates such as this show that the claims of Theorem 3.10 cannot be improved by replacing  or ≈2 by ≈. 3.3. Graded null and unit elements Graded generalizations of unit and null elements can again be obtained by replacing = in classical definitions by ↔. Definition 3.12. In FCT, we define the following graded properties of a binary connective c  L2 and a truth value η ∈ L:   Unit(c, η) =df (∀αβ) (η c α ↔ α) & (β c η ↔ β) Unit element   Null(c, η) =df (∀αβ) (η c α ↔ η) & (β c η ↔ η) Null element It is possible to study also the left- and right-argument variants of the “unit-ness” and “null-ness” degree, as done in [19]. Since these variants turn out to play a less significant rôle in theorems on graded dominance, we employ just the both-side variants here. In the following theorem, Claims (C13) and (C14) show a graded uniqueness of unit and null elements; (C15) shows a graded incompatibility of the properties of being a unit and a null of the same connective; and (C16) is a graded version of the well-known fact that if a monotone connective has the unit 1, then it has the null 0.

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11

Fig. 1. Graphs of the fuzzy connectives introduced in Example 3.11. Upper row: the Łukasiewicz t-norm tŁ (left) and its upper shift ws tŁ (right). Lower row: wr tŁ (left) and wn tŁ (right).

Theorem 3.13. FCT proves: (C13) (C14) (C15) (C16)

Null(c, η), Null(c, ζ ) ⇒ (η ↔ ζ )2 Unit(c, η), Unit(c, ζ ) ⇒ (η ↔ ζ )2 Null(c, η), Unit(c, η) ⇒ 0 LRMon(c), Unit(c, 1) ⇒ Null(c, 0)

Proof. (C13)

(C15)

(C16)

The claim follows from the following implications (and the transitivity of ↔): Null(c, η)

⇒

η c ζ ↔ η,

ζ cη↔η

Null(c, ζ )

⇒

η c ζ ↔ ζ,

ζ cη↔ζ

The proof of (C14) is analogous. The claim follows from the following implications, the transitivity of ↔, and the fact that (1 ↔ 0) = 0: Unit(c, η)

⇒

η c 0 ↔ 0,

1cη↔1

Null(c, η)

⇒

η c 0 ↔ η,

1cη↔η

Clearly 0 → 0 c α and 0 → α c 0 are theorems. The converse implications are derived as follows: first we obtain 0 c α → 0 c 1, α c 0 → 1 c 0 by LRMon(c) and the theorem α ≤ 1. Then Unit(c, 1) and the transitivity of implication completes the proof. 2

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12

The following theorem shows the congruence of the unit and null elements w.r.t. closeness of the connectives and truth degrees (in the sense of ≈ or  and ↔, resp.). Theorem 3.14. FCT proves: (C17) (C18) (C19) (C20)

Unit(c, η), c ≈2 d ⇒ Unit(d, η) Null(c, η), c ≈2 d ⇒ Null(d, η) Unit(c, η), LRCng(c), (η ↔ ζ )2 ⇒ Unit(c, ζ ) Null(c, η), LRCng(c), (η ↔ ζ )4 ⇒ Null(c, ζ )

Proof. The proof of the first two claims is straightforward. (C19) (C20)

By LRCng(c) and (η ↔ ζ )2 we obtain η c α ↔ ζ c α, β c η ↔ β c ζ . An application of Unit(c, η) then completes the proof. By LRCng(c) and (η ↔ ζ )2 we obtain η c α ↔ ζ c α, β c η ↔ β c ζ . Then by Null(c, η) we obtain η ↔ ζ c α, η ↔ β c ζ and the proof is concluded by using (η ↔ ζ )2 . 2

Observe that the double occurrence of η in the defining formula of null elements causes in (C20) a greater sensitivity of the lower estimate of the “nullness degree” to small (in the sense of ↔) changes in η, compared to the sensitivity of the lower estimate of the “unitness degree” to the same changes in η. Besides illustrating the graded notions of unit and null elements, the following example demonstrates the tightness of the general estimates given by the previous theorems. Example 3.15. Consider the Łukasiewicz t-norm tŁ and its compositions with the connectives ws , wr , and wn (see Example 3.11). It can be easily computed that in standard Łukasiewicz logic, Unit(tŁ , η) = η2 (i.e., η∗Ł η) and Null(tŁ , η) = (¬η)2 , for all η ∈ [0, 1]. Thus, for instance, 0.9 is still a 0.8-unit (and 0.1 a 0.8-null) of the Łukasiewicz t-norm in standard Łukasiewicz logic.6 Similarly we can compute, e.g., the degrees to which 1 is a unit and 0 is a null for tŁ distorted by a 0.1-large shift, rounding, and noise: Unit(ws tŁ , 1) = Null(ws tŁ , 0) = 0.92 = 0.8 and Unit(wr tŁ , 1) = Unit(wn tŁ , 1) = 0.952 = 0.9 in standard Łukasiewicz logic. 3.4. Graded idempotence, commutativity, and associativity In this section we shall investigate graded versions of the properties of idempotence, commutativity, and associativity of binary connectives. Definition 3.16. In FCT, we define the following graded properties of a binary connective c: L2 → L: Idem(c) =df (∀α)(α c α ↔ α)

Idempotence

Com(c) =df (∀αβ)(α c β ↔ β c α)     Ass(c) =df (∀αβγ ) (α c β) c γ ↔ α c (β c γ )

Commutativity Associativity

These graded properties again arise by replacing the (crisp) = in the defining formulae of the corresponding crisp properties by the (graded) equivalence connective of fuzzy logic. Notice that by (C21) of Theorem 3.17 below, it is immaterial whether we define graded commutativity with implication or equivalence. (C22) of Theorem 3.17 gives a similar result for the associativity of sufficiently commutative connectives. Observe also that the commutativity of c: L2 → L (i.e., c regarded as a crisp binary operation on L) is in fact the graded symmetry (see [10, Definition 18.6.1]) of c  L2 (i.e., c regarded as a binary fuzzy relation on L). 6 If separate degrees of left-unit and right-unit elements were considered, a more intuitive result that 0.9 is a left unit of t to degree 0.9 (and Ł similarly for null elements) would be obtained. However, since the elements are required to act as both-side units and nulls in Definition 3.12, only the degree of 0.9 ∗Ł 0.9 = 0.8 is obtained for Unit(tŁ , 0.9) in standard Łukasiewicz logic.

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13

(C24) and (C25) of Theorem 3.17 show that partial commutativity and associativity are abundant: in particular, all connectives with less than full “difference” (in the sense of →) between their height and plinth are at least partially commutative and associative: thus, e.g., all subnormal connectives in Łukasiewicz models have non-zero degrees of commutativity and associativity. Theorem 3.17. FCT proves: (C21) (C22) (C23) (C24) (C25)

Com(c) ⇔ (∀αβ)(α c β → β c α) Com2 (c) ⇒ Ass(c) ↔ (∀αβγ )((α c β) c γ ) → (α c (β c γ )) Com(c) ⇔ c ≈ c−1 ⇔ c ⊆ c−1 Hgt(c) → Plt(c) ⇒ Com(c) Hgt(c) → Plt(c) ⇒ Ass(c)

Proof. (C21)

(C22) (C23) (C24)

(C25)

The left-to-right direction holds trivially (a fortiori). Conversely, (∀αβ)(α c β → β c α) implies by specification α c β → β c α as well as β c α → α c β, and so it implies (α c β → β c α) ∧ (β c α → α c β), i.e., (α c β ↔ β c α), whence (∀αβ)(α c β → β c α) → Com(c) is obtained by generalization. One direction of the claim holds a fortiori. To obtain the converse direction, use commutativity twice to rearrange the arguments on both sides of implication. The first equivalence follows directly from the definition and the second from (C21). First observe that Hgt(c) → Plt(c), i.e., (∃αβ)(α c β) → (∀α  β  )(α  c β  ), is equivalent to (∀αβα  β  ) (α c β → α  c β  ) by quantifier shifts valid in MTL. Then specify β for α  and α for β  in the latter formula to obtain Com(c) by (C21). Proceed as in (C24), only specify α c β for α; γ for β; α for α  ; and β c γ for β  . 2

As expected, commutativity converts left-properties to right-properties and vice versa. The following theorem shows that these conversions are ensured if graded commutativity is assumed twice. Theorem 3.18. FCT proves: (C26) Com2 (c) ⇒ LCng(c) ↔ RCng(c) (C27) Com2 (c) ⇒ LMon(c) ↔ RMon(c) Proof. (C26)

It is sufficient to derive LCng(c) → RCng(c), as the converse follows by symmetry and the equivalence is obtained by min-conjunction. The proof follows easily from the fact that α c γ ↔ β c γ implies γ c α ↔ γ c β by applying Com(c) to both sides of the former equivalence.

The proof of (C27) is analogous.

2

The following theorem shows the transmission of these properties to connectives that are close in the sense of ≈ or : Theorem 3.19. FCT proves: (C28) Idem(c), c ≈ d ⇒ Idem(d) (C29) Com(c), c  d ⇒ Com(d) (C30) Ass(c1 ), c1 ≈4 c2 , LCng(ci ), RCng(cj ) ⇒ Ass(c2 ), for i, j ∈ {1, 2}

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Proof. (C28) (C29)

By respectively using c ≈ d and Idem(c) we obtain: α d α ⇔ α c α ⇔ α. By d ⊆ c, Com(c), and c ⊆ d, respectively, we obtain the following chain of implications: αdβ

(C30)

⇒ αcβ

⇒

β cα

⇒

β dα

Observe that Com(c) in the form of (C21) has to be used, as we would only obtain ≈2 rather than  in (C29) from the form of Definition 3.16. The case i = j = 1 is proved by the following chain of equivalences: (α c2 β) c2 γ

⇔

(α c2 β) c1 γ

by c1 ≈ c2

⇔

(α c1 β) c1 γ

by c1 ≈ c2 and LCng(c1 )

⇔

α c1 (β c1 γ )

by Ass(c1 )

⇔

α c1 (β c2 γ )

by c1 ≈ c2 and RCng(c1 )

⇔

α c2 (β c2 γ )

by c1 ≈ c2

The other cases only differ in the order of replacing c1 and c2 , which determines whether the left and right congruence of c1 or c2 is used. 2 The assertions of the following theorem are generalizations of well-known basic properties of t-norms (see, e.g., [3]). (C31) of Theorem 3.20 corresponds to the fact that the minimum is the greatest (so-called strongest) tnorm. In combination with (C31), (C32) of Theorem 3.20 generalizes the basic fact that the minimum is the only idempotent t-norm. Theorem 3.20. FCT proves: (C31) Mon(c), Unit(c, 1) ⇒ c ⊆ ∧ (C32) Idem(c), LMon(c) ∧ RMon(c) ⇒ ∧ ⊆ c Proof. (C31)

First we derive α c β → β by Mon(c) and Unit(c, 1): αcβ

(C32)

⇒

1cβ

by Mon(c) and the theorem α ≤ 1

⇒

β

by Unit(c, 1).

Analogously we derive α c β → α. Thus α c β → α ∧ β by the theorem’s premises. By prelinearity, we can take two crisp cases, α ≤ β and β ≤ α. If α ≤ β, we have: α∧β

⇒

α

⇒

αcα

by Idem(c)

⇒

αcβ

by RMon(c)

Analogously we have α ∧ β → α c β by Idem(c) and LMon(c) if β ≤ α.

2

Example 3.21. Consider again the Lukasiewicz t-norm tŁ and its distortions wn tŁ , wr tŁ by noise and rounding of Example 3.5. It can be easily seen, e.g., that Idem(tŁ ) = 0.5 (as the Euclidean distance of α∗Ł α from α is at worst 0.5 for α = 0.5) and that Com(wn tŁ ) = 0.9 with probability 1, showing the tightness of (C32) and (C29), respectively; (C30) yields the estimates Ass(wr tŁ ) ≥ 0.8 and Ass(wn tŁ ) ≥ 0.8. Remark 3.22. It can be observed that the traditional non-graded classes of fuzzy connectives can be defined by requiring the full satisfaction of some of the properties defined in Definitions 3.6, 3.12, and 3.16. In particular, a connective c is a (non-graded)

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t-norm

iff

 Com(c),  Ass(c),  LMon(c),  Unit(c, 1)

uninorm

iff

 Com(c),  Ass(c),  LMon(c), (∃η)  Unit(c, η)

binary aggregation operator

iff

 Mon(c), 1 c 1 = 1, 0 c 0 = 0

15

Furthermore, in standard Łukasiewicz logic, c is a (non-graded) quasicopula iff

 Unit(c, 1),  Null(c, 0),  MonCng(c).

Idempotent binary aggregation operators are those which additionally satisfy  Idem(c); commutative quasicopulas those which also satisfy  Com(c); etc. Observe that quasicopulas can in our setting be generalized not just in a graded manner, but also to analogous operators that satisfy MonCng or Cng with respect to an equivalence ↔ other than standard Łukasiewicz as a measure of distance. We shall, however, not pursue this direction here and shall now turn to the graded notion of dominance relation. 4. Graded dominance The traditional non-graded notion of the dominance relation between binary fuzzy connectives (see, e.g., [16,17,26]) is defined by the condition   (∀αβγ δ) (α d γ ) c (β d δ) ≤ (α c β) d (γ c δ) By replacing the crisp ≤ with → (i.e., by deleting the  hidden by Convention 2.1 in ≤ that appears in the non-graded definition), we obtain the following notion of graded dominance between binary fuzzy connectives. As usual, the traditional notion of dominance is expressible as the graded notion satisfied to degree 1, i.e., prepended with . Definition 4.1. The graded relation  of dominance between binary connectives is defined as follows:   c  d =df (∀αβγ δ) (α d γ ) c (β d δ) → (α c β) d (γ c δ) We shall first investigate general properties of the graded notion of dominance and then give semantic examples of computation or estimation of the dominance degrees for particular t-norms. 4.1. General properties of graded dominance The following theorem shows how graded dominance is transmitted to ≈-close connectives: Theorem 4.2. FCT proves: (D1) c  d, c ≈3 c , Cng(d) ⇒ c  d (D2) c  d, d ≈3 d , Cng(c) ⇒ c  d Proof. Claim (D1) is proved by the following chain of implications: (α d γ ) c (β d δ)

⇔

(α d γ ) c (β d δ)

by c ≈ c

⇒

(α c β) d (γ c δ)       αc β d γ c δ

by c  d

⇒

The proof of (D2) is analogous.

by c ≈2 c , Cng(d).

2

The following theorem generalizes the known fact of self-dominance of commutative associative fuzzy connectives to several graded variants:

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Theorem 4.3. FCT proves, for any i ∈ {1, 2}: (D3) (D4) (D5) (D6) (D7) (D8)

 Com(c),  Ass(c) ⇒ c  c  Com(ci ),  Ass(ci ), Cng(ci ), c1 ≈3 c2 ⇒ c1  c2  Com(ci ),  Ass(ci ), Mon(ci ), c1 ⊆ c2 , c2  c1 ⇒ c1  c2 Com(c), Ass4 (c), LRCng(c) ⇒ c  c Com(ci ), Ass4 (ci ), Cng2 (ci ), c1 ≈3 c2 ⇒ c1  c2 Com(ci ), Ass4 (ci ), LRCng(ci ), Mon(ci ), c1 ⊆ c2 , c2  c1 ⇒ c1  c2

Proof. (D3)

(D4)

(D5)

The proof is in fact classical (non-graded), but shall be adapted in (D6) to obtain a graded version. From  Com(c) we obtain γ c β = β c γ , so (by the axioms of identity) also (γ c β) c δ = (β c γ ) c δ. Thence by  Ass(c) (used twice) we obtain γ c (β c δ) = β c (γ c δ), and so also (by the axioms of identity) α c (γ c (β c δ)) = α c (β c (γ c δ)). Applying  Ass(c) twice then completes the proof. We shall prove the claim for i = 1; the proof for i = 2 just uses (D2) instead of (D1). By (D3) we obtain that  Com(c1 ) and  Ass(c1 ) imply c1  c1 , and by (D1) we further obtain that c1  c1 , c1 ≈3 c2 , and Cng(c1 ) imply c1  c2 . We shall prove the theorem for i = 1; the proof for i = 2 is analogous. From c2  c1 we get (α c2 γ ) ≤ (α c1 γ ) and (β c2 δ) ≤ (β c1 δ). Then we have the following chain of implications: (α c2 γ ) c1 (β c2 δ)

(D6)

⇒

(α c1 γ ) c1 (β c1 δ)

by Mon(c1 )

⇒

(α c1 β) c1 (γ c1 δ)

by  Com(c1 ) and  Ass(c1 ), using (D3)

⇒

(α c1 β) c2 (γ c2 δ)

by c1 ⊆ c2 .

The proof is analogous to that of (D3), just using the graded assumptions and LCng(c) instead of the first use of the identity axiom and RCng(c) instead of the second.

The proofs of (D7) and (D8) are analogous to those of (D4) and (D5), respectively, only using (D6) instead of (D3). 2 Theorem (D6) can be informally explained as saying that self-dominance (also known as Aczél’s property of bisymmetry) holds not only for t-norms (or any commutative and associative connective: cf. (D3) of Theorem 4.3 reconstructing this fact in FCT), but to a fair degree also for all connectives that are fairly commutative, very associative, and fairly left- and right-congruent. (D4) and (D7) of Theorem 4.3 generalize the result for a pair of connectives that may not be identical, but are still very close to each other (i.e., when c1 ≈3 c2 holds to a fairly large degree). (D5) and (D8) of Theorem 4.3 have no counterparts among known results; they provide us with bounds for the degree to which (c1  c2 ) holds, where the assumption (c1 ⊆ c2 ) & (c2  c1 ) would be obviously useless in the crisp non-graded framework (as it necessitates that c1 and c2 coincide anyway). Notice that (D6)–(D8) of Theorem 4.3 show that the crisp preconditions of  Com and  Ass can be relaxed by replacing them with graded ones, if some of the connectives in question are sufficiently congruent with equivalence. (Note, however, the heavy dependence of the estimated degree of dominance on the degree of associativity of one of the connectives, due to the quadruple assumption of associativity in these theorems.) Example 4.4. Let us again consider the Lukasiewicz t-norm tŁ and its distortions ws tŁ , wn tŁ , wr tŁ by shifting, noise and rounding introduced in Example 3.5. By (D3) we know that tŁ  tŁ . (D1) and (D2) of Theorem 4.3, in conjunction with the previously calculated degrees of its preconditions, yield the following estimates: (wr tŁ  tŁ ) ≥ 0.85, (wn tŁ  tŁ ) ≥ 0.85, (tŁ  wr tŁ ) ≥ 0.85, (tŁ  wn tŁ ) ≥ 0.85, and (ws tŁ  tŁ ) ≥ 0.7. For the degree of tŁ  ws tŁ , (D5) of Theorem 4.3 provides a better estimate than (tŁ  ws tŁ ) ≥ 0.7 yielded by (D1): namely, (tŁ  ws tŁ ) ≥ 0.9, as (ws tŁ ⊆ tŁ ) = 0.9 and all the remaining premises of (D5) are true to degree 1 for these two connectives.

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17

The following theorems deal with interactions between dominance and units, generalizing the known non-graded fact that dominance implies crisp inclusion (or pointwise order) of connectives. Read contrapositively, they provide bounds to the degree of dominance from the (usually known or at least more easily calculable) degrees of subsethood of the connectives and their units. Theorem 4.5. FCT proves the following graded properties of dominance: (D9) (D10) (D11) (D12) (D13)

 Unit(c, η),  Unit(d, η), c  d ⇒ c ⊆ d Unit(c, η), Unit(d, η), Cng(c), Cng(d), c  d ⇒ c ⊆ d Unit(c, η), Unit(d, ζ ), Cng(c), Cng3 (d), c  d, (η ↔ ζ )2 ⇒ c ⊆ d Unit(c, η), Unit(d, ζ ), Cng(c), Cng(d), c  d, η c η ↔ η, ζ d ζ ↔ ζ ⇒ η → ζ Unit2 (c, η), Unit2 (d, ζ ), Cng(c), Cng(d), c  d ⇒ η → ζ

Proof. (D9) (D10)

By c  d we have (α d η) c (η d δ) → (α c η) d (η c δ). From the assumptions α d η = α c η = α and η d δ = η c δ = δ we get the required α c δ → α d δ. Analogously to the previous proof, we derive: (α d η) c (η d δ) → (α c η) d (η c δ)

(D11) (D12)

⇒ (α d η) c (η d δ) → α d δ

by Unit(c, η) (whence α c η ↔ α, η c δ ↔ δ) and Cng(d)

⇒ αcδ→αdδ

analogously by Unit(d, η) and Cng(c)

The claim follows from (D10) by (C19). (Note that even the weaker assumption Cng(d) & LRCng(d) could be used instead of Cng3 (d).) The claim is proved by the following chain of implications and equivalences: η

(D13)

by c  d

⇔

ηcη

by the assumption

⇔

(ζ d η) c (η d ζ )

by Unit(d, ζ ) and Cng(c)

⇒

(ζ c η) d (η c ζ )

by c  d

⇔

ζ dζ

by Unit(c, η) and Cng(d)

⇔

ζ

by the assumption

This claim is just a weaker (but simpler to formulate) corollary of (D12).

2

The following theorem shows preservation of graded dominance under compositions. Theorem 4.6. Let α e β =df (α a β) c (α b β) and α f β =df (α a α) c (β b β) for all α, β ∈ L. Then FCT proves: (D14) (D15) (D16) (D17)

d  c, (d  a), (d  b), Mon(c) ⇒ d  e d  c, (d  a), (d  b), Mon(c) ⇒ d  f d  c, d  a, d  b, MonCng(c) ⇒ d  e d  c, d  a, d  b, MonCng(c) ⇒ d  f

Proof. (D14)

The claim is proved by the following chain of implications: (α e γ ) d (β e δ)     ⇔ (α a γ ) c (α b γ ) d (β a δ) c (β b δ)     ⇒ (α a γ ) d (β a δ) c (α b γ ) d (β b δ)

by the definition of e by d  c

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18

⇒



   (α d β) a (γ d δ) c (α d β) b (γ d δ)

⇔ (α d β) e (γ d δ) (D15)

by (d  a), (d  b), Mon(c) by the definition of e

Analogously to (D14), the claim is proved by the following chain of implications: (α e γ ) f (β e δ)     ⇔ (α a α) c (γ b γ ) d (β a β) c (δ b δ)     ⇒ (α a α) d (β a β) c (γ b γ ) d (δ b δ)     ⇒ (α d β) a (α d β) c (γ d δ) b (γ d δ) ⇔ (α d β) f (γ d δ)

by the definition of f by d  c by (d  a), (d  b), Mon(c) by the definition of e

The proofs of (D16) and (D17) are analogous, only the use of d  a, d  b enforces the use of MonCng instead of just Mon. 2 Theorem 4.7. FCT proves the following properties of graded dominance: (D18) (D19) (D20) (D21) (D22)

& & & & &

 c, Mon(c) ⇒ c  →  c, Mon(c) ⇒ c  ↔  c, Mon(c), Unit(c, 1) ⇒ LCng(c), and analogously for RCng(c)  c, Mon(c), & ⊆ c ⇒ Cng(c) 2 c, Mon(c),  Unit(c, 1) ⇒ Cng(c)

Proof. (D18)

(D19) (D20) (D21) (D22)

Clearly ((α → β) & α) ≤ β and ((γ → δ) & γ ) ≤ δ are theorems. Thus by Mon(c) we obtain ((α → β) & α) c((γ → δ) & γ ) → β cδ. As &  c, we obtain ((α → β)c(γ → δ)) & (α cγ ) → β cδ, whence by residuation (α → β) c (γ → δ) → (α c γ → β c δ). The proof is analogous to (D18). By Unit(c, 1) we obtain α ↔ β ⇔ (α ↔ β) c 1 ⇔ (α ↔ β) c (γ ↔ γ ). Furthermore, by (D19) we obtain (α ↔ β) c (γ ↔ γ ) ⇒ (α c γ ↔ β c γ ) from &  c and Mon(c). By (D19) we obtain (α ↔ β) c (γ ↔ γ ) ⇒ (α c γ ↔ β c γ ) from &  c and Mon(c). The assumption & ⊆ c completes the proof. The proof follows from the previous claim and (D9) of Theorem 4.5. 2

Theorem 4.8. FCT proves the following properties of graded dominance with respect to ∧: (D23) (D24) (D25) (D26) (D27) (D28)

Mon(c) ⇒ c  ∧  Unit(c, 1) ⇒ (∧  c) ≤ (∧ ⊆ c) Unit(c, 1), Cng(c), ∧  c ⇒ ∧ ⊆ c  Mon(c),  Unit(c, 1) ⇒ (∧ ⊆ c) = (∧  c) LMon(c) ∧ RMon(c) ⇒ (∧  c) ↔ (∀αβ)((α c 1) ∧ (1 c β) → α c β) & (∀αβ)(α c β → (α c 1) ∧ (1 c β))  Mon(c) ⇒ (∧  c) ↔ (∀αβ)((α c 1) ∧ (1 c β) = α c β)

Proof. (D23) (D24) (D25)

From α ∧ γ ≤ α and β ∧ δ ≤ β we obtain (α ∧ γ ) c (β ∧ δ) → (α c β) by Mon(c). Analogously we obtain (α ∧ γ ) c (β ∧ δ) → (γ c δ) and the proof is done. The claim follows from (D9) by  Unit(∧, 1). The claim follows from (D10) by  Unit(∧, 1) and  Cng(∧).

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(D26)

(D27)

19

We only need to derive (∧ ⊆ c) ≤ (∧  c), as the converse inequality follows from (D24). First observe that from  Mon(c) and  Unit(c, 1) we obtain c  ∧ by (C31). Since furthermore  Com(∧),  Ass(∧), and  Mon(∧), we can use (D5) to complete the proof. Left to right: From ∧  c we obtain (α c 1) ∧ (1 c β) ⇒ (α ∧ 1) c (1 ∧ β) ⇔ α c β. Since α c β → α c 1 by RMon(c) and α c β → 1 c β by LMon(c), we have derived α c β → (α c 1) ∧ (1 c β) from the premises of the theorem. Right to left: Starting from α ≤ β, which implies α ≤ α ∧ β, by LMon(c) we obtain α c 1 → (α ∧ β) c 1, and so (α c 1) ∧ (β c 1) → (α ∧ β) c 1. The same claim can be proved starting from β ≤ α, thus by prelinearity we have: LMon(c) ⇒ (α c 1) ∧ (β c 1) → (α ∧ β) c 1. Analogously we can prove RMon(c) ⇒ (1 c γ ) ∧ (1 c δ) → 1 c (γ ∧ δ). Thus together (with a little rearranging) we have:     LMon(c) ∧ RMon(c) ⇒ (α c 1) ∧ (1 c γ ) ∧ (β c 1) ∧ (1 c δ) → (α ∧ β) c 1 ∧ 1 c (γ ∧ δ) (1) The claim then follows from the following chain of implications: (α c γ ) ∧ (β c δ)

(D28)

⇒ ⇒

(α c 1) ∧ (1 c γ ) ∧ (β c 1) ∧ (1 c δ)     (α ∧ β) c 1 ∧ 1 c (γ ∧ δ)

by (1)

⇒

(α ∧ β) c (γ ∧ δ)

by the first part of the assumption

The claim is a direct corollary (by -necessitation) of (D27).

by the second part of the assumption

2

(D23) of Theorem 4.8 is a graded generalization of the well-known fact that the minimum dominates any aggregation operator [17]. (D26) of Theorem 4.8, illustrated by Example 4.9 below, demonstrates a rather surprising fact: that the degree to which a monotonic binary operation with the neutral element 1 dominates the minimum is nothing else but the degree to which it is larger. (D27) of Theorem 4.8 is an alternative characterization of operators dominating the minimum; for its non-graded version (D28) see [17, Proposition 5.1]. Example 4.9. (D26) of Theorem 4.8 can be utilized to easily compute degrees to which standard t-norms on the unit interval dominate the minimum. It can be easily shown that for any t-norm t,   (∧ ⊆ t) = inf x ⇒ t(x, x) x∈[0,1]

holds, i.e. that the largest “difference” (in the sense of →) of a t-norm t from the minimum can always be found on the diagonal. In standard Łukasiewicz logic, this degree is, for instance, 0.75 for the product t-norm t and 0.5 for the Łukasiewicz t-norm tŁ itself.7 So we can infer that in standard Łukasiewicz logic, the product t-norm t dominates the minimum with a degree of 0.75, while the Łukasiewicz t-norm tŁ dominates the minimum with a degree of 0.5. 4.2. The degree of dominance between tŁ and ordinal sums of tŁ In this subsection we give a semantic example of estimating the degree of dominance between particular standard t-norms on the real unit interval [0, 1]. In this subsection we work in an arbitrarily given standard (i.e., [0, 1]-valued) model of FCT. By TŁ we denote the class of all ordinal sums of the Łukasiewicz triangular norm tŁ (see [3] for properties of tŁ and the notion of ordinal sum of t-norms). The diagonal of a binary operation c is the map dg c (α) =df (α c α). The members of TŁ are known [3] to be the only t-norms t with the property   (α t β) = (α ∧ β) ∧ α [dg t ◦ m] β where m stands for arithmetic mean and α [dg t ◦ m] β ≡ dgt (α m β). By (D25), letting a = c = ∧, b = dg t ◦ m, and d = tŁ we have the lower estimate: tŁ  [dg t ◦ m]

⇒

tŁ  t.

7 Recall that t (α, β) = α · β and t = max(α + β − 1, 0), for all α, β ∈ [0, 1]. See, e.g., [3] for basic properties of standard t-norms. Ł 

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For the sake of brevity, it will be useful to denote l(α, β, γ , δ) =df (α tŁ β)[dg t ◦ m] (γ tŁ δ),     r(α, β, γ , δ) =df α [dg t ◦ m] γ tŁ β [dg t ◦ m] δ . Clearly, l and r are, respectively, the left and right side in the dominance inequality of tŁ  [dg t ◦ m]; thus we can write r ⊆ l for tŁ  [dg t ◦ m]. In what follows we analyze the degree of dominance and show that it boils down to a graded property of dg t resemblant of subadditivity. Proposition 4.10. For every t ∈ TŁ we have  

tŁ  [dg t ◦ m] ↔ (∀αβ) dg t (α) tŁ dg t (β) → dg t (α tŁ β) . Proof. Observe that the right-hand side of our claim is actually the infimum of r(α, β, α, β) → l(α, β, α, β) over all α, β ∈ [0, 1]. Therefore the proof can be carried out by showing that, given any quadruple α, β, γ , δ ∈ [0, 1], there exist α, ¯ β¯ ∈ [0, 1] such that



¯ α, ¯ → l(α, ¯ α, ¯ . r(α, β, γ , δ) → l(α, β, γ , δ) ≥ r(α, ¯ β, ¯ β) ¯ β, ¯ β) This inequality obviously entails that the infimum of the right-hand side has to be less or equal to the infimum of the left-hand side. On the other hand, given that every value attained by the right-hand side of the latter inequality is attained also by the left-hand side, the converse inequality between the infima has to hold as well. For the sake of clarity, we will prove the existence of such α¯ and β¯ in three consecutive substitutions of variables in the dominance inequality. In each step we replace the actual quadruplet of variables by a new one in such a way that the degree of dominance does not increase. The consecutive substitutions are8 :     α¯ = α  m γ  α  = α ∨ (1 − β) α  = α  + β  − β  m δ    β  = β  m δ  β = β β¯ = β      γ¯ = α  m γ  γ  = γ ∨ (1 − δ) γ  = γ  + δ  − β  m δ    δ  = β  m δ  δ = δ δ¯ = δ  1. Clearly α  ≥ α and γ  ≥ γ , which, by the monotony of all operations in question, yields the inequality r(α, β, γ , δ) ≤ r(α  , β  , γ  , δ  ). On the other hand, it is straightforward to check that (α  tŁ β) = (α tŁ β) and (γ  tŁ δ) = (γ tŁ δ), meaning that l(α, β, γ , δ) = l(α  , β  , γ  , δ  ). As r does not decrease while l stays constant, the value of the dominance inequality does not increase. 2. The first issue we have to be concerned about is the non-negativity of α  and β  . Looking back at the first step, it is clear that α  + β  > 1 and γ  + δ  > 1. The non-negativity follows by the fact that m attains values within the unit interval [0, 1]. By the definition of the Łukasiewicz t-norm, tŁ (α  , β  ) = α  + β  − 1 = α  + β  − 1 = tŁ (α  , β  ) and, analogously, tŁ (γ  , δ  ) = tŁ (γ  , δ  ). Hence the left-hand side of the dominance inequality stays constant under this substitution. Further, given the definition of m it is immediate that β  + δ  = 2(β  m δ  ). Therefore α  + γ  = α  + γ  and β  + δ  = β  + δ  . As a consequence (using the definition of m once again) the right-hand side of the dominance inequality stays constant under this substitution. All in all, the degree of dominance stays intact. 3. Reiterating some of the considerations of the previous step, it is immediate why r does not change its value. From the previous step we know that α  + β  ≥ 1, γ  + δ  ≥ 1, and β  = δ  . These three facts, when considered ¯ = α¯ + β¯ − 1 and together, yield α¯ + β¯ = (α  m γ  ) + β  ≥ 1 and analogously γ¯ + δ¯ ≥ 1, whence (α¯ tŁ β) ¯ ¯ (γ¯ tŁ δ) = γ¯ + δ − 1. Therefore, expanding the definition of l yields:   

   ¯ γ¯ , δ) ¯ = dgt α  m γ  + β  − 1 m α  m γ  + δ  − 1 . l(α, ¯ β, 8 Notice that since in this subsection we are working in the standard [0, 1]-valued semantics of higher-order fuzzy logic, we are not restricted to the syntax of FCT and can freely use the arithmetical operations + and − on the interval [0, 1].

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21

In view of the definition of m, the latter expression further equals          

dgt α  + β  − 1 m γ  + δ  − 1 = dgt α  tŁ β  m γ  tŁ δ  = l α  , β  , γ  , δ  . ¯ The overall change of the dominance degree is nil. Finally, observe that α¯ = γ¯ and β¯ = δ.

2

Till now, all our considerations have been independent on the choice of the background logic. Further on, let us consider the degree of dominance in models over standard Łukasiewicz logic, i.e., the logic of the Łukasiewicz t-norm tŁ . In this logic, (α →Ł β) ↔ ((1 − β) →Ł (1 − α)) for every α, β ∈ [0, 1]. The dual to the Łukasiewicz t-norm tŁ is the Łukasiewicz t-conorm sŁ (α, β) = 1 − ((1 − α) tŁ (1 − β)). We will also use the dual diagonal dgt (α) = 1 − dg t (1 − α). For the degree in Proposition 4.10 we obtain

(∀αβ) dg t (α) tŁ dg t (β) →Ł dg t (α tŁ β)       

⇔ (∀αβ) 1 − dgt (1 − α) tŁ 1 − dgt (1 − β) →Ł 1 − dgt 1 − (α tŁ β) by the definition of dgt

  

⇔ (∀αβ) dg t (1 − α)[1 − sŁ ]dgt (1 − β) →Ł 1 − dgt (1 − α) sŁ (1 − β) by the definition of sŁ



 ¯  ¯ ¯ ⇔ (∀α¯ β) dg t (α)[1 ¯ − sŁ ]dg t (β) →Ł 1 − dg t (α¯ sŁ β) by the substitution

   ¯ ¯ ¯ ¯ sŁ dg t (β) by the properties of →Ł ⇔ (∀α¯ β) dg t (α¯ sŁ β) →Ł dg t (α) Since the t-conorm sŁ is a truncated addition on the unit interval, the last property is in fact a graded notion of subadditivity of dgt . In general, for a fully monotone unary operation u we can define its degree of subadditivity as:   SubAdd(u) =df (∀αβ) u(α sŁ β) → u(α) sŁ u(β) . Thus in standard Łukasiewicz logic, for every t ∈ TŁ we have:   SubAdd dgt ⇒ tŁ  t. 5. Application to graded properties of fuzzy relations In this section we shall apply graded dominance to graded properties of fuzzy relations. The following definitions generalize the graded notions from [10,15] by considering an arbitrary fuzzy connective c in place of strong conjunction. Definition 5.1. In FCT, we define the following graded properties of binary fuzzy relations: Refl(R) =df (∀x)Rxx

reflexivity

Sym(R) =df (∀xy)(Rxy → Ryx)

symmetry

Transc (R) =df (∀xyz)(Rxy c Ryz → Rxz)

c-transitivity

Extc (A, R) =df (∀xy)(Ax c Rxy → Ay)

c-extensionality of A w.r.t. R

AntiSymc (R, E) =df (∀xy)(Rxy c Ryx → Exy)

c-antisymmetry of R w.r.t. E

Furthermore we define the class operation c given by the connective c as follows: (P c Q)x1 . . . xn =df P x1 . . . xn c Qx1 . . . xn , for tuples x1 . . . xn of an arbitrary arity. (Thus, e.g., c is strong intersection if c = &, weak union if c = ∨, etc., of fuzzy classes or fuzzy relations.) The following theorems show the importance of graded dominance for graded properties of fuzzy relations. Theorem 5.2 is a graded generalization of the well-known theorem by De Baets and Mesiar that uses dominance to characterize preservation of transitivity by aggregation [16, Theorem 2]. By (R4), in monotone operators with the null element 0 (e.g., t-norms), the degree of graded dominance c  d is exactly the degree to which c-transitivity is preserved by d-intersections.

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22

Theorem 5.2. FCT proves: c  d, Mon(d),  Transc (R),  Transc (S) ⇒ Transc (R d S) c  d, Mon(d), Cng(d), Transc (R), Transc (S) ⇒ Transc (R d S)  Null(c, 0), (∀RS)( Transc (R) &  Transc (S) → Transc (R d S)) ⇒ c  d Mon(d),  Null(c, 0) ⇒ (c  d) ↔ (∀RS)( Transc (R) &  Transc (S) → Transc (R d S))

(R1) (R2) (R3) (R4) Proof. (R1)

(R2)

We need to derive Rxz d Sxz from (Rxy d Sxy) c (Ryz d Syz). Now the latter implies (Rxy c Ryz) d (Sxy c Syz) by c  d, which in turn implies the required Rxz d Sxz by Mon(d), since we have Rxy c Ryz ≤ Rxz and Sxy c Syz ≤ Sxz by  Transc (R) and  Transc (S), respectively. Analogously as in the proof of (R1) we derive: (Rxy d Sxy) c (Ryz d Syz)

(R3)

⇒ (Rxy c Ryz) d (Sxy c Syz)

by c  d

⇒ Rxz d Sxz

by Transc (R), Transc (S), Mon(d), Cng(d)

Take any α, β, γ , δ ∈ L. Fix three elements a = b = c = a and define two relations (see Definition 2.3 for the notation used): R =df {α/ab, β/bc, α c β/ac} S =df {γ /ab, δ/bc, γ c δ/ac}

(R4)

Since  Null(c, 0), obviously  Transc (R) and  Transc (S). Thus we can infer Transc (R d S), whence by specification we obtain (Rab d Sab) c (Rbc d Sbc) → (Rac d Sac). To complete the proof, use the definitions of R and S (by which Rab = α, Sab = β, etc.). The claim is a corollary of (R1) and (R3). 2

Example 5.3. Let R, S be two min-transitive fuzzy relations; i.e., Trans∧ (R) = Trans∧ (S) = 1. Recall Example 4.9, by which (∧  tŁ ) = 0.5 and (∧  t ) = 0.75 in standard Łukasiewicz logic. Since furthermore Mon(tŁ ) = Mon(t ) = 1, (R1) of Theorem 5.2 yields the following lower bounds for the min-transitivity degrees of the tŁ -resp. t -intersections R tŁ S and R t S in standard Łukasiewicz logic: Trans∧ (R tŁ S) ≥ 0.5 and Trans∧ (R t S) ≥ 0.75. These estimates are in fact tight, as seen from the following instances: Consider R = {1/ab, 0.5/bc, 0.5/ac} and S = {0.5/ab, 1/bc, 0.5/ac}. Clearly, both R and S are min-transitive, but their tŁ -intersection R tŁ S = {0.5/ab, 0.5/bc, 0/ac} and t -intersection R t S = {0.5/ab, 0.5/bc, 0.25/ac} are not. Nevertheless, Trans∧ (R tŁ S) = (0.5 ∧ 0.5 ⇒Ł 0) = 0.5 and Trans∧ (R t S) = (0.5 ∧ 0.5 ⇒Ł 0.25) = 0.75, which are exactly the bounds yielded by (R1) in standard Łukasiewicz logic. The similitude between the defining formulae of Transc (R) and Extc (A, R) makes it possible to adapt the results of Theorem 5.2 for graded extensionality: Theorem 5.4. FCT proves: (R5) (R6) (R7) (R8)

c  d, Mon(d),  Extc (A, R),  Extc (B, S) ⇒ Extc (A d B, R d S) c  d, Mon(d), Cng(d), Extc (A, R), Extc (B, S) ⇒ Extc (A d B, R d S)  Null(c, 0), (∀ABRS)( Extc (A, R) &  Extc (B, S) → Extc (A d B, R d S)) ⇒ c  d Mon(d),  Null(c, 0) ⇒ (c  d) ↔ (∀ABRS)( Extc (A, R) &  Extc (B, S) → Extc (A d B, R d S))

Proof. The proofs are analogous to those of Theorem 5.2. As an example we shall show the proof of (R6): (Ax d Bx) c (Rxy d Sxy)

⇒

(Ax c Rxy) d (Bx c Sxy)

by c  d

⇒

Ay d By

by Extc (A, R), Extc (B, S), Mon(d), Cng(d)

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The definitions of R, S in the proof of (R4) need to be changed as follows for the proof of (R8). We only fix two elements b = c and define: A =df {α/b, α c β/c},

R =df {β/bc}

B =df {γ /b, γ c δ/c},

S =df {δ/bc}

The rest of the proof is analogous to (R4).

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Remark 5.5. Theorems 5.2 and 5.4 can be proved jointly if we define R c S =df R ◦c S ⊆ S, where (R ◦c S)xz =df (∀y)(Rxz c Syz). Then clearly Transc (R) = (R c R) and Extc (A, R) = (R c RA ), where RA xy =df (x = a) & Ay for an arbitrary fixed element a (serving as a dummy argument to increase the arity of A, cf. [29]). Consequently, e.g., both (R2) and (R6) are instances of the more general theorem c  d, Mon(d), Cng(d), R1 c S1 , R2 c S2 ⇒ (R1 d R2 ) c (S1 d S2 ), proved in the similar manner as above. Finally, the following theorem describes the preservation of several graded properties by symmetrizations of fuzzy relations. In the non-graded case, the commutativity of an operator trivially implies the symmetry of the symmetrization of a fuzzy relation by this operator. In the graded case, (R9) of Theorem 5.6 states that the degree to which a symmetrization is actually symmetric is bounded from below by the degree to which the aggregation operator c is commutative. (R10)–(R12) of Theorem 5.6 are similar graded generalizations of well-known non-graded facts [16,31,32]. Theorem 5.6. FCT proves: (R9) (R10) (R11) (R12)

Com(c) ⇒ Sym(R c R −1 ) Refl2 (R), & ⊆ c ⇒ Refl(R c R −1 ) d ⊆ c ⇒ AntiSymd (R, R c R −1 )  Transd (R), d  c, Mon(c) ⇒ Transd (R c R −1 )

Proof. Claims (R9)–(R11) follow easily from the definitions. Claim (R12) is a simple corollary of (R1).

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6. Conclusions In this paper, we have introduced several graded properties of unary and binary fuzzy connectives and investigated their properties. Unlike traditional non-graded properties, which either are or are not possessed by a fuzzy connective c: Ln → L, the graded properties we consider can be satisfied to a larger or lesser degree. Formulae of higher-order fuzzy logic MTL have been used to define the graded generalizations of traditional properties, and the apparatus of higher-order fuzzy logic has been used to derive theorems on these graded notions. In particular, graded versions of the properties of monotony, ↔-congruence (which is a generalization of the 1-Lipschitz property), commutativity, associativity, idempotence, null and unit elements, and the relation of dominance between binary fuzzy connectives have been studied. Graded theorems of the syntactic form ϕ1 , . . . , ϕn ⇒ ψ (in the notation of Convention 2.1) have been produced, which relative to a given logic of any left-continuous t-norm ∗ express the semantic fact that the combination of the truth degrees of the premises by ∗ gives a lower bound for the conclusion ψ: i.e., that ϕ1 ∗ ∗ · · · ∗ ϕn ∗ ≤ ψ∗ , where ϕ∗ is the truth degree of the condition ϕ evaluated in the logic of ∗. Thus, e.g., Theorem (C31) on page 14: Mon(c), Unit(c, 1) ⇒ c ⊆ ∧, says that the degree to which a fuzzy connective c is monotone (according to Definition 3.6, under the interpretation of logical symbols in the logic of ∗), combined by ∗ with the degree to which 1 is the unit element of c (according to Definition 3.12 in the same logic), is at most the degree of the graded inclusion (of Definition 2.3) of c in ∧. More than 70 theorems of this form have been proved in the paper. Moreover, a few examples of exact calculations or lower estimates of the dominance degrees between particular t-norms have been provided, and an application of graded dominance to the transmission of graded transitivity and extensionality of fuzzy relations has been given. These results demonstrate that the graded theory of fuzzy connectives is viable and that higher-order fuzzy logic is a suitable tool for the development of the theory. The approach thus opens the way for using graded properties of fuzzy connectives in all kinds of applications where the traditional crisp properties can be imperfectly satisfied.

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