Formalization of human categorization process using interpolative Boolean algebra. (English) Zbl 1395.91386

Summary: Since the ancient times, it has been assumed that categorization has the basic form of classical sets. This implies that the categorization process rests on the Boolean laws. In the second half of the twentieth century, the classical theory has been challenged in cognitive science. According to the prototype theory, objects belong to categories with intensities, while humans categorize objects by comparing them to prototypes of relevant categories. Such categorization process is governed by the principles of perceived world structure and cognitive economy. Approaching the prototype theory by using truth-functional fuzzy logic has been harshly criticized due to not satisfying the complementation laws. In this paper, the prototype theory is approached by using structure-functional fuzzy logic, the interpolative Boolean algebra. The proposed formalism is within the Boolean frame. Categories are represented as fuzzy sets of objects, while comparisons between objects and prototypes are formalized by using Boolean consistent fuzzy relations. Such relations are directly constructed from a Boolean consistent fuzzy partial order relation, which is treated by Boolean implication. The introduced formalism secures the principles of categorization showing that Boolean laws are fundamental in the categorization process. For illustration purposes, the artificial cognitive system which mimics human categorization activity is proposed.


91E10 Cognitive psychology
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Cohen, H.; Lafebre, C., Handbook of Categorization in Cognitive Science, (2005), New York, NY, USA: Elsevier, New York, NY, USA
[2] Bruner, J. S.; Goodnow, J. J.; Austin, G. A., A Study of Thinking, (1956), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[3] Goodwin, G. P.; Johnson-Laird, P. N., Mental models of Boolean concepts, Cognitive Psychology, 63, 1, 34-59, (2011)
[4] Feldman, J., A catalog of Boolean concepts, Journal of Mathematical Psychology, 47, 1, 75-89, (2003) · Zbl 1040.91093
[5] Rosch, E.; Lloyd, B. B., Principles of categorization, Cognition and Categorization, 27-48, (1978), Hillsdale, NJ, USA: Lawrence Erlbaum Associates, Hillsdale, NJ, USA
[6] Hampton, J. A., Concepts as prototypes, Psychology of Learning and Motivation: Advances in Research and Theory, 46, 79-113, (2006)
[7] Nosofsky, R. M.; Zaki, S. R., Exemplar and prototype models revisited: response strategies, selective attention, and stimulus generalization, Journal of Experimental Psychology: Learning Memory and Cognition, 28, 5, 924-940, (2002)
[8] Rosch, E., Slow lettuce: categories, concepts, fuzzy sets, and logical deduction, Concepts and Fuzzy Logic, 89-120, (2011), The MIT Press
[9] Zadeh, L. A., Fuzzy sets, Information and Computation, 8, 3, 338-352, (1965) · Zbl 0139.24606
[10] Zadeh, L. A., A note on prototype theory and fuzzy sets, Cognition, 12, 3, 291-297, (1982)
[11] Radojevic, D. G., Fuzzy set theory in the Boolean frame, International Journal of Computers Communications & Control, 3, 3, 121-131, (2008)
[12] Radojević, D. G., New [0,1]-valued logic: a natural generalization of Boolean logic, Yugoslav Journal of Operations Research, 10, 2, 185-216, (2000) · Zbl 0965.03027
[13] Zadeh, L. A., Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man, and Cybernetics, 3, 28-44, (1973) · Zbl 0273.93002
[14] Radojević, D., Real-valued realizations of boolean algebras are a natural frame for consistent fuzzy logic, Studies in Fuzziness and Soft Computing, 299, 559-565, (2013)
[15] Zadeh, L. A., Is there a need for fuzzy logic?, Information Sciences, 178, 13, 2751-2779, (2008) · Zbl 1148.68047
[16] Zadeh, L. A., From computing with numbers to computing with words—from manipulation of measurements to manipulation of perceptions, IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, 46, 1, 105-119, (1999) · Zbl 0954.68513
[17] Zadeh, L. A., A new direction in AI—toward a computational theory of perceptions, AI Magazine, 22, 1, 73-84, (2001)
[18] Fodor, J.; Roubens, M., Fuzzy Preference Modelling and Multicriteria Decision Support, (1994), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0827.90002
[19] Osherson, D. N.; Smith, E. E., On the adequacy of prototype theory as a theory of concepts, Cognition, 9, 1, 35-58, (1981)
[20] Kamp, H.; Partee, B., Prototype theory and compositionality, Cognition, 57, 2, 129-191, (1995)
[21] Johnson-Laird, P. N., Mental Models: Towards a Cognitive Science of Language, Inference, and Consciousness, (1983), Cambridge, Mass, USA: Harvard University Press, Cambridge, Mass, USA
[22] Radojević, D., Real sets as consistent Boolean generalization of classical sets, From Natural Language to Soft Computing: New Paradigms in Artificial Intelligence, 150-171, (2008), Bucharest, Romania: Editing House of Romanian Academy, Bucharest, Romania
[23] Milošević, P.; Petrović, B.; Radojević, D.; Kovačević, D., A software tool for uncertainty modeling using Interpolative Boolean algebra, Knowledge-Based Systems, 62, 1-10, (2014)
[24] Nosofsky, R. M., Exemplar-based approach to relating categorization, identification, and recognition, Multidimensional Models of Perception and Cognition, 363-393, (1992), New York, NY, USA: Lawrence Erlbaum Associates, New York, NY, USA
[25] Medin, D. L.; Schaffer, M. M., Context theory of classification learning, Psychological Review, 85, 3, 207-238, (1978)
[26] Radojević, D., Logical aggregation based on interpolative boolean algebra, Mathware & Soft Computing, 15, 1, 125-141, (2008) · Zbl 1152.03322
[27] Radojević, D., Interpolative relations and interpolative preference structures, Yugoslav Journal of Operations Research, 15, 2, 171-189, (2005) · Zbl 1109.03055
[28] Bache, K.; Lichman, M., UCI Machine Learning Repository, (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.