This chapter introduces the reader to Part II of the book, describing the fuzzy logic framework where the core methodological ideas of the book best fit. The appropriate background and fundamental concepts are epitomized, so to surface the space where the fuzzy logic modeling is placed. In this way, the comprehension of the role and potentialities of the fuzzy logic concept and the way they could be transferred to tackle real problems in the educational context (discussed in Part III), is facilitated.
TopFuzzy Logic Is Not Fuzzy!
Fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision and approximate reasoning […] Paradoxically, one of the principal contributions of fuzzy logic – a contribution which is widely unrecognized – is its high power of precisiation of what is imprecise. This capability of fuzzy logic suggests, as was noted earlier, that it may find important applications in the realms of economics, linguistics, law and other human-centric fields. (Zadeh, 2008, pp. 2753-2754)
The above quotation by Zadeh (2008) opens the door for entering the fuzzy world using, ostensibly, a kind of contradictory statement, as it reveals the duality of meaning of the term “fuzzy logic” (FL). In a wide sense, FL is much more than a logical system, and different branches can be observed (Cabrera et al., 2009):
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Logical systems aiming at formalizing approximate reasoning, i.e., FL in narrow sense,
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General theories of fuzzy sets, such as L-fuzzy sets, bipolar fuzzy sets, intuitionistic fuzzy sets, research on t-norms and/or copulas as adequate extensions of the intersection of fuzzy sets,
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The epistemic approach, knowledge representation and reasoning using FL, natural language, information systems, fuzzy databases (Dubois & Prade, 1991), and
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The relational approach, or the study of fuzzy relations and, more generally, fuzzy dependencies (Belohlávek & Vychodil, 2006). This aspect focuses on the representation and handling of imprecisely defined functions and relations and it is exactly this facet of FL that plays a pivotal role in its applications to system analysis and control.
The origin of FL perhaps can be placed in the introduction of multiple-valued logics by Jan Lukasiewicz in 1920, yet it is commonly accepted that it emerged from the theory of fuzzy sets, introduced by Lotfi Zadeh in 1965 (Zadeh, 1965). In the latter paper, the formalization of vagueness was based on fuzzy subsets (in the sense of a predicate that applies to a certain degree, not in absolute terms), underlying the notion of soft membership, in which objects might neither belong nor not belong to a set and that there may be borderline cases. Another contribution in the establishment of FL was that from Joseph Goguen with his “logic of inexact concepts” (Goguen, 1969). In this work he connotes:
We suggest a method for constructing and studying models of the way we use words; and we use the word ‘concept’ metaphorically in discussing meaning. ‘Exact concepts’ are the sort envisaged in pure mathematics, while ‘inexact concepts’ are rampant in everyday life. This distinction is complicated by the fact that whenever a human being interacts with mathematics, it becomes part of his ordinary experience, and therefore subject to inexactness. Ordinary logic is much used in mathematics, but applications to everyday life have been criticized because our normal language habits seem so different. (Goguen, 1969, p. 325)
Since then, the theory of FL has been constantly growing and, moreover, has been applied to very diverse fields, spanning from control theory to artificial intelligence, setting, nowadays, fuzziness manipulation and FL as a mainstream theory; the following sections try to explore the reasons behind this further.
TopAristotle Vs. Zadeh: A Battle For (Im)Precision And (Un)Certainty
It is impossible, then, that “being a man” should mean precisely not being a man, if “man” not only signifies something about one subject but also has one significance […] And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call “man”, and others were to call “not-man” but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact […] it is, then, impossible that it should be at the same time true to say the same thing is a man and is not a man. (Aristotle (384-322 BC), Metaphysics, Book Γ, Part 4, (Transl. by W. D. Ross), Jones, 2012, pp. 49-50)
In a nutshell, in fuzzy logic everything, including truth, is a matter of degree. (Zadeh, 1988, p. 84)