Schweizer B. and Sklar A. (1983) Probabilistic Metric Spaces, North-Holland, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to obtain a phenotype # from # Menger [49] suggested six decades ago that, due to inherent uncertainties of measurements, the distance d(x, y) should be replaced by a probability distribution P (x, y; d) describing the probability that the distance between x and y is at least d; see e.g. [50]. More recently, probabilistic convergence spaces have been introduced [51] where (F , x) q # means that the filter Shortening of stacks Elongation of stacks Opening of constrained stacks Closing of constrained stacks Figure 3. Fontana Schuster pretopology. Contraction and elongation ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North Holland, New York, 1983.

....In the next paragraph we supply some explanation about this choice. 3.2. The aggregation of attributes Besides of the logical arguments announced in [7] we will try here to show on two examples why this operator is more suitable than others classically used: the means [17] and the t norms [18 19]. Let us first compare the uninorm to a t norm. Let us imagine that we have the following description for the shot number 11: Shot 11 Actor 1 Actor 2 Camera Motion Type 2 Main background Color = Black 0.7 0.9 0.2 0.8 Figure 1. Description of Shot Number 11 Using a t norm will discard ....

. Schweizer B. and Sklar A., Probabilistic metric spaces, North Holland, New York, 1983.

....only if 1 (x) 2 (x) for all x F(E) becomes a partially ordered set, and this order corresponds to that of the lattice structure (F (E) min, max) defined by means of the pointwise intersection and union of fuzzy sets given by 2 = min 2 ) 2 = max 2 ) 2.2. According to [5], a function T : 0, 1] 0, 1] is called a triangular norm (t norm) if it satisfies T1 T5 below. A function S : 0, 1] 0, 1] is called a triangular conorm (tconorm) if it satisfies S1 S5 below. Associativity: T1. T (x, T (y, z) T (T (x, y) z) S1. S(x, S(y, z) S(S(x, y) z) ....

Schweizer B., Sklar A. Probabilistic Metric Spaces, North Holland,Amsterdam, 1983.

....dual operator, which corresponds to augmenting a t conorms. 1. Introduction The concept of a triangular norm was introduced by Menger [1 ] in order to generalize the triangular inequality of a metric. The current notion of a t norm and its dual operation (tconorm) is due to Schweizer and Sklar [2,15]. Both of these operations can also be used as a generalization of the Boolean logic connectives to multi valued logic. The t norms generalize the conjunctive AND operator and the t conorms generalize the disjunctive OR operator. This situation allows them to be used to define the intersection ....

. Schweizer B. and Sklar A., Probabilistic metric spaces, North Holland, New York, 1983.

....Connectives The idea of a triangular norm (t norm) first arose as a means of generalizing the triangular inequality of a metric. A slightly different modern definition of a t norm and its dual operator, the triangular co norm (t conorm) is largely a result of work done by Schweizer and Sklar[44] [45] and acts as a generalization of Boolean logical operators in the multi valued fuzzy domain. The t norm operator generalizes the Boolean operator of conjunction and similarly, the t conorm generalizes the 55 Table 3.5: Some simple t norms and associated t conorms. operation of disjunction. ....

B. Schwcizcr and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983.

....i# T is continuous and xTx x for all x in (0, 1) A binary operation S on [0, 1] is called a triangular conorm (shortly, t conorm) if there exists a t norm T such that xSy = 1 x)T (1 for all x, y in [0, 1] and is denoted by T # . The t norms and t conorms were studied in detail in [18]. The most commonly used t norm and t conorm are = max, which are deduced directly from the natural order of real number. Definition 2. Let S and T be t norm or t conorm, respectively, and let A = a ij ) mp , B = b ij ) pn be fuzzy matrices. Define AB = c ij ) mn c ij = a i1 T b 1j ....

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983, pp.54-77.

....3D flow simulation [4] In conjunction with parallel coordinates we extend the interactively specified brush by a certain, user defined percentage to accommodate a smooth gradient of DOI values at its borders. When combining smooth brushes through logical operators, we apply the L ukasiewicz norm [16], known from fuzzy logic as a comparably large norm [14] i.e. a norm which (when applied repeatedly) only slowly converges to and therefore better conserves the transitional DOI gradients, for computing AND as well as OR combinations. See figure 6 for an example of smoothly brushed parallel ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. Noth-Holland, New York, 1983.

....is called isomorphic to the original t norm f (a; b) Isomorphic operations provide numerous new examples of t norms and t conorms. The complete description of all possible [0; 1] based logical operations (which uses rescaling and isomorphisms) has been given, in effect, in [17] see also [13, 15, 20, 22]) It turns out that every t norm f (a; b) can be described as follows: ffl we subdivide the interval [0; 1] into subintervals; ffl the restriction of the t norm f (a; b) to each of these subintervals is isomorphic either to the algebraic t norm a Delta b, or to max(a b Gamma 1; 0) or ....

B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.

....is the eukasiewicz w t norm) and R 2 is Min transitive. R is only 1 2 reflexive and R 2 is only v v v O reflexive. Thus R and R 2 are C consequence operators for =1 and =0 v v respectively and is not a ct consequence relation for any ct 0. 6. On the other hand, if is a continuous t norm [10], the residuated implication I (b a) sup z v(a) z v(b) is a preorder. In particular, for minimum, product and Luckasiewicz t norms are respectively obtained iMin = 1 v v(b) if v(a) v(b) iProd otherwi s e, v = 1 if v(a) v(b) v(b) v(a) otherwise, IW(b a) Min (1, ....

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, 1983.

.... t(x is at least S) supf S (y) j y xg t(x is at most S) supf S (y) j y xg Logical operations on the truth values are defined using tnorms and t conorms, i.e. commutative, associative, and non decreasing binary operations on the unit interval with neutral elements 1 and 0, respectively [20, 10]. These operations can be defined in various ways; we restrict ourselves to the well known examples min and max: TM (x;y) min(x;y) SM (x;y) max(x;y) Then conjunctions and disjunctions of fuzzy predicates can be defined as follows: t(A(x) B(x) T t(A(x) t(B(x) t(A(x) B(x) S ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....certainly monqp ) All of these belong to the class of norms, which seems to capture what one would expect of a reasonable conjunction operator. The dual concept of norm is that of an norm, which expresses the essential properties of fuzzy disjunction operators (cf. Schweizer Sklar [21]) Theorem 2. In every DFS , a. a nq C n is the identity truth function; b. a a is a strong negation operator; 12 c. a a is a norm; d. 6 a C a a 6 a C , i.e. a is the dual norm of a under a , e. 6 , C a 6 a C The fuzzy disjunction ....

B. Schweizer and A. Sklar. Probabilistic metric spaces. North-Holland, Amsterdam, 1983.

....nature of the vagueness concepts appearing in real world problems (see R. L. De Mantaras [6] In this work, by the term fuzzy logic we mean a specific [0; 1] valued propositional logic whose connectives are interpreted via triangular norms. Recall (see, for instance, Schweizer and Sklar [12], Butnariu and Klement [1] that a triangular norm (t norm for short) is a function T : 0; 1] Theta[0; 1] 0; 1] which is symmetric (T (x; y) T (y; x) associative (T (x; T (y; z) T (T (x; y) z) has 1 as unit element (T (x; 1) x) and is monotone (x 1 x 2 implies T (x 1 ; y) T (x 2 ....

Schweizer, B. and Sklar A., Probabilistic Metric Spaces. North Holland, New York, 1983.

....a Delta : Delta a(n times) lim a n = 0 for all a 1. So, for ab, we get a meaningless result D (x) 0 for all x. It turns out that we get the same meaningless result for all Gammaoperations different from min. Let s formulate this result in precise terms. Definition (Schweizer 1983) [57]. An Gammaoperation (t Gammanorm) is a continuous, symmetric, associative, monotonic operation f : 0; 1] Theta [0; 1] 0; 1] for which f (1; x) x. An Gammaoperation is called Archimedian if f (x; x) x for all x 2 (0; 1) and strict if it is strictly increasing, as the function of ....

....Usually, three types of operations are used: ffl min; ffl strict operations; ffl Archimedian operations. Proposition 8. If f is an Archimedian or a strict operation, then for every a 2 (0; 1) lim n 1 f (a; a) n times) 0: 29 Proof of Proposition 8. It is known (Schweizer 1983) [57] that every strict operation is isomorphic to ab, namely, there exists a continuous monotonic function for which (f (a; b) a) b) Then, f (a; a) a) n 0, so f (a; a) 0. For Archimedian operations, similarly, g(f (a; b) min(g(a) g(b) g(0) for some ....

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B. Schweizer and A. Sklar. Probabilistic metric spaces (North Holland, N.Y., 1988).

....= 1 ffl 0 Phi = ffl Phi is commutative and associative ffl moreover oe(A [ B) oe(A) Phi oe(B) for any disjoint events A and B. Such pseudo additive measures have been introduced by Dubois and Prade [12] and Weber[39] when Phi is a triangular conorm in the sense of Schweizer and Sklar [32]. Clearly adequate candidates for Phi are maximum and the bounded sum (if L is numerical) so that decomposable measures include possibility and probability measures. Axiom DM can be called decomposable monotonicity. By duality, any uncertainty ordering that obeys A1 A2 and A3 and DDM can be ....

....20 ffl moreover ae(A B) ae(A) fi ae(B) for any events A and B such that A [ B = S. Such dual pseudo additive measures are of the form ae(A) nL (oe(A) where nL is an involutive order reversing map of L. Operation fi can be taken as a triangular norm in the sense of Schweizer and Sklar[32]. Clearly adequate candidates for fi are minimum and the Lukasiewicz conjunction (max(0; a b Gamma 1) if L is numerical) so that dual pseudo additive measures include necessity and probability measures. However the above relaxation of the probabilistic framework is still too restrictive to ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....1 ; c 2 ; OR(f 1 ; f 2 ) Abduction w = AND(f 1 ; c 1 ; f 2 ; c 2 ) w = AND(f 1 ; c 1 ; c 2 ) Induction w = AND(f 1 ; c 1 ; f 2 ; c 2 ) w = AND(c 1 ; f 2 ; c 2 ) 4. Extend the Boolean operators AND, OR, and NOT from f0, 1g to [0, 1] according to the study of Tnorm and T conorm [2, 6, 17]. The extended Boolean operators in NARS are: AND(x; y) x y OR(x; y) x y x y NOT (x) 1 x where the rst two are applied only when x and y are independent to each other, meaning that the value of one provides no information on the value of the other. When these operators are applied ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....is called isomorphic to the original t norm f (a; b) Isomorphic operations provide numerous new examples of t norms and t conorms. The complete description of all possible [0; 1] based logical operations (which uses rescaling and isomorphisms) has been given, in effect, in [16] see also [12, 14, 19, 21]) It turns out that every t norm f (a; b) can be described as follows: ffl we subdivide the interval [0; 1] into subintervals; ffl the restriction of the t norm f (a; b) to each of these subintervals is isomorphic either to the algebraic t norm a Delta b, or to max(a b Gamma 1; 0) or ....

B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York, 1983.

....result gives a representation in terms of automorphisms (Ovchinnikov and Roubens 1991) Theorem 2. 3 Any strong negation N can be represented by an automorphism of [0# 1] as N (x) 1 (1 ; x) Triangular norms (t norms) are developed as tools to use in probabilistic metric spaces (cf. Schweizer and Sklar (1983)) Weber (1983) proposed to use them as connectives in fuzzy set theory. Although, in general, t norms are not necessarily in [0# 1] all continuous t norms are in [0# 1] In this study only continuous t norms are considered. A continuous t norm is defined as a symmetric, associative, ....

....y) n ;1 (T (n(x)#n(y) 1) Note that since t norms are associative, T (x# y# z) T (T (x# y)#z) T (x# T (y# z) is well defined. 2.1 Generators of Continuous Archimedean t norms In this section we briefly present additive generators of continuous Archimedean t norms. For more details see Schweizer and Sklar (1983). The following is a representation theorem of Ling (1965) See Schweizer and Sklar (1983) for historical comments on this representation. Theorem 2.4 A t norm T is continuous and Archimedean if and only if there exists a continuous and strictly decreasing function g : 0# 1] Re with ....

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Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam.

....to the fuzzy field are t norms and t conorms. A t norm T is a semigroup of the unit interval (associative, commutative, with identity 1) which is non decreasing in each place. Archimedean t norms are such that #a # (0, 1) aTa a. Any Archimedean continuous t norm T can be written as follows: (Schweizer and Sklar 1983) aT b = # 1 (min(#(0) #(a) #(b) where # : 0, 1] # [0, #(0) is a continuous and decreasing function, such that #(1) 0. Similar to Holder norms, families of t norms T p can be written as T p (x) # 1 p n # i=1 # p (x i ) # (24) for which # p (0) #. For our purpose, we only ....

Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces, North--Holland.

....fuzzy eld instead of min and max. A t norm T is a semigroup of the unit interval (associative, commutative, with identity 1) which is non decreasing in each place. Continuous Archimedean t norms are such that 8a 2 (0; 1) aTa a. Any Archimedean continuous t norm T can be written as follows: (Schweizer and Sklar 1983) aT b = 1 (min( 0) a) b) 2) where : 0; 1] 0; 0) is a continuous and decreasing function, such that (1) 0. Similar to H older norms, parameterized families of strict t norms T p , where p 2 R is a parameter, can be written as T p (x) 1 p n X i=1 p (x i ) ....

Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces, North{Holland.

....in B 1 is smaller than the degree of confidence in B 2 , then 3 our confidence in A 1 B 1 must be smaller (or at least equal, but not larger) than our confidence in A 2 B 2 . A complete description of operations that satisfy these properties has been given, in effect, in [9] see also [6, 8, 12, 14]) It turns out that for every t norm, we can subdivide the interval [0; 1] into subintervals on each of which t norm is isomorphic either to the algebraic t norm a Delta b, or to max(a b Gamma 1; 0) or to min(a; b) and f (a; b) min(a; b) when a and b belong to two different ....

B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, N.Y., 1983.

....values, given that other components take other values. The dependence structure of the risks is contained within F. Copulas represent a way of trying to extract the dependence structure from the joint distribution and to extricate dependence and marginal behaviour. It has been shown by Sklar (see Schweizer and Sklar (1983)) that every joint distribution can be written as F (x 1 ; x n ) C(F 1 (x 1 ) F n (x n ) 6 for a function C that is known as the copula of F . A copula may be thought of in two equivalent ways: as a function (with some technical restrictions) that maps values in the unit ....

Schweizer, B., and A. Sklar (1983): Probabilistic Metric Spaces. North{ Holland/Elsevier, New York.

....3 is true because the sum (2) can be interpreted as P[a 1 X 1 b 1 ; a n X n b n ] which is non negative. For any continuous multivariate distribution the representation (1) holds for a unique copula C. If F 1 ; F n are not all continuous it can still be shown (see Schweizer and Sklar (1983), Chapter 6) that the joint distribution function can always be expressed as in (1) although in this case C is no longer unique and we refer to it as a possible copula of F . The representation (1) and some invariance properties which we will show shortly, suggest that we interpret a copula ....

....continuity of the transformations T i is necessary for general random variables (X 1 ; X n ) t since, in that case, F 1 T i (X i ) T i F 1 X i . In the case where all marginal distributions of X are continuous it suces that the transformations are increasing (see also Chapter 6 of Schweizer and Sklar (1983)) As a simple illustration of the relevance of this result, suppose we have a probability model (multivariate distribution) for dependent insurance losses of various kinds. If we decide that our interest now lies in modelling the logarithm of these losses, the copula will not change. Similarly if ....

Schweizer, B., and A. Sklar (1983): Probabilistic Metric Spaces. North{Holland/Elsevier, New York.

....the tuple of connectives always contains at least a conjunction interpreted by a continuous triangular norm and that the choice of this triangular norm determines the interpretation of the remaining connectives, too. Triangular norms were introduced in the framework of probabilistic metric spaces [18], based on ideas first presented in [12] and they are applied in several fields, e.g. in fuzzy sets [20] fuzzy logics [2, 7, 19] and their applications, but also in the theory of generalized measures [1] Extensive overviews on triangular norms can be found in [8, 18] 2 Basic fuzzy logical ....

.... of probabilistic metric spaces [18] based on ideas first presented in [12] and they are applied in several fields, e.g. in fuzzy sets [20] fuzzy logics [2, 7, 19] and their applications, but also in the theory of generalized measures [1] Extensive overviews on triangular norms can be found in [8, 18]. 2 Basic fuzzy logical operations In this section we repeat the essential definitions of fuzzy logical operations. The conjunction is usually interpreted by a triangular norm (t norm for short) i.e. a commutative, associative, non decreasing operation T : 0; 1] 2 [0; 1] with a neutral ....

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Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

.... at least S) sup S (y) y # x t(x is at most S) sup S (y) y # x Logical operations on the truth values are defined using t norms and t conorms, i.e. commutative, associative, and non decreasing binary operations on the unit interval with neutral elements 1 and 0, respectively [20, 10]. These operations can be defined in various ways; we restrict ourselves to the well known examples min and max: TM (x, y) min(x, y) SM (x, y) max(x, y) Then conjunctions and disjunctions of fuzzy predicates can be defined as follows: t(A(x) # B(x) T # t(A(x) t(B(x) # t(A(x) ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....= p. 2. TNOR is a bijection from FUNCT(2,MRHC2) onto FUNCT(2,MLHC2) 3. RIMP is the inverse mapping of TNOR and vice versa. In a forthcoming paper we will investigate which properties of t and p are translated by RIMP and TNOR analogously to theorems 3.1 and 4.1, respectively. See also [1, 2, 8, 15, 19 21]. In a forthcoming second paper we will study relations between QL implications on the one hand and negations, T norms, and S norms on the other hand, following the philosophy presented in this paper. 10 Acknowledgement The author wishes to thank ULRICH FIESELER for fruitful scientific ....

B. SCHWEIZER and A. SKLAR. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....asserts the existence of a minimal and a maximal element in set A. Hence, given the weak ordering and the boundaries, one can replace the set A by the familiar interval notation #e; u# The following lemma demonstrates some of the consequences of axioms imposed on a bounded ordered semigroup [14]. Lemma 1 Let A = hA; #; #i be a bounded ordered semigroup with bounds e and u.Then A also satisfies the following conditions for all a; b 2 A: i) a # b # sup#a; b#, ii) u # a # a # u # u, iii) a # a # a. In [14, Section 5.3] a function defined on a closed real interval #a; e#, endowed ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....not strict is called nilpotent. In combination with Proposition 1.2, a continuous t norm T is Archimedean if and only if ffi T (x) x for all x 2 ]0; 1[ Problems concerning diagonals of continuous t norms appeared in several works. Recall, e.g. the famous open problem of Schweizer and Sklar [7] whether the continuity of the diagonal implies the continuity of the underlying t norm. Mayor and Torrens [6] have characterized the t norms determined by means of their diagonal, ffi, via T (x; y) max(0; ffi(max(x; y) Gamma jx Gamma yj) Further, B ezivin and Tom as [1] proved that a strict ....

....continuous t norms. In each case, the class of all t norms with a given diagonal is constructively characterized. Note that the only diagonal of a continuous t norm having a unique underlying t norm is the identity, ffi(x) x; in this case, the corresponding t norm is the minimum t norm TM [4, 7]. We shall often use the following representation theorem for continuous Archimedean t norms (see, e.g. 4, 5, 7] Theorem 1.4 : Let T be a continuous Archimedean t norm. There is an additive generator of T , i.e. a continuous strictly decreasing function f : 0; 1] 0; 1] such that 8x; y ....

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Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York, 1983.

....containing infinitary conjunctions, and we show that they are semantically stronger than all the other logics studied in this paper. Key words: Fuzzy logic, many valued logic, Frank t norm 1 Frank t norms Triangular norms were introduced in the framework of probabilistic metric spaces [33, 32, 34], based on ideas first presented in [22] and they are applied in several fields, e.g. in fuzzy sets [35] fuzzy logics [2, 14, 28] and their applications, but also in the theory of generalized measures [1, 19] and nonlinear differential and difference equations [27] A triangular norm (t norm ....

.... , i.e. for all 0 2 [0; 1] lim 0 T = T 0 ; lim 0 S = S 0 : For each 2 [0; 1] the Frank t norm T and the Frank t conorm S are dual to each other, and they solve the functional equation T (x; y) S(x; y) x y: 4) It was shown in [9] that, together with their ordinal sums (see [34]) these are the only pairs of continuous t norms and t conorms solving the functional equation (4) Extensive overviews on Frank and other t norms can be found in [18, 34] 2 Propositional fuzzy logics based on Frank t norms A many valued logic with a continuum of truth values modelled by the ....

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B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....of fuzziness. By doing so, measurement theoretic frameworks to discuss semantics of fuzzy set theory are obtained. In Section 2, basic definitions, and representation and uniqueness results for algebraic structures, called ordered semigroups, are given. Mainly the results of (Fuchs 1963) and (Schweizer Sklar 1983) are translated in terms of ordered algebraic structures as is customary in the measurement theory literature. Then, two related but different measurement problems are stated, membership measurement and property ranking. It is shown that, although the first problem received much attention in the ....

....asserts the existence of a minimal and a maximal element in set A. Hence, given the weak ordering and the boundaries, one can replace the set A by the familiar interval notation #e; u# The following lemma demonstrates some of the consequences of axioms imposed on a bounded ordered semigroup (Schweizer Sklar 1983). Lemma 1. Let A = hA; #; #i be a bounded ordered semigroup with bounds e and u.Then A also satisfies the following conditions for all a; b 2 A: i) a # b # sup#a; b#, ii) u # a # a # u # u, iii) a # a # a. In (Schweizer Sklar 1983, Section 5.3) a function defined on a closed real ....

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Schweizer, B. & Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam.

....(2( 1 ; n ) Meaning 2 (t P ( 1 ) t P ( n ) The unit interval [0; 1] equipped with a triangular norm, forms a commutative, fully ordered semigroup with neutral element 1 and annihilator 0. Triangular norms were introduced in the framework of probabilistic metric spaces [34, 33, 35], based on ideas first presented in [23] and they are applied in several fields, e.g. in fuzzy sets [36] fuzzy logics [2, 14, 29] and their applications, but also in the theory of generalized measures [1, 20] and nonlinear differential and difference equations [28] To repeat the essential ....

.... and the family of Frank t conorms (S ) 2[0;1] is strictly increasing with respect to the parameter (see [1] Both families are continuous with respect to , i.e. for all 0 2 [0; 1] lim 0 T = T 0 ; lim 0 S = S 0 : Extensive overviews on Frank and other t norms can be found in [18, 35]. 2 Fuzzy logics with residual implications A reasonable way of constructing connectives in fuzzy logics is to start with a (left) continuous t norm T and to use the residuum (R implication, see [4, 8, 7, 12, 31, 32, 30] defined by R T (x; y) sup fz 2 [0; 1] j T (x; z) yg: 2) as the ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....with margins whose domain is S 1 S 2 . Sn . Let x = x 1 , x 2 , xn ) and y = y 1 , y 2 , yn ) be any points in S 1 S 2 . Sn . Then H(x) H(y) # n X k=1 H k (x k ) H k (y k ) For the proof, see Schweizer and Sklar (1983) [15]. Definition 3. An n dimensional subcopula is a function C # with the following properties: 3 2 Copulas 1. DomC # = S 1 S 2 . Sn , where each S k is a subset of I containing 0 and 1; 2. C # is grounded and n increasing; 3. C # has margins C # k , k = 1, 2, n, which ....

Schweizer B. and Sklar A. (1983) Probabilistic Metric Spaces, North-Holland, New York.

....uncertainty correspond to three types of dependence in these structures. 2 Preliminaries triangular norms Triangular norms are usually viewed as fuzzy generalizations of the Boolean conjunction. However, they were studied in the early sixties in the area of probabilistic metric spaces (see [22]) and some of them even earlier. Let us recall their basic properties relevant to our considerations. A t norm (triangular norm) is an operation T : 0; 1] 2 [0; 1] which is commutative, associative, monotone in each component, and which satisfies the boundary condition T (1; u) u (see e. ....

....and some of them even earlier. Let us recall their basic properties relevant to our considerations. A t norm (triangular norm) is an operation T : 0; 1] 2 [0; 1] which is commutative, associative, monotone in each component, and which satisfies the boundary condition T (1; u) u (see e.g. [3, 22]) The function : u 7 1 Gamma u is the standard fuzzy negation. The dual t conorm to T is the operation S : 0; 1] 2 [0; 1] defined by the (de Morgan) formula S(u; v) T ( u; v) 1 Gamma T (1 Gamma u; 1 Gamma v) The Frank family of t norms T s , s 2 [0; 1] will play an essential ....

Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York, 1983.

.... formalized by means of the Compositional Rule of Inference [Zad 73] A generalized (pointwise) expression for the Compositional Rule of Inference is the following: w in the universe of B and u in the universe of A B (w) s u t(A (u) I(A(u) B(w) 2) where t denotes a t norm [Men 42] ScS 83] Web 83] s a t conorm, s u a s based supremum in the universe of A and I(A(u) B(w) denotes an implication operation [TrV 85] If the A B rule of eq. 1) were to be applied to linguistic variables it would turn into T 1g T Nk , where the indices mean to say, that from the g th ....

Schweizer B., Sklar A.: "Probabilistic Metric Spaces". North Holland, Amsterdam, 1983

....results are given. A very brief review of the theory of measurement is given in Section 2.2. In Section 2.3, basic definitions and representation and uniqueness results of algebraic structures called ordered semigroups are given. This is an abstract treatment and mainly the results of [17] and [34] are translated in terms of ordered algebraic structures as is customary in the measurement theory literature. In Section 3, the basic operations on the unit interval are reviewed. Definition of the triangular conorms is given. Triangular conorms are candidate numerical representations for the ....

....#e; u#. As usual, we write #e; u# for the open interval fa : u # a # eg, #e; u# for the interval open on the right, fa : u # a # eg and #e; u# for the interval open on the left, fa : u # a # e.g. The following lemma summarizes the consequences of axioms of a bounded ordered semigroup [34] 7 . Lemma 1 Let A = hA; #; #i be a bounded ordered semigroup. Then A also satisfies the following conditions for all a; b 2 A: i) a # b # sup#a; b# (ii) u # a # a # u # u for all a 2 A 8 . iii) a # a # a for all a 2 A. Proof: First, observe that a # b # e ....

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B. Schweizer and A. Sklar, Probabilistic Metric Spaces (North-Holland, Amsterdam, 1983) 24

....to which the rule output can be applied to the overall FC output. The main inference issues for the FC are: the definition of the fuzzy predicate evaluation, which is usually a possibility measure (Zadeh, 1978) the LHS evaluation, which is typically a triangular norm (Schweizer and Sklar, 1963, Schweizer and Sklar, 1983, Bonissone, 1987) the conclusion detachment, which is normally a triangular norm or a material implication operator; and the rule output aggregation, which is usually a triangular conorm for the disjunctive interpretation of the rule base, or a triangular norm for the conjunctive case. Under ....

....are based on a fuzzy valued representation of uncertainty and imprecision. Typically they use Linguistic Variables (Zadeh, 1978, Zadeh, 1979) to represent different information granularities and Triangular norms to propagate the fuzzy boundaries of such granules (Schweizer and Sklar, 1963, Schweizer and Sklar, 1983, Dubois and Prade, 1984, Bonissone, 1987, Bonissone and Decker, 1986, Bonissone et al. 1987b) The basic inferential mechanism used in fuzzy reasoning systems is the generalized modus ponens (Zadeh, 1979) which makes use of inferential chains (syllogisms) 12.4.1 Triangular norms: A Review ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North Holland, New York, 1983.

....fuzzy sets with their membership functions. The collection T is called the B generated tribe. C1. We fix a t norm, i.e. a binary operation T : 0; 1] 2 [0; 1] which is commutative, associative, nondecreasing, and satisfies the boundary condition T (a; 1) a for all a 2 [0; 1] see [18]) We take the standard fuzzy negation 0 : 0; 1] 0; 1] defined by a 0 : 1 Gamma a, and the t conorm S: 0; 1] 2 [0; 1] dual to T , i.e. S(a; b) T (a 0 ; b 0 ) 0 . C2. We extend the operations T , 0 , S to operations T , c , S on T (i.e. on fuzzy sets) pointwise: ....

B. Schweizer, A. Sklar. Probabilistic Metric Spaces. North-Holland, New York, 1983.

....C(x; 1) x and C(1; y) y for all x; y 2 I; and 2. Monotonicity) if 0 x 1 x 2 1 and 0 y 1 y 2 1, then C(x 2 ; y 2 ) Gamma C(x 1 ; y 2 ) Gamma C(x 2 ; y 1 ) C(x 1 ; y 1 ) 0: The idea of a copula was introduced by A. Sklar in response to a question from M. Fr echet. Sklar proved (see [12, 13]) that if H was the joint distribution function of two random variables, X and Y , and F and G were the distribution functions of X and Y respectively, then one could always find a copula C such that H(x; y) C(F (x) G(y) Furthermore, the copula is uniquely determined on range(F ....

Schweizer, B. and A. Sklar, "Probabilistic Metric Spaces," Elsevier North Holland , New York, 1983.

....each element x in X the value 1 whenever x belongs to A, and the value 0 otherwise. In order to generalize the set theoretical operations like intersection and union (or the corresponding Boolean logical operations conjunction and disjunction, respectively) we need triangular norms and conorms [6, 7, 8]: A triangular norm (t norm) is a binary operation on [0; 1] i.e. a function T : 0; 1] 2 [0; 1] which is commutative, associative, monotone in both components, and satisfies the boundary condition T (x; 1) x: If T is a t norm, then the dual triangular conorm (t conorm) S : 0; 1] 2 ....

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....of fuzziness. By doing so, measurement theoretic frameworks to discuss semantics of fuzzy set theory are obtained. In Section 2, basic definitions, and representation and uniqueness results for algebraic structures, called ordered semigroups, are given. Mainly the results of (Fuchs 1963) and (Schweizer Sklar 1983) are translated in terms of ordered algebraic structures as is customary in the measurement theory literature. Then, two related but different measurement problems are stated, membership measurement and property ranking. It is shown that, although the first problem received much attention in the ....

....asserts the existence of a minimal and a maximal element in set A. Hence, given the weak ordering and the boundaries, one can replace the set A by the familiar interval notation [e; u] The following lemma demonstrates some of the consequences of axioms imposed on a bounded ordered semigroup (Schweizer Sklar 1983). Lemma 1. Let A = hA; Phii be a bounded ordered semigroup with bounds e and u.Then A also satisfies the following conditions for all a; b 2 A: i) a Phi b sup(a; b) ii) u Phi a a Phi u u, iii) a Phi a a. In (Schweizer Sklar 1983, Section 5.3) a function defined on a ....

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Schweizer, B. & Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam.

....For each pair of nodes ff; fi 2 U , we can calculate several degrees of conditional independency, one for each other node in the model, Dep(ff; fijfl) 8fl 2 U Gamma ff Gamma fi. In order to have a single measure of dependency, we could aggregate them in some way; any triangular norm Omega [37], such as, for example, the minimum or the product, could be an appropriate conjunctive operator. So, we define the global degree of dependency Dep t (ff; fi) by means of Dep t (ff; fi) O fl2U Gammaff Gammafi Dep(ff; fijfl) To preserve the strongest dependencies compatible with a singly ....

B. Schweizer, A. Sklar, Probabilistic Metric Spaces (North-Holland, New York, 1983).

....between set theory and classical logic. In that context, the fuzzy set intersection and union correspond to connectives AND and OR of fuzzy logic respectively. In this paper we use the terms union or disjunction of fuzzy sets and intersection or conjunction of fuzzy sets interchangeably. norms [31, 32, 33, 34] is borrowed from the statistical metric spaces literature to model the fuzzy set intersection [11, 3, 41] Although the aim of such research is to generalize the concepts of intersection and union of fuzzy sets to more abstract settings, the practical reason for doing so remained mainly vague. ....

....and results are given. A very brief review of the theory of measurement is given in Section 2.2. In Section 2.3, basic definitions and representation and uniqueness results of algebraic structures called ordered semigroups are given. This is an abstract treatment and mainly the results of [17] and [34] are translated in terms of ordered algebraic structures as is customary in the measurement theory literature. In Section 3, the basic operations on the unit interval are reviewed. Definition of the triangular conorms is given. Triangular conorms are candidate numerical representations for the ....

[Article contains additional citation context not shown here]

B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

....be defined and studied in this section. The representation theorem for fuzzy preorders will be used to extend the induction principle to a Multivalued Logic. Like in section 1, the relation between fuzzy preorders and fuzzy CC systems is investigated. In the following will denote a fixed t norm [10]. Definition 7. 4] Given a fuzzy preorder I on F,a fuzzy subset t of F is said to be a t set (t for true) if it is closed under Modus Ponens, i.e. for each a,b F, it is verified: t(a) I(a,b) t(b) Analogously, a fuzzy subset f of F is said to be a f set (f for false) if it is ....

B. Schweizer and A. Sklar, "Probabilistic Metric Spaces". North-Holland 1983.

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Schweizer B. and Sklar A. (1983) Probabilistic Metric Spaces, North-Holland, New York.

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Schweizer, B. and Sklar, A. : Probabilistic Metric Spaces. North-Holland, 1983.

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B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North Holland, 1983.

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B. Schweizer and A. Sklar. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

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. Schweizer B. and Sklar A., Probabilistic metric spaces, North Holland, New York, 1983.

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Schweizer, B., and Sklar, A., 1983, Probabilistic Metric Spaces, Elsevier Science, New York.

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B. Schweizer and A. Sklar, Probabilistic metric spaces (North Holland, New York, 1983).

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B. SCHWEIZER and A. SKLAR. Probabilistic Metric Spaces. North-Holland, Amsterdam, 1983.

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