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These are excerpts and elaborations from my book "The Nature of Consciousness"
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Fuzzy Logic 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. Duhems principle is analogous to Heisenbergs 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. Back to the beginning of the chapter "Common Sense" | Back to the index of all chapters |
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