# Is Fuzzy Logic a Pseudoscience or a Valid Mathematical Branch?

• jostpuur
In summary, the conversation discusses the controversial topic of fuzzy logic and its applications in mathematics, specifically in control theory and image recognition. While some argue that it is a superior approach, others reject it in favor of more traditional methods like probability theory. However, fuzzy logic has been proven to be useful in certain situations, such as highly complex or nonlinear systems, and can even exhibit human-like behavior when combined with expert systems. The conversation also touches on the overhyping of fuzzy logic and the initial skepticism towards it from control engineers. Ultimately, both classical mathematical logic and fuzzy logic can be used to explain each other, making it a matter of convenience and practicality in choosing which approach to use.
jostpuur
When I first heard about the fuzzy logic, some guys told me that it would replace all "classical mathematics", being superior to it, but scientists are unfortunately reluctant to accept it yet. I immediately concluded, that fuzzy logic would be pseudo science.

I later learned, that fuzzy logic is in fact a real branch of mathematics, and then concluded that these people who had explained fuzzy logic to me were merely exceptionally incompetent. However, these people had not come up with their claims on their own. They had some book from where they were reading those claims.

Anyone had similar experiences? What's really the situation with fuzzy logic?

I don't see the big deal about it. 'Fuzzy logic' is my normal way of thinking. It's not like it's something new...

Fuzzy logic is not a "replacement" of mathematics, it is just a slightly different way of approaching certain problems. Fuzzy logic is extremely useful in e.g. image recognition, control theory and similar problems where there sometimes are no "yes or no" solutions to problems, i.e. the problems are "fuzzy", hence the name.

It is also possible to combine fuzzy logic with e.g. expert systems, in some applications (control systems) this give rise to algorithms that can exhibit almost "human" behavior.

I interpret your answers as that you do not have similar experiences of the mystification of fuzzy logic as I had.

What is this:

Fuzzy logic is controversial in some circles and is rejected by some control engineers and by most statisticians who hold that probability is the only rigorous mathematical description of uncertainty.
(http://en.wikipedia.org/wiki/Fuzzy_logic)

Is that a straw man argument?

Fuzzy set theory is, at its core, a version of set theory that replaces the concept of set measurement being discrete with the concept of fuzzy membership. As a consequence, the axiom of the excluded middle also must be discarded. Throwing out the law of the excluded middle means throwing out axiomatic probability theory.

There's nothing wrong with throwing out axioms; an axiom is an assumption, after all.
Seeing what results when one replaces some axioms with others is one of things theoretical mathematicians do. Replacing the parallel postulate with something else yields the non-Euclidean geometries.

The non-Euclidean geometries are very useful in some domains and are essential for the understanding of much of modern physics. However, that doesn't mean that non-Euclidean geometry will replace Euclidean geometry, "being superior to it". Euclidean geometry remains more useful than non-Euclidean geometries in many, many domains. The same goes for fuzzy set theory / fuzzy logic. It is very useful in some domains. The classical theories opposed by fuzzy logic remain useful (more useful) in many, many domains.

Their are three key problems with fuzzy set theory (and fuzzy logic). The first is its name. Fuzzy logic has a prior connotation in English. Fuzzy logic has caught on in Japan much more so than in the West in part because the derogatory association with fuzzy-minded thinking is absent. In fact, fuzzy logic jibes nicely with some Eastern philosophies.

The second problem is that, like the parallel postulate, the axioms discarded by fuzzy set theory are very useful and very powerful. In throwing those axioms to the wayside one must also throw out the tools that are a consequence of those axioms.

The third problem is that fuzzy logic has been overhyped. Statements such as the following are mild examples of the hyperbole associated with fuzzy logic:
jostpuur said:
some guys told me that it would replace all "classical mathematics", being superior to it
f95toli said:
this give rise to algorithms that can exhibit almost "human" behavior.

Just to clarify, by "human" I meant: "Can be used in situations where more conventional methods did not work very well and therefore often required human intervention".
In control theory fuzzy logic can sometimes be used where e.g. ordinary PID regulation fails. A good example would be highly non-linear or very complex systems where rule based fuzzy logic is sometimes easier to implement than more "mathematical methods" such as PID, especially when combined with expert systems since it makes it possible to design systems that behaves as if it they are "experienced" and react accordingly (i.e. the behaviour is more "human" than conventional methods).

jostpuur said:
What is this:
...
Is that a straw man argument?
It's an example of a statement on Wikipedia that has been flagged with "verification needed".

jostpuur said:
What is this:
Fuzzy logic is controversial in some circles and is rejected by some control engineers and by most statisticians who hold that probability is the only rigorous mathematical description of uncertainty.
(http://en.wikipedia.org/wiki/Fuzzy_logic)

Is that a straw man argument?
No, its just presented as yet another wiki truth without much justification.

The control engineers I know use two words to reject fuzzy control techniques: Controllability and stability. Proving these "ilities" is a big part of what control engineers do. Coming up with the control mechanism is the easy part of their job. Proving that it will stand up to whatever evils Murphy's law throws at the plant and the control is the hard part of their job. When fuzzy control first came into being, the practitioners scoffed at the ideas of controllability and stability. "Nobody proves controllability and stability for real-world control systems." In actuality they scoffed at the concepts because they couldn't prove stability and controllability. The situation may have changed since those early days. First impressions are very important. The first impression of fuzzy control by control engineers in both the petrochemical and aerospace industry was very negative.

u can formulate any branch of mathematics using fuzzy logics. because classical mathematical logics (set theory) can be explained by mean of reducing fuzzy logic to it. but also the converse its true.

remember that any branch of mathematics can be formulated in terms of set theory (classical mathematical logics)

so, focusing in one or the other (from to which one starts to attack the problem) it´s just a mater of
convenience or being practical. (i.e. start from convenient axioms, which in fact can be obtained from the
other formulation by mean of deduction).

best regards
rayo

Atm adaptive logic or logic that learns from its mistakes is part of computer programming, fuzzy logic is the next step. If 1 do x if 0 do y if 0/1 do z. So it's not exactly out of the loop anyway. Don't know about maths, I haven't studied it to that high a level.

With quantum computing, it works on fuzzy bits called qbits, so it's very useful there, no doubt. Trouble is Quantum computing is in its infancy.

Fuzzy logic is about getting computers to use algorithms that adapt to conditions not based on normal logic. Just like humans do, we are not Vulcans, and neither are logical answers useful in all situations.

Last edited:
Schrodinger's Dog said:
With quantum computing, it works on fuzzy bits called qbits, so it's very useful there, no doubt. Trouble is Quantum computing is in its infancy.
I don't think that is quite right. I can't think of a single "fuzzy" quantum algorithm. Quantum computing is, however, related to another branch of logic; namely reversible computing which until the advent of quantum computing was (as far as I know) mainly used as "toy" systems for theorists and had no real applications.

f95toli said:
I don't think that is quite right. I can't think of a single "fuzzy" quantum algorithm. Quantum computing is, however, related to another branch of logic; namely reversible computing which until the advent of quantum computing was (as far as I know) mainly used as "toy" systems for theorists and had no real applications.

I agree, what I should of said is that it may be very useful there, if we can get quantum computing(QC) off the drawing board.

So as you say they have no real applications yet. But fuzzy logic in computing now does seem to have at least have some use in computing with only 1/0.

QC it´s a branch of fuzzy logics, but using complex (instead of real) probabilities. and some others axioms
as well. (for example schroedinger equation, etc.)

best regards
rayo

## What is fuzzy logic?

Fuzzy logic is a type of mathematical logic that allows for partial truths and degrees of uncertainty. It is based on the idea that things can exist in varying degrees of membership to a set or category, rather than just being a part of it or not.

## How does fuzzy logic differ from traditional logic?

Traditional logic, also known as Boolean logic, operates on binary values of true or false. Fuzzy logic, on the other hand, allows for degrees of truth between 0 and 1. This allows for more complex and nuanced reasoning, especially in situations where there is ambiguity or uncertainty.

## What are the applications of fuzzy logic?

Fuzzy logic has a wide range of applications in fields such as artificial intelligence, control systems, decision making, and image processing. It is particularly useful in situations where precise and deterministic solutions are not possible or practical.

## What are the advantages of using fuzzy logic?

Fuzzy logic can handle imprecise and uncertain data, which makes it more suitable for real-world problems. It also allows for more flexible and adaptive decision making. In addition, fuzzy logic is easier to understand and implement compared to other forms of mathematical logic.

## Are there any limitations to fuzzy logic?

While fuzzy logic has many advantages, it is not a perfect solution for all problems. It can be computationally expensive and may not always provide the most accurate results. Additionally, fuzzy logic requires expert knowledge to define the membership functions and rules, which can be challenging in some applications.

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