From Free Will Debate to Embodiment of Fuzzy Logic into Washing Machines: On Fuzzy and Rough Sets Approaches to Vagueness Modeling

Deep scientific ideas have at least one distinctive property: they can be applied both by philosophers in abstract fundamental debates and by engineers in concrete practical applications. Mathematical approaches to modeling of vagueness also possess this property. Problems connected with vagueness have been discussed at the beginning of XXth century by philosophers, logicians and mathematicians in developing foundations of mathematics leading to clarification of logical semantics and establishing of mathematical logic and set theory. Those investigations led also to big step in the history of logic: introduction of three-valued logic. In the second half of XXth century some mathematical theories based on vagueness idea and suitable for modeling vague concepts were introduced, including fuzzy set theory proposed by Lotfi Zadeh in 1965 [16] and rough set theory proposed by Zdzisşaw Pawlak in 1982 [4] having many practical applications in various areas from engineering and computer science such as control theory, data mining, machine learning, knowledge discovery, artificial intelligence. Concepts in classical philosophy and in mathematics are not vague. Classical theory of concepts requires that definition of concept C hast to provide exact rules of the following form: