Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
These are excerpts and elaborations from my book "The Nature of Consciousness"
One of the major breakthroughs in inexact reasoning came in 1965 when the Azerbaijani mathematician Lotfi Zadeh invented "Fuzzy Logic". Zadeh applied Lukasiewicz's multi-valued logic to sets. In a multi-valued logic, propositions are not only true or false but can also be partly true and partly false. A set is made of elements. Elements can belong to more than one set (e.g., I belong both to the set of authors and to the set of Italians) but each element either belongs or does not belong to a given set (I am either Italian or not). Zadeh's sets are "fuzzy" because they violate this rule. An element can belong to a fuzzy set "to some degree", just like Lukasiewicz's propositions can be true to some degree (and not necessarily completely true).
The main idea behind Fuzzy Logic is that things can belong to more than one category, and they can even belong to opposite categories, and that they can belong to a category only partially. For example, I belong both to the category of good writers and to the category of bad writers: I am a good writer to some extent and a bad writer to some other extent. In more precise words, I belong to the category of good writers with a certain degree of membership and to the category of bad writers with another degree of membership. I am not fully into one or the other. I am both, to some extent.
Fuzzy Logic goes beyond Lukasiewicz's multi-valued logic because it allows for an infinite number of truth values: the degree of “membership” can assume any value between zero and one.
Zadeh's theory of fuzzy quantities implicitly assumes that things are not necessarily true or false, but things have degrees of truth. The degree of truth is, indirectly, a measure of the coherence between a proposition about the world and the state of the world. A proposition can be true, false, or… vague with a degree of vagueness.
Fuzzy Logic can explain paradoxes such as the one about removing a grain of sand from a pile of sand (when does the pile of sand stop being a pile of sand?). In Fuzzy Logic each application of the inference rule erodes the truth of the resulting proposition.
Fuzzy Logic is also consistent with the principle of incompatibility stated at the beginning of the 20th century by the French physicist Pierre Duhem: the certainty that a proposition is true decreases with any increase of its precision. The power of a vague assertion rests in its being vague: the moment we try to make it more precise, it loses some of its power. A very precise assertion is almost never certain. For example, “today is a hot day” is certainly true, but its truth rests on the fact that I used the very vague word “hot”. If now I restate it as “today the temperature is 36 degrees”, the assertion is not certain anymore. Duhem’s principle is analogous to Heisenberg’s principle of uncertainty: precision and uncertainty are inversely proportional. Fuzzy Logic models vagueness and reflects this principle.
While mostly equivalent to Probability Theory (as proven by the US mathematician Bart Kosko), Fuzzy Logic yields different interpretations. Probability measures the likelihood of something happening (e.g., whether it is going to rain tomorrow). Fuzziness measures the degree to which it is happening (e.g., how heavily it is raining today). And, unlike probabilities, Fuzzy Logic deals with single individuals, not populations. Probability theory tells you what are the chances of finding a tall person in a crowd, whereas fuzzy logic tells you to what degree that person is tall.
Technically, a fuzzy set is a set of elements that belong to a set only to some extent. Each element is characterized by a degree of membership. An object can belong (partially) to more than one set, even if they are mutually exclusive, in direct contrast with one of the pillars of classical Logic: the "law of the excluded middle". Each set can be a subset of another set with a degree of membership. A set can even belong (partially) to one of its parts. Degrees of membership also imply that Fuzzy Logic admits a continuum of truth values from zero to one, unlike classical Logic that admits only true or false (one or zero).
In Kosko's formulation, a fuzzy set is a point in a unitary hypercube (a multi-dimensional cube whose faces are all one). A non-fuzzy set (a traditional set) is one of the vertexes of such a cube. The paradoxes of classical Logic occur in the middle points of the hypercube. In other words, paradoxes such as the liar's or Russell's can be interpreted as "half truths" in the context of Fuzzy Logic.
A fuzzy set's entropy (which could be thought of as its "ambiguity") is defined by the number of violations of the law of non-contradiction compared with the number of violations of the excluded middle. Entropy is zero when both laws hold, is maximum in the center of the hypercube. Alternatively, a fuzzy set's entropy can be defined as a measure of how a set is a subset of itself.
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