Rough Sets and Sorites Paradox

Abstract

We discuss the rough set approach to approximation of vague concepts. There are already published several papers on rough sets and vague concepts staring from the seminal papers by Zdzislaw Pawlak. However, only a few of them are discussing the relationships of rough sets with the sorites paradox. This paper contains a continuation of discussion on this issue.

Keywords

• vagueness
• vague concept
• sorites paradox
• (adaptive) rough set

1 Introduction

The rough set (RS) approach was proposed by Professor Zdzislaw Pawlak in 1982 [ 35, 36, 40 ] as a tool for dealing with imperfect knowledge, in particular with vague concepts. Over the years many applications of methods based on rough set theory alone or in combination with other approaches have been developed.

The rough set approach seems to be of fundamental importance in arti cial intelligence and cognitive sciences, especially in machine learning, data mining and knowledge discovery from databases, pattern recognition, decision support systems, expert systems, intelligent systems, multiagent systems, adaptive systems, autonomous systems, inductive reasoning, commonsense reasoning, adaptive judgement, con ict analysis.

Rough sets have established relationships with many other approaches such as fuzzy set theory, granular computing, evidence theory, formal concept analysis, (approximate) Boolean reasoning, multicriteria decision analysis, statistical methods, decision theory, matroids have been clari ed. Despite the overlap with many other theories rough set theory may be considered as an independent discipline in its own right. There are reports on many hybrid methods obtained by combining rough sets with other approaches such as soft computing (fuzzy sets, neural networks, genetic algorithms), statistics, natural computing, mereology, principal component analysis, singular value decomposition or support vector machines.

In particular some relationships of the rough set approach with vague concepts were shown (see, e.g., [2, 3, 5, 13, 29, 31, 32, 37, 38, 41, 45, 49{51, 55]). However, the relationships with sorites paradox are not explored well yet. In this paper, we extend a discussion on this topic, especially presented in [ 25 ].

Let us also note that the relationships with vague concepts of other approaches to uncertainty such as fuzzy sets or graded consequence are elaborated in the literature (see, e.g., [7{13, 17, 28, 43, 44]).

This paper is structured as follows. In Sect. 2 we present some preliminaries on vague sets. Rudiments of rough sets, in particular approximations of concepts, are discusses in Sect. 3. The rough set approach to sorites paradox is presented in Sect. 4. The issue of higher order vagueness in rough sets is covered in Sect. 5. In Sect. 6, we present some constraints on induced classi ers for vague concepts related to the sorites paradox. They are making it possible to eliminate the contradiction characteristic to sorites paradox which is related to behaviour of the classi er when it passes through di erent approximation regions. Sect. 7 emphasizes the need for development the adaptive rough set approach. 2

Vague Sets

Mathematics requires that all mathematical notions (including set) must be exact, otherwise precise reasoning would be impossible. However, philosophers (see, e.g., [ 26 ] and recently computer scientists as well as other researchers have become interested in vague (imprecise) concepts. Moreover, in the XX century one can observe the drift paradigms in modern science from dealing with precise concepts to vague concepts, especially in the case of complex systems (e.g., in economy, biology, psychology, sociology, quantum mechanics).

Almost all concepts we are using in natural language are vague [ 1, 6 ]. Therefore, common sense reasoning based on natural language must be based on vague concepts and not on classical logic. Interesting discussion of this issue can be found in [ 45 ]. The idea of vagueness can be traced back to the ancient Greek philosopher Eubulides of Megara (ca. 400BC) who rst formulated so called \sorites" (heap) and \falakros" (bald man) paradox (see, e.g., [ 26 ]). There is a huge literature on issues related to vagueness and vague concepts in philosophy (see, e.g., [4, 14, 19, 26, 27, 46{48]).

Vagueness is often associated with the boundary region approach (i.e., existence of objects which cannot be uniquely classi ed relative to a set or its complement) which was rst formulated in 1893 by the father of modern logic, German logician, Gottlob Frege (1848-1925) (see [ 15 ]). According to Frege (see Grundgesetze der Arithmetik, vol. ii, Sect.56 [ 15, 16 ]) the concept must have a sharp boundary:

To the concept without a sharp boundary there would correspond an area that would not have any sharp boundary { line all around.

It means that mathematics must use crisp, not vague concepts, otherwise it would be impossible to reason precisely. However, vagueness in opinion of Ludwig Wittgenstein is an essential feature of language with semantics speci ed by 'language games'. A language is not a calculus with rigid rules that provide for all possible circumstances. There are many vague concepts in natural languages [ 1, 6 ]. One should also note that vagueness also relates to insu cient speci city, as the result of lack of feasible searching methods for sets of features adequately describing concepts.

Discussion on vague (imprecise) concepts in philosophy includes the following characteristic features of them [ 26 ]: (i) the presence of borderline cases, (ii) boundary regions of vague concepts are not crisp, (iii) vague concepts are susceptible to sorites paradox. In the sequel we discuss these issues in the RS framework. The reader can nd the discussion on application of the RS approach to vagueness in [ 45 ]. 3

Rough Set Based Concept Approximation

The starting point of rough set theory is the indiscernibility relation, which is generated by information about objects of interest (de ned later in this section as signatures of objects). The indiscernibility relation expresses the fact that due to a lack of information (or knowledge) we are unable to discern some objects employing available information (or knowledge). This means that, in general, we are unable to deal with each particular object but we have to consider granules (clusters) of indiscernible objects as a fundamental basis for our theory.

>From a practical point of view, it is better to de ne basic concepts of this theory in terms of data. Therefore we will start our considerations from a data set called an information system.

Suppose we are given a pair A = (U; A) of non-empty, nite sets U and A, where U is the universe of objects, and A { a set consisting of attributes, i.e. functions a : U ! Va, where Va is the set of values of attribute a, called the domain of a. The pair A = (U; A) is called an information system (see, e.g., [ 34 ]). Any information system can be represented by a data table with rows labeled by objects and columns labeled by attributes. Any pair (x; a), where x 2 U and a 2 A de nes the table entry consisting of the value a(x).

Any subset B of A determines a binary relation INDB on U , called an indiscernibility relation, de ned by x INDB y if and only if a(x) = a(y) for every a 2 B; (1) where a(x) denotes the value of attribute a for object x:

Obviously, INDB is an equivalence relation. The family of all equivalence classes of INDB, i.e., the partition determined by B, will be denoted by U=INDB, or simply U=B; an equivalence class of INDB, i.e., the block of the partition U=B, containing x will be denoted by B(x) (other notation used: [x]B or more precisely [x]INDB ). Thus in view of the data we are unable, in general, to observe individual objects but we are forced to reason only about the accessible granules of knowledge (see, e.g., [ 33, 36, 42 ]).

If (x; y) 2 INDB we will say that x and y are B-indiscernible. Equivalence classes of the relation INDB (or blocks of the partition U=B) are referred to as B-elementary sets or B-elementary granules. In the rough set approach the elementary sets are the basic building blocks (concepts) of our knowledge about reality. The unions of B-elementary sets are called B-de nable sets.

For B A we denote by InfB(x) the B-signature of x 2 U , i.e., the set f(a; a(s)) : a 2 Bg. Let IN F (B) = fInfB(s) : s 2 U g. Then for any objects x; y 2 U the following equivalence holds: xINDBy if and only if InfB(x) = InfB(y).

The indiscernibility relation will be further used to de ne basic concepts of rough set theory. Let us de ne now the following two operations on sets X U LOWB(X) = fx 2 U : B(x)

Xg;

UPPB(X) = fx 2 U : B(x) \ X 6= ?g; assigning to every subset X of the universe U two sets LOWB(X) and UPPB(X) called the B-lower and the B-upper approximation of X, respectively. The set BNB(X) = UPPB(X)

LOWB(X); will be referred to as the B-boundary region of X:

If the boundary region of X is the empty set, i.e., BNB(X) = ?, then the set X is crisp (exact) with respect to B; in the opposite case, i.e., if BNB(X) 6= ?, the set X is referred to as rough (inexact) with respect to B. Thus any rough set, in contrast to a crisp set, has a non-empty boundary region.

Thus a set is rough (imprecise) if it has nonempty boundary region; otherwise the set is crisp (precise). This is exactly the idea of vagueness proposed by Frege.

Let us observe that the de nition of rough sets refers to data (knowledge), and is subjective, in contrast to the de nition of classical sets, which is in some sense an objective one.

Due to the granularity of knowledge, rough sets cannot be characterized by using available knowledge. Therefore with every rough set we associate two crisp sets, called lower and upper approximation. Intuitively, the lower approximation of a set consists of all elements that surely belong to the set, whereas the upper approximation of the set constitutes of all elements that possibly belong to the set, and the boundary region of the set consists of all elements that cannot be classi ed uniquely to the set or its complement, by employing available knowledge. 4

Approximations of Concepts and Sorites Paradox

Let us consider the heap paradox. 1. 10,000 grains of sand is a heap of sand. 2. 10,000 grains of sand is a heap of sand, then 9999 grains of sand is a heap of sand. 3. 9999 grains of sand is a heap of sand, then 9998 grains of sand is a heap of sand. (2) (3) (4) 4. : : : 5. Conclusion. 1 grain of sand is a heap of sand.

For a given set X by card(X) we denote the cardinality of X. Let us consider the sequence of collections of grains of sand: x1; : : : ; xi; xi+1; : : : ; xN ; such that card(xi) card(xi+1) = 1 for i = 1; : : : ; N 1.

It is worthwhile mentioning that the concept of heap is vague. This concept may be perceived di erently by di erent agents. Now let us consider an agent ag having a decision system A = (U; A; d), where U fx1; : : : ; xi; xi+1; : : : ; xN g is a family of collections of grains and A is a set of conditional attributes over U . The decision d assigns to each x 2 U the decision d(x) equal to 1 if x is a heap and 0, otherwise. This decision is made, e.g., by another agent agdec on the basis of some attributes (usually di erent from attributes from A). We denote by H the decision class fx 2 U : d(x) = 1g and by H the decision class fx 2 U : d(x) = 0g. In particular, the decision d is assigned to each xi from the considered sequence. The agent ag de nes a partition of U using the the lower approximation of H, i.e., LOWA(H), the boundary region of H, i.e., BNA(H), and the lower approximation of H, i.e., LOWA( H).

In our example, we assume that x1 2 LOWA(H) and xN 2 LOWA( H).

By a bounce we understand any i such that one of the following conditions is satis ed: (i) xi 2 LOWA(H) & xi+1 2 BNA(H), (ii) xi 2 LOWA(H) & xi+1 2 LOWA( H), (iii) xi 2 BNA(H) & xi+1 2 LOWA( H).

Now, we explain why such bounces may occur.

Let us consider the rst case. The two remaining cases are analogous. One could argue that we have a problem because there exists i such that xi 2 LOWA(H) and xi+1 2 BNA(H) but the di erence between cardinalities xi and xi+1 is negligible (card(xi) card(xi+1) = 1) from the point of view of the concept heap. Observe that in the rough set approach the agent ag using the decision system A is perceiving objects (i.e., in our example collections of grains) by means of attributes from A. Let us assume that A = fcardg and the conditional attribute card assigns to any collection of grains x 2 U its cardinality. The decision d is taken by another agent agdec and it may be based, e.g., on the basis of a shape of collection of grains. For example, d(x) = 1 if the shape of x is trapezoidal with su ciently large ratio of the trapezoid hight to the length of the longest parallel sides of the trapezoid, and 0, otherwise. It may happen in U that the decision made by the agent agdec for all collections of grains from the elementary granule A(xi) (with the same cardinality, say n) are equal to 1, i.e., all collections of grains from A(xi) have the relevant trapezoidal shape accepted by d as the positive examples of the concept heap. However, in case of A(xi+1) (consisting of collections of grains with the same cardinality equal to n 1) there are in U collections x; y of grains such that d(x) = 1 (i.e., x 2 H) and d(y) = 0 (i.e., y 2 H). This explains that the considered case of bounce is possible despite the fact that the di erence between card(xi) and card(xi+1) looks negligible from the point of view of the concept heap.

>From the above considerations, we conclude that in general one can assume that for any i:

xi 2 LOWA(H) implies xi+1 2 LOWA(H) [ BNA(H) [ LOWA( H); (5) instead of xi 2 LOWA(H) implies xi+1 2 LOWA(H): (6)

Analogous conditions may be formulated when we change the condition in the predecessor from the lower approximation of H, to the boundary region of H, or to the lower approximation of the complement of H.

Of course, in some cases some arguments of the alternative on the right hand side may be eliminated. For example, in some cases of decision table A of agent ag in Eq. 5 may be eliminated on the right had side of implication the third argument of the alternative. 5

Higher Order Vagueness and Rough Sets: Toward

Adaptive Rough Sets

In [ 26 ], it is stressed that boundaries of vague concepts are not crisp. In the de nition presented in this chapter, the notion of boundary region is de ned as a crisp set BNB(X). However, let us observe that this de nition is relative to the subjective knowledge expressed by attributes from B. Di erent sources of information may use di erent sets of attributes for concept approximation. Hence, the boundary region can change when we consider these di erent views. Another reason for boundary change may be related to incomplete information about concepts. They are known only on samples of objects [ 18 ]. Hence, when new objects appear again the boundary region may change. >From the discussion in the literature it follows that vague concepts cannot be approximated with satisfactory quality by static constructs such as induced membership inclusion functions, approximations or models derived, e.g., from a sample. Understanding of vague concepts can be only realized in a process in which the induced models are adaptively matching the concepts in dynamically changing environments. This conclusion seems to have important consequences for further development of rough set theory in combination with fuzzy sets and other soft computing paradigms for adaptive approximate reasoning. For further details the reader is referred, e.g., to [ 49, 50, 56 ].

>From the above considerations it follows that for dealing with higher order vagueness one should consider an extension of the rough set approach to the adaptive rough set approach. In this approach, approximations of a vague concept are considered over a family of decision systems fAtgt2T , where T is a set of indices, e.g., time points. Hence, we obtain a family of the lower approximations, upper approximations and boundary regions of the considered vague concept which are changing, e.g., over time (see Figure 1).

It is worthwhile mentioning that the elements of this family are obtained through interaction with the environment what is pointing to the necessity of adaptive strategy RS1

RS2

RS3

RS4

RS5 ...

time 1 embedding the adaptive rough set approach in the framework of interactive granular computing and WisTech program (see, e.g., [22{24, 21]). 6

Constraints on Bouncing Between Di erent

Approximation Regions

If one would like to obtain some constraints on bouncing collections of sand grains between di erent approximation regions of the concept 'to be a heap of sand' assuming that succeeding collections are obtained by individually removing one grain from the preceding ones, then more details on rough set based approximations should be considered. For example, one may require that the changes of membership functions on consecutive collections of sand grains are below a given threshold. Let us consider an illustrative example to explain this issue in more detail.

First of all, one should note that usually information about approximated concept is partial, e.g., provided by a sample of cases 'for' and 'against' a given concepts. Hence, in the rough set approach were developed methods for inductive extensions of approximation spaces from samples U of objects represented by decision systems on the universe U of all objects [ 39, 30 ].

In Figure 2 is presented a simple example of classi er for a concept C U . The classi er is induced from a given partial information about C represented by a decision system Ad = (U; A; d) with the set of objects U U and the decision d equal (or almost equal) on U to the restriction to U of the characteristic function of C. The classi er represents an approximation of the characteristic function of the concept C U .

The procedure of con ict resolution shown in Figure 2 between induced decision rules matching a given new case x (belonging to an extension U of U , i.e., U U ) (and perceived as a signature of x, i.e., InfA(x)) may be realized using arguments 'for' and 'against' membership in C determined by these rules. The arguments are aggregated using weights wk as it is presented in Figure 3 what nally leads to the classi er computing the membership function C for a given concept C.

Now, one can consider the membership function H as an approximation of the characteristic function of the vague concept `to be a heap of sand' H U 1  C 2  C 3  C G1  (1,2,3) x

Match 1  C 2  C 3  C 4  C G2  (1,2,3,4)

Conflict_res i input granule ((1, 2, 3),(4, 5, 6, 7))

matching granule

Conflict_res (Match(InfA(x),G1,G2)) induced (analogously as above) from a partial information represented by a decision system Ad = (U; A; d), where d is a characteristic function of a vague concept H U restricted to the sample U U . We use H in considerations concerning the paradox of heap of sand (Figure 4). Note that the induced approximations of the concept H are now de ned as follows. The lower approximation of H is de ned by LOWA(H ) = fx 2 U : H (x) = 1g, the upper approximation of H is de ned by UPPA(H ) = fx 2 U : H (x) > 0 _ H (x) = undef inedg and the boundary region of H : BNA(H ) = UPPA(H ) n LOWA(H ).

In the considered example, we assume that the induced approximation of H represented by H is consistent with a given sequence x1; : : : ; xN , i.e., for any xi and xi+1 representing consecutive collections of sand grains after individually removing one grain in each step, we have H (xi) = 1 if xi 2 H and

H (xi) = 0 if xi 2= H : In Figure 4 is presented a simple property of behavior of the model of H on elements of a sequence x1; : : : ; xN . If some additional constraints concerning weights are satis ed than one can see that the boundary region cannot be omitted. Moreover, using the assumption about a `bounce size' of the membership values in passing from xi to xi+1, one can see that it can be necessary for the considered sequence to 'spend more time' in the boundary region before going out of it. One can specify such assumptions about 'bounce size' using the following constraints for induced classi ers: wC (xi) wC (xi+1) and wC (xi+1) wC (xi) , where is a given threshold bounding bounces in degrees of memberships of xi and xi+1.

xi

xi+1 x1,..., xi , xi1,..., xN ;  H (x1)  1,  H (xN )  0 If then xi  LOWA (H  )  xi1  UPPA (H  ) wC ( xi1)  wC ( xi ), wC ( xi1)  wC ( xi ) there exists i0 : xi0  BN A (H  ) xi  BN A (H  )  xi1  UPPA (U  \ H  ) xi  LOWA (U  \ H  )  xi1  LOWA (U  \ H  ) 1 We discussed the sorites paradox in the rough set approach. We have added a discussion on possible new constraints which should be added and preserved by approximations of vague concepts. These constraints are related to behavior of induced classi ers approximating vague concepts on sequences of objects considered in the sorites paradox for these vague concepts. We have also pointed the necessity of development of the adaptive rough set approach.

There are numerous logical approaches to vagueness (see, e.g., [ 20, 52, 54 ]). However, from the above considerations it follows that adaptive logic based on rough sets can be relevant for the outlined approach. It is worthwhile mentioning here the following sentences from [ 53 ]:

Aristotle's man of practical wisdom, the phronimos, does not ignore rules and models, or dispense justice without criteria. He is observant of principles and, at the same time, open to their modi cation. He begins with nomoi established law and employs practical wisdom to determine how it should be applied in particular situations and when departures are warranted. Rules provide the guideposts for inquiry and critical re ection. We plan to develop a logical approach to vagueness based on rough sets and adaptive judgement [21{24].

Acknowledgments. This work was partially supported by the Polish National Science Centre (NCN) grants DEC-2011/01/D /ST6/06981, DEC-2013/09/B/ST6/01568 as well as by the Polish National Centre for Research and Development (NCBiR) under the grant DZP/RID-I-44 / 8 /NCBR/2016.

References

1. G. P. Baker , P. M. S. Hacker (Eds.), Wittgenstein: Understanding and Meaning: Volume 1 of an Analytical Commentary on the Philosophical Investigations , Part II : Exegesis xx1- 184 , Blackwell, 2005 . (2nd Edition) .

2. M. Banerjee , M. K. Chakraborty , Foundations of vagueness: A category-theoretic approach , Electronic Notes in Theoretical Computer Science 82 ( 4 ) ( 2003 ) 10 { 19 .

3. J. G. Bazan , A. Skowron , R. Swiniarski , Rough sets and vague concept approximation: From sample approximation to adaptive learning, Transactions on Rough Sets V ( 2006 ) 39 { 62 .

4. M. Black , Vagueness: An exercise in logical analysis , Philosophy of Science 4 ( 4 ) ( 1937 ) 427 { 455 .

5. Z. Bonikowski , U. Wybraniec-Skardowska , Vagueness and roughness . Transactions on Rough Sets IX ( 2008 ) 1 { 13 .

6. L. C. Bums , Vagueness. An Investigation into Natural Languages and the Sorites Paradox , Kluwer, 1991 .

7. D. Dubois , F. Esteva , L. Godo , H. Prade , An information-based discussion of vagueness , In: Proceedings of the 10th IEEE International Conference on Fuzzy Systems , Melbourne, Australia, December 2- 5 , 2001 , IEEE, IEEE Computer Science Press, 2001 , 781 { 784 .

8. D. Dubois , L. Godo , H. Prade , F. Esteva , An information-based discussion of vagueness , In: H. Cohen , C. Lefebvre (Eds.), Handbook of Categorization in Cognitive Science , Elsevier, Amsterdam, 2005 , pp. 892 { 913 .

9. D. Dubois , J. Lang , H. Prade , Handling uncertainty, context, vague predicates, and partial inconsistency in possibilistic logic , In: D. Driankov , P. W. Eklund , A. L. Ralescu (Eds.), Fuzzy Logic and Fuzzy Control , IJCAI '91, Workshops on Fuzzy Logic and Fuzzy Control , Sydney, Australia, August 24 , 1991 , Proceedings, LNCS 833, Springer 1991, pp. 45 { 55 .

10. D. Dubois , H. Prade , Modeling uncertain and vague knowledge in possibility and evidence theories , In: R. D. Shachter , T. S. Levitt , L. N. Kanal , J. F. Lemmer (Eds.), UAI '88: Proceedings of the Fourth Annual Conference on Uncertainty in Arti cial Intelligence , Minneapolis, MN, USA, July 10 - 12 , 1988 , North-Holland, 1988 , pp. 303 { 318 .

11. D. Dubois , H. Prade , Fuzzy sets { a convenient ction for modeling vagueness and possibility , IEEE Transactions on Fuzzy Systems 2 ( 1 ) ( 1994 ) 16 { 21 .

12. D. Dubois , H. Prade , Modeling uncertain and vague knowledge in possibility and evidence theories , CoRR abs/1304 .2349. URL http://arxiv.org/abs/1304.2349

13. S. Dutta , S. Basu , M. K. Chakraborty , Many-valued logics, fuzzy logics and graded consequence: A comparative appraisal , In: K. Lodaya (Ed.), Logic and Its Applications , 5th Indian Conference, ICLA 2013 , Chennai, India, January 10 - 12 , 2013 , Proceedings, Springer, LNCS 7750 , Springer, 2013 , 197 { 209 .

14. K. Fine , Vagueness, truth and logic, Synthese 30 ( 1975 ) 265 { 300 .

15. G. Frege, Grundgesetzen der Arithmetik 2, Verlag von Hermann Pohle, 1903 .

16. P. Geach , M. Black (Eds.), Translations from the Philosophical Writings of Gottlob Frege, Blackwell , 1960 .

17. J. A. Goguen , The logic of inexact concepts , Synthese 19 ( 1968 -69) 325 { 373 .

18. T. Hastie, R. Tibshirani , J. H. Friedman , The Elements of Statistical Learning: Data Mining, Inference, and Prediction , Springer-Verlag, 2001 .

19. C. G. Hempel, Vagueness and logic, Philosophy of Science 6 ( 1939 ) 163 { 180 .

20. D. Hyde , Vagueness, Logic, and Ontology, Ashgate, 2008 .

21. A. Jankowski , Complex Systems Engineering: Wisdom for Saving Billions Based on Interactive Granular Computing (in preparation) , Springer, 2016

22. A. Jankowski , A. Skowron , A Wistech paradigm for intelligent systems, Transactions on Rough Sets VI ( 2007 ) 94 { 132 .

23. A. Jankowski , A. Skowron , Wisdom technology: A rough-granular approach , In: M. Marciniak , A . Mykowiecka (Eds.), Bolc Festschrift , Springer, Heidelberg, LNCS 5070, Springer, 2009 , 3 { 41 .

24. A. Jankowski , A. Skowron , R. W. Swiniarski , Interactive complex granules, Fundamenta Informaticae 133 ( 2-3 ) ( 2014 ) 181 { 196 .

25. A. Jankowski , A. Skowron , Rough sets and vague concepts , Analele Universitatii din Bucuresti. Seria Informatica LXII(3) ( 2015 ) 119 { 133 .

26. R. Keefe , Theories of Vagueness, Cambridge Studies in Philosophy, Cambridge, UK, 2000 .

27. R. Keefe , P. Smith , Vagueness: A Reader, MIT Press, Massachusetts, MA, 1997 .

28. J. Lawry , D. Dubois , A bipolar framework for combining beliefs about vague propositions , In: G. Brewka, T. Eiter , S. A . McIlraith (Eds.), Principles of Knowledge Representation and Reasoning: Proceedings of KR 2012 , Rome, Italy, June 10-14, 2012 , AAAI Press, 2012 .

29. S. Marcus, The paradox of the heap of grains, in respect to roughness, fuzziness and negligibility , In: L. Polkowski , A . Skowron (Eds.), Proceedings of RSCTC' 1998 , Warsaw, June 22-26, 1998 , LNAI 1424, Springer-Verlag, 1998 , 19 { 23 .

30. H.S. Nguyen , A. Skowron , Rough Sets: From Rudiments to Challenges . In: A. Skowron , Z. Suraj (eds.), Rough Sets and Intelligent Systems - Professor Zdzislaw Pawlak in Memoriam, Intelligent Systems Reference Library 42 , Springer 2013 , 75 - 173 .

31. E. Orlowska, Semantics of vague concepts , In: G. Dorn, P. Weingartner (Eds.), Foundation of Logic and Linguistics , Plenum Press, 1984 , 465 { 482 .

32. E. Orlowska, Reasoning about vague concepts , Bulletin of the Polish Academy of Sciences, Mathematics 35 ( 1987 ) 643 { 652 .

33. S. K. Pal , L. Polkowski , A . Skowron (Eds.), Rough-Neural Computing : Techniques for Computing with Words , Cognitive Technologies , Springer-Verlag, 2004 .

34. Z. Pawlak , Information systems - Theoretical foundations, Information Systems 6 ( 1981 ) 205 { 218 .

35. Z. Pawlak , Rough sets, International Journal of Computer and Information Sciences 11 ( 1982 ) 341 { 356 .

36. Z. Pawlak , Rough Sets: Theoretical Aspects of Reasoning about Data, System Theory, Knowledge Engineering and Problem Solving 9 , Kluwer Academic Publishers, 1991 .

37. Z. Pawlak , Vagueness and uncertainty: A rough set perspective , Computational Intelligence: An International Journal 11 ( 1995 ) 217 { 232 .

38. Z. Pawlak , Vagueness - A rough set view . In: J. Mycielski , G. Rozenberg , A . Salomaa (Eds.), Structures in Logic and Computer Science , Springer, LNCS 1261 , Springer, 1997 , 106 { 117 .

39. Z. Pawlak , A. Skowron , Rough sets: Some extensions , Information Sciences 177(1) ( 2007 ) 28 { 40 .

40. Z. Pawlak , A. Skowron , Rudiments of rough sets, Information Sciences 177(1) ( 2007 ) 3 { 27 .

41. L. Polkowski , M. Semeniuk-Polkowska: Boundries, borders, fences, hedges . Fundamenta Informaticae 129 ( 1-2 ) ( 2014 ) 149 { 159 .

42. L. Polkowski , A. Skowron , Rough mereological calculi of granules: A rough set approach to computation , Computational Intelligence: An International Journal 17 ( 3 ) ( 2001 ) 472 { 492 .

43. H. Prade , A two-layer fuzzy pattern matching procedure for the evaluation of conditions involving vague quanti ers , Journal of Intelligent and Robotic Systems 3 ( 2 ) ( 1990 ) 93 { 101 .

44. H. Prade , C. Testemale , Generalizing database relational algebra for the treatment of incomplete/uncertain information and vague queries , Information Sciences 34(2) ( 1984 ) 115 { 143 .

45. S. Read, Thinking about Logic: An Introduction to the Philosophy of Logic , Oxford University Press, 1994 .

46. G. Ronzitti (Ed.), Vagueness: A Guide , Logic, Epistemology, and The Unity of Science Series , vol. 19 , Springer, 2011 .

47. B. Russell , Vagueness, The Australian Journal of Psychology and Philosophy 1 ( 1923 ) 84 { 92 .

48. S. Shapiro (Ed.), Vagueness in Context, Clarendon Press, 2006 .

49. A. Skowron , Rough sets and vague concepts , Fundamenta Informaticae 64 ( 1-4 ) ( 2005 ) 417 { 431 .

50. A. Skowron , R. Swiniarski , Rough sets and higher order vagueness , In: D. Slezak , G. Wang , M. Szczuka , I. Duntsch, Y. Yao (Eds.), Proceedings of RSFDGrC' 2005 , Regina, Canada, August 31-September 3 , 2005 , Part I , LNAI 3641, SpringerVerlag , 2005 , 33 { 42 .

51. D. Slezak , P. Wasilewski , Foundations of rough sets from vagueness perspective , In: A. E. Hassanien , Z. Suraj , D. Slezak , P. Lingras (Eds.), Rough Computing: Theories, Technologies and Applications , IGI Global, 2007 , 1 { 37 .

52. K. Tanaka , F. Berto , E. Mares , F. Paoli (Eds.), Paraconsistency: Logic and Applications , Springer, 2013 .

53. L. P. Thiele , The Heart of Judgment: Practical Wisdom, Neuroscience, and Narrative, Cambridge University Press, 2010 .

54. P. Verdee , S. van der Waart van Gulik , A generic framework for adaptive vague logics , Studia Logica 90 ( 2008 ) 385 { 405 .

55. M. Wolski, Science and semantics: A note on vagueness , In: A. Skowron , Z. Suraj (Eds.), Rough Sets and Intelligent Systems. Professor Zdzislaw Pawlak in Memoriam, Series Intelligent Systems Reference Library 43 , Springer 2013 , 623 { 643 .

56. Q. Zhang , J. Wang , G. Wang , H. Yu , The approximation set of a vague set in rough approximation space , Information Sciences 300 ( 2015 ) 1 { 19 .