103

On Fuzzy Sets Philosophical Foundations

Luis Adrian Urtubey

103.1 The Emergence of Fuzzy Sets

Doubtlessly the motivations for the development of fuzzy logic are deemed closelyassociated with technologically biased concerns. In his 1962 paper -where the termfuzzy is supposed to appear for the first time in its current usage- Zadeh was primar-ily attentive to the emergence and evolution of system theory as well as its impacton the field of electric engineering [9], [10]. Moreover when Zadeh uses the termfuzzy in this paper, he is not cared about inanimate systems, but he is mostly trou-bled above animate or biological systems, which are generally orders of magnitudemore complex than man-made systems. It brought himself to claim that it appearsnecessary to count on a radically different class of mathematics, one that accountsfor fuzzy or cloudy quantities which are not describable in terms of probability dis-tributions. Moreover even in the case of man-made systems it turns out apparent theneed of such innovation.1

Ironically, Zadeh also referred there to the fact that in most practical cases thea priori data as well as the criteria by which a system is judged are far from be-ing precisely specified or having accurately known probability distributions. I sayironically, because later on one of the most frequent criticism on the application offuzzy sets to deal with vagueness and imprecision -specially from the philosophicalside- has been focussed on the very assignment of determined quantities by meansof fuzzy sets membership functions.2

As someone coming from philosophy, I would like to address here some reflec-tions about fuzzy sets from a philosophical standpoint, far from these criticisms.These are not about completely new issues, but I think that they will show themselvesto be significant to look into other connections between philosophy and fuzziness.

103.2 What Logic Has to Do With It

This is a question that many logicians must have likely formulated when they wereconfronted with the increasing application of fuzzy logic. For some philosophically-minded logicians, fuzzy logic seemed to be appropriate for treating the logic ofvagueness. It produced immediate reactions against fuzzy logic led by logicians

1 Rudolf Seising considers the rise and evolution of fuzzy logic over these years. See [5].2 Carl W. Entemann has dealt with some of this criticisms in [2].

R. Seising et al. (Eds.): On Fuzziness: Volume 2, STUDFUZZ 299, pp. 713–717.DOI: 10.1007/978-3-642-35644-5_103 © Springer-Verlag Berlin Heidelberg 2013

714 103 On Fuzzy Sets Philosophical Foundations

who remain stubbornly faithful to classical bivalent logic.3 One can say that manygenerations up to now have philosophically grew up under the influence of the un-favorable reception of fuzzy logic by some notorious philosophers and logicians. Inever had sympathy for this attitude, much less if one considers for example, that itis likely to draw a parallel between the rise of Aristotle’s syllogistic -according to theinterpretation of some prestigious scholars- and fuzzy logic beginnings. This is oneof the lines I want to pursue here.

Specialists in Aristotle’s logic have paid attention to the influence of geometry andthe theory of proportions in the origins of his syllogistic. In particular, Robin Smithdares say that Aristotles syllogistic theory is more properly regarded as mathematicsthan as logic as understood by most contemporary logicians [6].

Leaving aside most philological details, it turns out that many techniques andideas which Aristotle makes use of in the theory of syllogism are borrowed fromharmonic theory, i.e. from the mathematics of music. Moreover many clues splitthroughout Aristotles Prior Analitics would suggest a mathematical context muchmore than an argumentative one. If things went this way, while working on hissyllogistic, Aristotle could have had in mind something else: a mathematical theoryof epistemology, as R. Smith have called it, ultimately derived from Plato’s Theory ofForms. The famous passage in Plato’s Republic VI, where mathematics is describedas in some way an image of true knowledge, had already suggested a theory of thistype.

103.3 Fuzzy Sets Epistemimological Foundations

Plausibly one may claim that the epistemological grounds of fuzzy sets have not re-ceived as much attention as their logic-semantical conundrums. Philosophically theproblem posed by an epistemology of fuzzy sets might be tackled from a Kantian per-spective, as a consequence of the thesis that properties in itself cannot be perceived,but only the individual phenomena. Kant’s Critique established that human searchfor knowledge is possible because there are phenomena, but at the same time, it intro-duced an insuperable dichotomy between a subject who perceives the phenomenonon the one side and the phenomenal object on the other. The property, the Kantianthink in itself, remains impenetrable to the perceiving subject. Consequently the trueperceivable phenomenon is the individuation; the split of properties and the ongoingparticipation an individual x has in a property P. The language, as a universal humanmean of representation, aims just at representing these perceptions. Therefore, evenin language, it cannot be either a truly clear-cut determination of predicates by setsof individuals because properties inexorably escape that sort of determination.

Such a conclusion is the main contempt of the philosophy of knowledge of ArthurSchopenhauer, one of the most prominent post-kantian and anti-idealist philosophersof the XIXth century. From the beginning, human search for knowledge confrontsa subject with an object that marks subjects limitations. It is Schopenhauer main

3 For a recent publication covering that issues see [3].

103.4 A Literary Digression on Borges and “The Zahir” 715

concern that this opposition never is over. There is no way, including dialectic, ofovercoming this limitation human beings are damned to, therein the Schopenhaue-rian pessimistic conclusion that see human existence as striving continuously in asenseless battle [4].

Recently Manuel Tarrazo has also stressed this connection of Schopenhauer epis-temic framework with the existing concept of fuzziness [7]. That being the case, afuzzy set can be seen epistemologically as an accurate and mathematically elaboratedmean to represent the split of properties in many individuals.

According to this interpretation, a fuzzy set contains infinite instances of possibleindividual perceptions concerning certain property. Notably all these perceptions canonly involve individuals and then fuzzy sets represent properties only indirectly, bygiving the individual different degrees of participation. In this way properties areimplicitly included in the fuzzy representation of the world. Consequently, degreesof membership in a fuzzy set, have to be naturally explained by the impossibility ofexpressing properties in a direct way.

Zadeh’s computing with words and perceptions approach introduced in the lastyears in soft computing, has achieved a more epistemological interpretation of fuzzysets [12]. Notably E. Trillas has also emphasized in different places the relationshipbetween the use of words and fuzzy sets stemming from the worldly interaction of asubject with objects and properties [8].

103.4 A Literary Digression on Borges and “The Zahir”

It is noteworthy that the story called The Zahir, by the Argentinian writer JorgeLuis Borges, has an appealing connection with the split-property-approach to fuzzysets considered above. The story refers the existence of a fantastic object, the Za-hir, which has the power of representing universal properties in a single object. InBuenos Aires at the time of the story -Borges said, not lacking in fine irony- thezahir is a twenty-cent coin that the story-teller incidentally obtains in a corner-bar-and-grocery-store: “I asked the owner for an orange gin; with the change I was giventhe Zahir; ...The thought struck me that there is no coin that is not the symbol of allthe coins that shine endlessly down throughout history and fable”. [1]

There have been many different objects which embodied the Zahir at differenttimes all around the world. Among them, “in Gujarat, at the end of the eighteenthcentury, Zahir was a tiger” ... “a magic tiger that was the perdition of all who sawit”. The figure of this tiger was painted in a palace: In “the jail at Nighur, therewas a cell whose floor, walls and vaulted ceiling was covered by a drawing (in bar-baric colors that time, before obliterating had refined) of an infinite tiger. It was atiger composed of many tigers, in the most dizzying of ways; it was crisscrossedwith tigers, striped with tigers, and contained seas and Himalayas and armies thatresembled other tigers”. [1]

716 References

Thinking about the story, one can also figure out a fuzzy set as a mean of express-ing something like the infinite tiger of that picture. Capturing the infinite non-perfectindividuals that are embraced by a property, a fuzzy set can represent the very prop-erty as it shows up.

103.5 Conclusion

As L. Zadeh used to say, Fuzzy Logic is not fuzzy. A fuzzy set is a very precisedevice to deal with the imprecision of human life. Fuzzy sets and Fuzzy Logic haveinnumerable affinities with several areas of human knowledge [11]. I have exploredsome relationships of fuzzy sets with mainstream epistemology and metaphysics. Ihave also shown that there are antecedents concerning the impact of technologicallymotivated developments, as may be the case of Fuzzy Logic, on theoretical under-pinnings. This fact has been illustrated by appealing to such a prestigious logicaltheory as Aristotle’s syllogistic.

Contrasting with the project of a mathematical epistemology of perfectly definedconcepts, which Aristotle might have had in mind while he was elaborating his syllo-gistic theory, it makes sense to see Fuzzy Logic now as the endeavor for developinga mathematical epistemology of commonplace imperfect intellectual constructs.

Acknowledgement. I owe special thanks to the Editors for inviting me to participatein this volume. This work was partially supported by grant PICT2007 BID-1606.

References

1. Borges, J.L.: Collected Fictions, Transl. by Andrew Hurley. Penguin Books U.K. (1999)2. Entemann, C.W.: Fuzzy Logic: Misconceptions and Clarifications. Artificial Intelligence

Review 17, 65–84 (2002)3. Ronzitti, G. (ed.): Vagueness: A Guide. Springer (2011)4. Schopenhauer, A.: The World as Will and as Representation, Transl. by E.F.J. Payne.

Dover (1958)5. Seising, R.: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and its Initial

Applications – Developments up to the 1970s, 1st edn. STUDFUZZ, vol. 216. Springer,Heidelberg (2007)

6. Smith, R.: The Mathematical Origins of Aristotle’s Syllogistic. Archive for History ofExact Sciences 19, 201–210 (1978)

7. Tarrazo, M.: Schopenhauer’s Prolegomenon to Fuzziness. Fuzzy Optimization and Deci-sion Making 3, 227–254 (2004)

8. Trillas, E.: On the Use of Words and Fuzzy Sets. Information Sciences 176, 1463–1487(2006)

9. Zadeh, L.A.: From Circuit Theory to System Theory. Proceedings of the IRE 50, 856–865(1962)

10. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)

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11. Zadeh, L.A.: Coping with the Imprecision of the Real World: An Interview with Lotfi A.Zadeh. Communications of the ACM 27(4) (April 1984)

12. Zadeh, L.A.: From Computing with Numbers to Computing with Words: From Manip-ulation of Measurements to Manipulation of Perceptions. IEEE Transactions on Circuitsand Systems I: Fundamental Theory and Applications 45(1), 105–111 (1999)