Think about the property of being tall. If someone is 5'4" (162.5 cm), they're not tall. If they're 6'8" (203.2 cm), they are tall. What happens in the middle?

In classical set theory, every person is either tall or not. As a result, there has to be some cutoff below which people aren't tall, and above which they are. But that doesn't match up with the way that we think about tallness. Someone who's 6' (182.9 cm) is kinda tall but not really.

Fuzzy logic and set theory is the math that allows you to reason precisely about properties like tallness, where some things have the property and other things don't and still other things kinda have it.

Now imagine trying to formulate tallness in the language of probability theory. If you say that a person who's 6' is 50% tall, and I show up with a large group of people of that exact height, then you'd have to claim that half of them are tall and half of them aren't in order for the law of large numbers to hold. Is that really what you want, or even coherent?

I'm not able to do an 'elevator pitch' because I don't actually work with fuzzy logic, but the fact is... the concept itself is just not the same as propability. Compare these two statements:

1. The color of this apple is kind of yellow, but there's also a little red here and there.

2. This apple may be yellow with p=0.8, and red with p=0.2.

Fuzzy sets allow you to use set theoretic tools over objects with varying degrees of membership. This is a useful construct when trying to reason over probabilistic evidence, and IMO can be a useful modelling tool.

> set theoretic tools over objects with varying degrees of membership

you can also do exactly that in probability theory by multiplying the probabilities. Is there any difference?

enruquito says>"you can also do exactly that in probability theory by multiplying the probabilities."

Have you got a concrete example?

i.e., show us the probability version precisely with mathematics and maybe someone can tell you the corresponding fuzzy version, if any.

Have you got a concrete example?

Fuzzy logic (FL) made it easier for control systems engineers to implement rule-based systems where rules were often extracted in human language from experts who had learned to manually control the systems.

The Japanese were quick to adopt FL and benefited enormously financially. Now FL is embedded almost everywhere.

Michio Sugeno built a fuzzy logic control system that flies a helicopter with a rotor blade missing! It is possible to do that with a conventional control system (that is, a mathematical solution must exist), but I believe the mathematics might be a bit more difficult.