IFSA-EUSFLAT 2009
On ‘family resemblances’ with fuzzy sets∗ E. Trillas, C. Moraga1
A. Sobrino2
1
European Centre for Soft Computing Mieres (Asturias), Spain 2 University of Santiago de Compostela Santiago de Compostela, Spain Email: {enric.trillas, claudio.moraga}@softcomputing.es,
[email protected]
Abstract— This paper takes into account the Wittgenstein’s idea on family resemblances as a particular crisp relation between some fuzzy sets, that is, between some predicates representable from its use. It is shown that all uses of the same predicate actually do have family resemblance, that some pairs of predicates cannot, and a numerical degree of family resemblance is introduced. Keywords— Degree of family resemblance, Family resemblance, Fuzzy sets.
sets, and a numerical degree for measuring such relation.
Usually, full-normalized fuzzy sets, µP ∈ [0, 1]X , those for which there are x, y ∈ X such that µP (x) = 1, µP (y) = 0, appear as ‘data’ in the modeling of fuzzy systems, and they are neither self-contradictory (µP ≤ µP ), nor negatively self contradictory (µP ≤ µP ). Of course, µP denotes the fuzzy set corresponding to ‘not p’. Points x verifying µP (x) = 1 can be taken as the prototypes of P in X, and points y verifying µP (y) = 0 as the anti-prototypes of P in X. 1 Introduction In its own nature, this paper is not a conclusive one, but Ludwig Wittgenstein headed two influential traditions in the only a tentative to reflect the potentially interesting subject of so-called philosophy of language, that were originated by the family resemblances shown by ‘data’ fuzzy sets. That is, his famous books, Tractatus logico-philosophicus (1922, [1]), by fuzzy sets with prototypes and anti-prototypes. and Philosophical Investigations (1953, [2]), respectively. The Tractatus does not properly deal with ordinary language, but with the logical analysis of propositions built up from atomic propositions, considered as ‘pictures’ of facts and keeping a strict correspondence with the world, understood through the totality of facts. Instead, the Philosophical Investigations in which Wittgenstein abandoned logical analysis, meant a shift in conferring a main role to the ways of designating facts, as an activity-oriented perspective on language. What is central at this respect, is that language does not primarily consist on describing the facts, but on playing ‘language’s games’, or ways of dynamically using words to define their meaning, and that are to be described. In order to fix the meaning of a word, to show how a word works, it should be placed in the context and environment it is used. Language is not yet defined through propositions that, now, in Wittgenstein’s view, come from their function in a language’s game, and to note the absence of boundaries for describing such use of words, Wittgenstein introduced the term ‘family resemblances’. Of course, Wittgenstein’s idea on ‘family resemblances’, is broader than the relation of family resemblance in next section.
2 Family resemblance of fuzzy sets Let P, Q, ... be predicates on a universe of discourse X, such that their use, or meaning, is described by fuzzy sets P, Q, ..., given by membership functions µP , µQ , ... in [0, 1]X . For each membership function µ in [0, 1]X , define the sets of • its 0-points, Z(µ) = {x ∈ X; µ(x) = 0} • its 1-points, S(µ) = {x ∈ X; µ(x) = 1} Definition 2.1 With X ⊂ R, the relation of family resemblance, fr ⊂ [0, 1]X × [0, 1]X , is defined by (µ, σ) ∈ fr if and only if, 1. Z(µ) ∩ Z(σ) = ∅, S(µ) ∩ S(σ) = ∅ 2. µ is non-decreasing in A ⊂ X iff σ is non-decreasing in A. 3. µ is decreasing in A ⊂ X iff σ is decreasing in A.
In this definition, both ‘decreasing’ and ‘non-decreasing’, are not in strict sense, but allowing some constant pieces that can Fuzzy logic manages the extensional meaning of predicates be taken as the first, or the second, by following what happens through its use, once captured by the corresponding member- before or after. ship functions. This paper is nothing else than a first approach, in the path towards Zadeh’s Computing With Words, to inNotice that the binary relation fr is only predicable between troduce ‘family resemblances’ between full-normalized fuzzy full-normalized fuzzy sets, that is, such that Z(µ) = ∅, and S(µ) = ∅. Denote ∗
This work has been partially supported by the Foundation for the Advancement of Soft Computing (Asturias, Spain), and CICYT (Spain) under project TIN2008-06890-C02-01 ISBN: 978-989-95079-6-8
F∗ (X) = {µ ∈ [0, 1]X − {0, 1}X ; Z(µ) = ∅, S(µ) = ∅}.
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IFSA-EUSFLAT 2009 It should be pointed out that fuzzy sets µ in {0, 1}X (crisp sets) are excluded, since such functions µ are neither decreasing, nor non-decreasing, but only piecewise constant with the values 0 or 1. In addition, in many cases if (µ, σ) ∈ fr, with µ, σ ∈ {0, 1}X , it should be µ = σ. Obviously, also constant fuzzy sets µr (µr (x) = r, with r ∈ [0, 1], x ∈ X) are not in F∗ (X), since Z(µr ) = ∅, or S(µr ) = ∅. Of course, if σ results from a translation of µ keeping Z(µ) ∩ Z(σ) = ∅, and S(µ)∩S(σ) = ∅, it is obvious that (µ, σ) ∈ fr.
F∗ (X)/fr does not exist. For example, in the above figures, it is σ ∈ [µ] ∩ [λ ]. Anyway, [µ] can be called the family of µ ∈ F∗ (X), and σ ∈ [µ] a relative of µ, although µ could have relatives in other families, like it happens in people’s families. Theorem 2.5 For no complement, µ of µ ∈ F∗ (X), is (µ, µ ) ∈ fr. Proof. If for x, y ∈ A, x ≤ y, and µ(x) ≤ µ(y), it is µ (y) ≤ µ (x).
Remark 2.2 Any non full-normalized fuzzy set, represented by a non-constant but continuous membership function µ ∈ Theorem 2.6 For no opposite, or antonym, µ of µ ∈ F∗ (X), [0, 1]R , can be re-scaled to a full-normalized one µ∗ by given by a symmetry α in X, µ = µ ◦ α, is (µ, µ) ∈ fr. µ − min(µ) µ∗ = max(µ) − min(µ) Proof. If x ≤ y, and µ(x) ≤ µ(y), it follows α(y) ≤ α(x), and µ(α(y)) ≤ µ(α(x)), or µ ◦ α(y) ≤ µ ◦ α(x), that is Obviously, it is Z(µ∗ ) = ∅, S(µ∗ ) = 0, that is µ∗ ∈ F∗ (R), µ(y) ≤ µ(x). (see [3]) and µ∗ is non-decreasing (decreasing) if and only if µ is nondecreasing (decreasing), but µ∗ and µ can differ in the reTheorem 2.7 Let u : X → X be a bijective mapping such spective slopes (provided they have derivatives, it is µ∗ = that µ / max(µ) − min(µ). • If x ≤ y, then u(x) ≤ u(y)
Example 2.3
• u−1 (Z(µ)) ⊂ Z(µ)
1. Fuzzy sets µ, σ in figure 1 verify (µ, σ) ∈ fr, since Z(µ)∩ Z(σ) = [0, 2] = ∅, and S(µ) ∩ S(σ) = {10} = ∅, and both are non-decreasing in X = [0, 10].
• u−1 (S(µ)) ⊂ S(µ)
2. Fuzzy set µ in figure 1, and fuzzy set λ in figure 2, verify for µ ∈ F∗ (X), X ⊂ R. It is (µ, µ ◦ u−1 ) ∈ fr. (µ, λ) ∈ / fr, since S(µ) ∩ S(λ) = ∅, and λ is decreasing in [0, 2], but µ is not. Proof. It is Z(µ) ∩ Z(µ ◦ u−1 ) = ∅, since x ∈ Z(µ), or µ(x) = 0, implies u−1 (x) ∈ Z(µ), or µ(u−1 (x)) = µ ◦ u−1 (x) = 0, it is, x ∈ Z(µ ◦ u−1 ). Analogously, x ∈ S(µ), σ λ µ or µ(x) = 1, that is, x ∈ S(µ ◦ u−1 ). Hence, Z(µ) ∩ Z(µ ◦ u−1 ) = ∅, and S(µ) ∩ S(µ ◦ u−1 ) = ∅. If µ is non-decreasing, 2 2 5 ‘x ≤ y ⇒ µ(x) ≤ µ(y), from u−1 (x) ≤ u−1 (y), follows µ(u−1 (x)) ≤ µ(u−1 (y)). Analogously, if µ is decreasing, so Figure 1: Figure 2: it is µ ◦ u−1 . In particular, if Q represented by µQ = µP ◦ u−1 is a synTheorem 2.4 fr is a reflexive and symmetric relation in onym of P (see[4]), then (µP , µQ ) ∈ fr. µ
F∗ (X).
Proof. It is immediate to check that fr does verify the reflex- Example 2.8 ive and symmetric properties. 1. Fuzzy sets µ, σ in figure 5 verify σ = µ ◦ α, with the As it is intuitive, crisp relation fr is not transitive. A counsymmetry α(x) = 10 − x, and represent two antonyms. terexample is given by membership functions µ, σ, λ in the Obviously, µ and µ ◦ α do not show family resemblance. figures 1
1 µ
µ
µ
λ
µ µ
σ
σ
µ
σ
µ σ
0
X
0
X
Figure 3:
Figure 4:
that verify (µ, σ) ∈ fr, and (σ, λ) ∈ fr, but (µ, λ) ∈ / fr, since Z(µ) ∩ Z(λ) = ∅. Hence, if sets [µ] = {σ ∈ F∗ (X); (µ,σ) ∈ fr} are not [µ] = F∗ (X), empty and cover F∗ (X), because it is ∗
2
8
Figure 5:
Figure 6:
−1 2. Fuzzy sets √ µ, σ in figure 6 verify σ = µ ◦ u , with u(x) = x, and show family resemblance.
µ∈F∗ (X)
they do not give a partition of F (X), and the quotient ISBN: 978-989-95079-6-8
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IFSA-EUSFLAT 2009 3 Degree of family resemblance
r1 Ovch(µ, σ) + r2 I(µ, σ) DR(µ, σ) = , Obviously, if σ1 , σ2 are in the family [µ], they do verify r1 + r2 (µ, σ1 ) ∈ fr, and (µ, σ2 ) ∈ fr, but this does not mean that with r1 , r2 ∈ R+ − {0}, σ1 , σ2 show the same extent of family resemblance with µ. A way of measuring such extent is by means of a convenient of which the only DR that is symmetric is the one with r1 = T -indistinguishability (see([5])) DR : fr → [0, 1], defining, r2 = 12 (arithmetic mean). In addition, and since W ≤ prod, Ovch is also W -transitive: W (Ovch(µ, σ), Ovch(σ, λ)) ≤ If (µ, σ) ∈ fr, the degree up to which µ resembles σ is Ovch(µ, σ)·Ovch(σ, λ) ≤ Ovch(µ, λ). Thus, also DR with DR(µ, σ) ∈ [0, 1]. r1 = r2 = 12 is W -transitive, since: 3.1 W (DR(µ, σ), DR(σ, λ)) = Any prod-indistinguishability E in F∗ (X), can be represented by Ovchinnikov’s theorem (see [6]) Ovch(µ, σ) + Ovch(σ, λ) − 1 + max(0, 2 f (µ) f (σ) , ), E(µ, σ) = Inf ( I(µ, σ) + I(σ, λ) − 1 f ∈F f (σ) f (µ) )≤ 2 with F a family of functions f : F∗ (X) → R+ − {0}. Ovch(µ + λ) I(µ + λ) + )≤ max(0, 2 2 Provided X is a closed interval in R, consider only the max(0, DR(µ, λ)) = DR(µ, λ), functions µ ∈ F∗ (X) that are Riemann-integrable in X, and such that both Z(µ) and S(µ) are intervals in X. Denote by F∗∗ (X) this subset of functions in F∗ (X), and take because Ovch(µ, σ) + Ovch(σ, λ) − 1 ≤ 6 max(0, Ovch(µ, σ)+Ovch(σ, λ)−1), and I(µ, σ)+I(σ, λ)− f1 (µ) = µ(x)dx, f2 (µ) = length of the interval X−Z(µ). 1 ≤ max(0, I(µ, σ) + I(σ, λ) − 1). Hence, we will take X
Then, Ovch(µ, σ) = min(
Ovch(µ, σ) + I(µ, σ) , 2 as the index of family resemblance. DR(µ, σ) =
f1 (µ) f1 (σ) f2 (µ) f2 (σ) , , , ), f1 (σ) f1 (µ) f2 (σ) f2 (µ)
for all µ, σ ∈ F∗∗ (X), is a prod-indistinguishability in the part of fr in F∗∗ (X) × F∗∗ (X).
Remark 3.1 Obviously, functions DR are not only applicable to pairs in fr, but also to all pairs in F∗∗ (X) × F∗∗ (X) 3.4
3.2
Let E be a T -indistinguishability relation. The crisp relation
If X = [a, b] ⊂ R, define 1 I(µ, σ) = 1 − | b−a
6
µ ≡ σ ⇔ E(µ, σ) > 0,
6 X
µ(x)dx −
σ(x)dx|, X
for all µ, σ ∈ F∗∗ (X). Function I is a W-indistinguishability (W Łukasiewicz t-norm), since:
is reflexive, and symmetric. Obviously, and provided T = min, or T = prod, it is also transitive. Nevertheless, if T = W , since “E(µ, σ) > 0 and E(σ, λ) > 0” is equivalent to the is the existence of ε > 0 such that “E(µ, σ) ≥ ε and E(σ, λ) ≥ ε”, it is
• I(µ, µ) = 1, and I(µ, σ) = I(σ, µ), for all µ, σ ∈ F∗∗ (X).
W (E(µ, σ), E(σ, λ)) ≥ W (ε, ε) = max(0, 2ε − 1), or
% 1 [| X µ(x)dx − • W (I(µ, σ), I(σ, λ)) = max(0, 1 − b−a % % % σ(x)dx|+| X σ(x)dx− X λ(x)dx|]) ≤ max(0, 1− X % % % 1 1 b−a | X µ(x)dx− X λ(x)dx|) = 1− b−a | X µ(x)dx− % λ(x)dx| = I(µ, λ), for all µ, σ, λ ∈ F∗∗ (X), because X % % of the triangular inequality, | X µ(x)dx − X λ(x)dx| ≤ % % % % | X µ(x)dx − X σ(x)dx| + | X σ(x)dx − X λ(x)dx|.
E(µ, λ) ≥ max(0, 2ε − 1). To have E(µ, λ) > 0, it is sufficient that 2ε − 1 > 0, or ε > 0.5. In this case, µ ≡ σ and σ ≡ λ imply µ ≡ λ (transitivity). Hence E(µ, σ) > 0.5 allows to take µ and σ as ‘equivalent’.
If 0.5 < Ovch(µ, σ), and 0.5 < I(µ, σ), it is µ ≡ σ for both T -indistinguishabilities E = Ovch and E = I. Thus, if in addition to (µ, σ) ∈ fr, is 0.5 < ε ≤ Ovch(µ, σ), and 0.5 < δ ≤ I(µ, σ), it is also 0.5 < ε+δ 2 ≤ DR(µ, σ), imply3.3 ing µ ≡ σ for E = DR. For example if 0.7 ≤ Ovch(µ, σ), Since it is Ovch(µ, σ) = I(µ, σ) and, in general, and 0.7 ≤ I(µ, σ), it is 0.7 ≤ DR(µ, σ). In these cases, it Ovch(µ, σ) < I(µ, σ), a better degree could be obtained with can be said that µ and σ show high family resemblance. a mean like, ISBN: 978-989-95079-6-8
308
IFSA-EUSFLAT 2009 When (µ, σ) ∈ fr, if e.gr., Ovch(µ, σ) ≥ 0.8 and I(µ, σ) ≥ 0.8 implying DR(µ, σ) ≥ 0.8, there is a so high family resemblance between µ and σ that µ and σ can be taken as strongly representing similar uses of the same predicate (linguistic label).
µ
µ
σ
2
3
4
Example 3.2
5
6
Figure 9:
1. Functions µ, σ in figure 7, do obviously verify (µ, σ) ∈ fr. For them, f1 (µ) = 5, f2 (µ) = 8, f1 (σ) = 4 and f2 (σ) = 7,
σ
µ
µ
µ
σ
2
3
8
9
hence, Ovch(µ, σ) = min( 54 , 45 , 87 , 78 ) = 0.8. 1 |5 − 4| = 0.9, it results Since, I(µ, σ) = 1 − 10
2
4. Functions µ, σ in figure 10, verify (µ, σ) ∈ fr. For them f1 (µ) = 1, f2 (µ) = 2, f1 (σ) = 2 and f2 (σ) = 4, hence, Ovch(µ, σ) = min( 12 , 21 , 24 , 42 ) = 0.5. Since I(µ, σ) = 1 −
7
8
hence, Ovch(µ, σ) = min( 52 , 25 , 64 , 46 ) = 25 = 0.4. 1 (|5 − 2|) = 0.7, it results Since, I(µ, σ) = 1 − 10
3. Functions µ, σ in figure 9, verify (µ, σ) ∈ fr. For them f1 (µ) = 5 + 1 = 6, f2 (µ) = 7, f1 (σ) = 4 + 2 = 6 and f2 (σ) = 8, hence, Ovch(µ, σ) = min( 66 , 78 , 87 ) = 78 = 0.88. 1 10 (|6
DR(µ, σ) =
σ
µ
− 6|) = 1, it results
0.88 + 1 = 0.94, 2
2
3
5
7
8
Figure 11: Since I(µ, σ) = 1 −
0.4 + 0.7 = 0.55, 2
µ and σ do not show high family resemblance.
Since I(µ, σ) = 1 −
0.5 + 0.9 = 0.7, 2
Figure 8:
DR(µ, σ) =
− 1) = 0.9, it results
5. Functions µ, σ in figure 11, verify (µ, σ) ∈ fr. For them f1 (µ) = 3 + 2·0.5 + 1 = 5, f2 (µ) = f2 (σ) = 10 + 5 = 7.75, hence, and f1 (σ) = 0.5 + 3·0.5 + 3·0.5 2 5 = 0.64. Ovch(µ, σ) = 7.75 µ
5
1 10 (2
DR(µ, σ) =
σ
3
7
6
µ and σ show high family resemblance.
2. Functions µ, σ in figure 8, verify (µ, σ) ∈ fr. For then, f1 (µ) = 4 + 2·0.5 = 5, f2 (µ) = 6, f1 (σ) = 2, f2 (σ) = 4,
µ
5
Figure 10:
0.8 + 0.9 = 0.85, 2
µ and σ show high family resemblance.
µ
4
Figure 7:
DR(µ, σ) =
3
1 10 (7.75
DR(µ, σ) =
− 5) = 0.725, it results
0.64 + 0.725 = 0.6825, 2
µ and σ do not show high family resemblance. 6. Functions µ, σ in figure 12, verify (µ, σ) ∈ fr. It is: f1 (µ) = 3, f2 (µ) = 4, f1 (σ) = 3 and f2 (σ) = 5.Then, Ovch(µ, σ) =
4 , I(µ, σ) = 1 5
That is, DR(µ, σ) = 0.9, and µ, σ show high family resemblance.
µ and σ show high family resemblance. ISBN: 978-989-95079-6-8
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IFSA-EUSFLAT 2009
5 Last remarks
µ
5.1
σ
3
4
5
6
7
8
Figure 12:
4 The case of membership functions with the same predicate. Given a predicate, or linguistic label, P on X, there is not a single fuzzy set generated by P , but each use of P in X, once reflected by its membership function µP (see [4]), defines a fuzzy set P by x ∈r P ⇔ µP (x) = r, for all x ∈ X, r ∈ [0, 1]. Nevertheless, all functions µP do have some 0-points, and some 1-points, in common, as well as they should either decrease or non-decrease simultaneously in the same parts of X. Hence, all µP should show some degree of family resemblance. This is the case for example, with P =small in X = [0, 10], if two of its uses are represented by the membership functions µ1P and µ2P in figure 13.
µ 1P
µ P2
2
5
8
Figure 13: It is clear that (µ1P , µ2P ) ∈ fr, and: 8 • Ovch(µ1P , µ2P ) = min( 55 , 10 8 , 10 ) = 0.8
• I(µ1P , µ2P ) = 1 −
1 10 (|5
− 5|) = 1.
In his Philosophical Investigations, Wittgenstein conceived language in a way close to how people manages it. Thus, the meaning of an imprecise predicate is not given by necessary and sufficient conditions, but is built up by similarity with its prototypes in the universe of discourse, like in the case of big in [0,10] with, at least, the prototype 10. Notwithstanding, for more complex predicates like P =beautiful in a set of art’s objects, it could be not clear the existence of prototypes and, since in such cases it could be S(µP ) = ∅, the study of the family resemblance for these predicates remains an open problem. 5.2 Certainly, to consider X ⊂ R is a restriction for this paper’s results. Nevertheless, most of the predicates fuzzy logic considers are those exhibiting a numerical characteristic, allowing to translate them into an interval in R. This is the case, for example, of P = tall in a big population X, that is translated into the interval [0, 2], in meters, by the numerical characteristic ‘Height’, and the predicate Q = big in such interval. That is, by the identification µtall = µbig ◦ Height, or µtall (x) = µbig (Height(x)), for all x in X, once the use of big in [0, 2] is chosen accordingly with that of tall interpreted, as it turn, by big height. There are not so obvious cases that, also, can be translated into an interval through a more complex process, consisting in identifying each x in X with an n-tuple of significative parts in x, x := (x1 , . . . , xn ), with numerical characteristics Chi (xi ) ∈ [ai , bi ], and taking Ch(x) = A(Ch1 (x1 ), . . . , Chn (xn )) ∈ [a, b], with some n-place aggregation function A. An example is given by P = beautiful in a set X of paintings, if each painting x can be partitioned in x = x1 ∪ . . . ∪ xn (xi ∩ xj = ∅, i = j), allowing to interpret the statement ‘x is P’ as the composite one (x1 is P ,. . . , xn is P ). Provided each component ‘xi is P ’ is numerically evaluable by a clearly explicitable characteristic Chi (xi ) ∈ R+ , the values of µP could be obtained by µP (x) =
r1 Ch1 (x1 ) + . . . + rn Chn (xn ) (ri ≥ 0), r1 + . . . + rn
Hence,
that are in the interval [min µP , max µP ]. n an Of course, A(a1 , . . . , an ) = r1 ar11+...+r +...+rn , is not the only way of reasonably aggregating the n-tuples (Ch1 (x1 ), . . . , Chn (xn )) ∈ Rn . For example, if ‘x is P’ is Examples 1 and 4 in section 3, do correspond to uses of interpretable by ‘x1 is P ’ and . . . and ‘xn is P ’, A can be P =big and P =around five, respectively. taken as a t-norm, provided Chi (xi ) ∈ [0, 1], 1 ≤ i ≤ n. Remark 4.1 In natural language, families [µ] must be ‘open’, but not ‘closed’ like they were defined in section 2. These fam6 Conclusions ilies are here static (sets), but in natural language they should be dynamical. With time, an element σ that was not in [µ], 6.1 but possibly with a not too low degree DR(σ, µ), could be in- The concept of family resemblances is here introduced as a cluded in [µ] and thus generating a new family, indeed, chang- crisp-binary relation for only a particular type of fuzzy sets in ing the relation fr. Actually, fr is not a permanent relation, in the real line, with 0-points and 1-points. Even more restrictive natural language it is a changing one. Only with families of is the class of Riemann-integrable membership functions, to resemblance taken as classical sets, human thought seems to which pairs a degree of family resemblance is assigned. Anybe impossible (see [7]). way, this paper should be viewed as only a first trial to con0.8 + 1 = 0.9, 2 and µ, σ show high family resemblance. DR(µ1P , µ2P ) =
sider Wittgenstein’s idea on family resemblances with fuzzy ISBN: 978-989-95079-6-8
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IFSA-EUSFLAT 2009 sets . For example, the definition of fr could be, perhaps, extended to other fuzzy sets without 1-points (non-normalized fuzzy sets), or without 0-points, as well as to pairs (µ, σ) with either µ, or σ in {0, 1}X . That is, to non-data fuzzy sets, to fuzzy sets resulting from computations with data ones. In addition, the selected degree DR cannot be considered as the definitive definition to measure to what extent there is family resemblance. More study on the subject is deserved since, for example, DR can be applied to pairs (µ, σ) ∈ / fr, and more accurate values can be obtained in some cases with non sym2 ·I(µ,σ) . metric functions DR(µ, σ) = r1 ·Ovch(µ,σ)+r r1 +r2 6.2
References [1] L. Wittgenstein. Tractatus logico-philosophicus. Routledge & Kegan Paul, London, 1961. [2] L. Wittgenstein. Philosophical Investigations. Basil Blackwell, London, 1973. [3] E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo, and E. Casti˜neira. Computing with Antonyms. In Forging New Frontiers: Fuzzy Pioneers I, volume 217 of Studies in fuzziness and soft computing, pages 133–153. Springer, Berlin, 2007. [4] E. Trillas. On a model for the meaning of predicates. A na¨ıve approach to the genesis of fuzzy sets. In Rudolf Seising, editor, Fuzzy sets theory. Philosophy and criticism, Studies in fuzziness and soft computing. Springer Verlag, forthcoming.
Of course, definition 2.1 could also be applied to crisp sets by [5] L. Valverde E. Trillas. An inquiry into indistinguishability opsimply allowing either µ or σ to only have ‘constant pieces’. erators. Aspects of Vagueness. H.J. Skala, S. Termini, E. Trillas At this respect, examples like the one in figure 14 are of (Eds.), pages 231–256. Reidel, Dordrecht, 1984. some interest. In such figure, µ is the trapezoidal fuzzy set [6] S.V. Ovchinnikov. Representation of transitive fuzzy relations. (3.8, 4, 6, 6.2), and σ does represent the crisp subset [4, 6]. Aspects of Vagueness. H.J. Skala, S. Termini, E. Trillas (Eds.), pages 105–118. Reidel, Dordrecht, 1984.
σ
3
4
[7] S. Pinker. Words and Rules. Basic Books, N.Y., 1999.
µ
5
6
7
8
Figure 14: Since Ovch(µ, σ) = 0.83, I(µ, σ) = 0.98, it follows DR(µ, σ) = 0.90, a big value suggesting a so high resemblance between µ and σ that, perhaps, could allow to identify µ with µP , P = Almost between 4 and 6. That is, µ could be ‘linguistically approached’ by the imprecise predicate P . 6.3 Maybe what is here introduced could be useful to approach the (pending) problem of ‘linguistic approximation’. Namely, given the output f : X → [0, 1] of a system of fuzzy rules, how to find a predicate P in X such that f could be identified with µP ? A possible way could come once identified a membership function µP (representing a use of the known predicate P ), such that, DR(f, µP ) > ε, and Sup|f − µP | < δ / fr. Then ‘P ’ can (for fixed ε > 0, δ > 0), even if (f, µP ) ∈ be called an (ε, δ)-linguistic approximation of f . For example, in 6.2, Almost between 4 and 6 is an (0.89, 0.2)-linguistic approximation of µ. 6.4 In this paper, the threshold 0.7 is taken for the sake of illustrating the idea of ‘high family resemblance’. Nevertheless, and although 0.7 seems to be a good enough value for the examples shown in figures 7 to 14, such threshold’s value still remains to be studied.
Acknowledgment The authors express their thanks to the two anonymous reviewers for their interesting and constructive remarks. ISBN: 978-989-95079-6-8
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