When discussing fuzziness in the context of Fuzzy Set Theory or Fuzzy Logic (BTW two different though related disciplines) it always occurs at some point in time that people start to confuse the language of FST with the things that want to be denoted by expressions in that languge. 

Of course, the language of FST is exact, by definition. Like all mathematics used to represent reality in some way. The fuzziness is in reality. We should leave it there. What FST tries to do, is simply: catch as best as possible the vagueness or fuzziness inherent in our observations and images of reality in a non-fuzzy language called FST or FL, actually a mathematical language, so that we can reason about it and derive interesting truths about reality. It is the old game of language versus reality, or meta-language versus language, whatever you like.

It may be that some people are not very much impressed and convinced by the formalism of Fuzzy Set Theory à la Zadeh. However, in the meantime, Fuzzy Logic à la Hájek has made enormous progress beyond the rather simple ideas of FST. And I have read some results in FL which make it clear that FL is even more powerful than ordinary Boolean logic, in the sense that certain FL's  have some important features which are lacking in traditional higher logics (things like completeness etc.)

Still it will always remain a not so easy empirical problem how to map and match the phenomena of the real world to the language of FL. First, there is not one Fuzzy Logic, but there is an entire taxonomy of subtly different Fuzzy Logics. Second, having settled on a particular Fuzzy Logic to use, you have to choose between alternative interpretations of the basic terms of that logic, e.g. how to you want to interpret the logical connective "OR" or "AND" in the fuzzy world.

Fuzzy Logic doesn't dictate what you have to choose in particular applications. But it tells you exactly, once you have made your choices, what are the consequences of your choice, so you may check if that makes sense in your domain. If yes, go ahead, if not, then look for another "matching rule".

I appreciate that as a very sensible principle, because it enables me to find out the best possible match in every special case, but many people seem to be irritated or confused by so much freedom - although, as stated above, natural languages are even more flexible and everybody accepts it.

And what if the special interpretation of FL by Zadeh doesn't make sense to you? Well, there are many alternatives to his approach still within the confines of FL. Or perhaps you have to invent a complete new type of language, in between the messy formalism of natural language and the too powerful and rich formalism of FL.

Please notify me if you have found such a precious tool!

Ambiguity in the expressions is not always due to an inherent fuzzy set of basic primordials. Rather, ambiguity is a contextual response to a polynomial stimuli. Sometimes fuzzy sets can produce clearly deciphered patterns in events and result sets.

Dear Juan,

thanks for sharing this interesting question. Your first statement is still ambiguous because I miss the exact definition of an "old man".

BTW: My age is 72 years, do I belong toyour first E?

I´m looking forward to the discussion.

Dear Hanno,

You are right, my age is 74 and I am a very young man, at least, regarding my activity.

By contrast, I know some old teenage men, who seem to be retired.

Dear Gokul,

You are right, but my question is the converse one. What I have stated is whether, in real world applications, fuzzy membership can be a consequence of expression ambiguity.

That is, your claim is "Fuzzy does not always lead to ambiguity" and my question is "can ambiguity lead to fuzzy membership".

Notice that, if the meaning of an expression is ambigous, hardly we can know whether if is true or false.

Jaun and Hanno

At 73 I fit in the middle of a set formed by us. I have spent most of my years avoiding fuzzy logic and sets, but there are advantages of lack of definition. If old man is conditioned on grey hair then age may not matter. If all conditions are included then over 55 forms the set. Truth depends upon bounds, even fuzzy ones. It may be important to the question asked to include ambiguity. Ambiguity will form the set.

A recent news article stated that three elderly men were involved in an accident. The oldest was 57. Was this based on age or condition? Had I been with them I am sure it would not have stated four elderly, unless the set was defined by age.

Dear Dexue,

I think you have pointed it out perfectly. If you have such an uncertain description for a feature like "old man", that you surely can´t assign any person to any how defined membership.

If you try in with numeric bounds for "old" like "old is between 70 and 80 years", you could use a matching fuzzy degree for example for Juan, Joseph and me.

I will try once more: If you define "old" is more then 80 years, you will have no success. If you add "people on average die with 90, you have defined an interval and you can try to use fuzzy algorithms again.

Languages were always intented to be able to make a perfect description of reality, but this was never the case. Fuzzy membership is one way to make languages more accurate, but it is also not a sufficient method. A child with spanish mother and african father is maybe 50 % european (????) but your example with old vs. young already shows that also including fuzzy membership we cannot realize a perfectly accurate language. I conjecture that no language bazed on a finite set of words can be perfectly accurate. [Including fuzzy membership does not change the hypothesis, we still have only a finite set of words. Just add the cifres, the comma and the % symbol to the words...]

If conditions of membership are stated clearly and unambiguously then there will not be what you called fuzzy membership, an object will be a member if it meets the stated conditions. In your example the expression "old" is ambiguous socially, professionally and naturally and the supplemental "very old" is not clear either, how very is very in the first place. Consider the concept of continuous functions on a closed interval [a,b]. We say a function f is continuous at any point  x₀ of (a,b) if ∀ε > 0,∃δ > 0:|x-x₀| < δ --> |f(x)-f(x₀)| < ε . This is unambiguous definition and condition of membership for a function defined on the interval and therefore  membership is not fuzzy. 

Let me start with a simple YES. Ambiguous expression leads to fuzzy membership function. A fuzzy membership function without ambiguity represents an interval (i.e. bigger than 90) which means a sharp switch from 0 ("not an old man") degree of truth to 1 ("an old man").  However, fuzzy membership function with ambiguity represents a fuzzy set which has no sharp switch from 0 to 1. for example, a man is "approximately old" if he has more than 70 years and he is "definitely old" if he has more than 85 years old. This means a man between 70 and 85 has an increasing degree of truth from 0 to 1 and thereafter the degree of truth continues as 1 up to infinity (i.e.150 years). numeric example: A man with 73 years old is an "old man" with a degree of truth =0.2 which means there is possibility of 20% to be and 80% not to be an "old man".

We often think in relative terms and our understanding of the world around us include adjectives (e.g., good, bad, cheap, and expensive) and adverbs which are indicators of vagueness or ambiguity. There are also determinative adjectives that are related to quantities (e.g., few, many, much, and several). These correspond to fuzzy quantifiers (or hedges). Adverbs, such as very and extremely are words that modify a verb or adjective. These modifiers are situated contextually and change depending on our knowledge of the contexts. Fuzzy logic tries to capture this context by allowing linguistic statements to hold a range of truth (values), a continuum rather than a singularity, hence the term “fuzzy.”

In classical (Boolean) logic the statement “John is old” is either completely true (i.e., 1) or completely false (i.e., 0), but there are clearly degrees of oldness, as well as an area between completely old and completely not old, which are not accounted for in classical logic. In fuzzy logic, someone who is partially old might be assigned a 0.3, someone who is very old might be assigned a 0.8. A word like “partially” may also imply a range of assignments, so “partially old” might be anywhere between, say 0.2 and 0.8. Hence, fuzzy logic can map overlapping linguistic meaning to a continuum of numbers representing a range of truth. To sum up, in fuzzy logic a proposition is not either true or false, but may be partly true (or partly false) to any degree. This degree is usually taken as a real number in the interval [0,1], i.e., the truth value of a sentence is not restricted to true or false

When discussing fuzziness in the context of Fuzzy Set Theory or Fuzzy Logic (BTW two different though related disciplines) it always occurs at some point in time that people start to confuse the language of FST with the things that want to be denoted by expressions in that languge. 

Of course, the language of FST is exact, by definition. Like all mathematics used to represent reality in some way. The fuzziness is in reality. We should leave it there. What FST tries to do, is simply: catch as best as possible the vagueness or fuzziness inherent in our observations and images of reality in a non-fuzzy language called FST or FL, actually a mathematical language, so that we can reason about it and derive interesting truths about reality. It is the old game of language versus reality, or meta-language versus language, whatever you like.

It may be that some people are not very much impressed and convinced by the formalism of Fuzzy Set Theory à la Zadeh. However, in the meantime, Fuzzy Logic à la Hájek has made enormous progress beyond the rather simple ideas of FST. And I have read some results in FL which make it clear that FL is even more powerful than ordinary Boolean logic, in the sense that certain FL's  have some important features which are lacking in traditional higher logics (things like completeness etc.)

Still it will always remain a not so easy empirical problem how to map and match the phenomena of the real world to the language of FL. First, there is not one Fuzzy Logic, but there is an entire taxonomy of subtly different Fuzzy Logics. Second, having settled on a particular Fuzzy Logic to use, you have to choose between alternative interpretations of the basic terms of that logic, e.g. how to you want to interpret the logical connective "OR" or "AND" in the fuzzy world.

Fuzzy Logic doesn't dictate what you have to choose in particular applications. But it tells you exactly, once you have made your choices, what are the consequences of your choice, so you may check if that makes sense in your domain. If yes, go ahead, if not, then look for another "matching rule".

I appreciate that as a very sensible principle, because it enables me to find out the best possible match in every special case, but many people seem to be irritated or confused by so much freedom - although, as stated above, natural languages are even more flexible and everybody accepts it.

And what if the special interpretation of FL by Zadeh doesn't make sense to you? Well, there are many alternatives to his approach still within the confines of FL. Or perhaps you have to invent a complete new type of language, in between the messy formalism of natural language and the too powerful and rich formalism of FL.

Please notify me if you have found such a precious tool!

Well, (oops!) on the contrary, I would say a big no. Fuzziness - in any sense of the word - does not have anything to do with ambiguity. Quite on the contrary. It deals with more vaild and therefore accurate measures. It means refination, detail, precision - over against the sheer 1 or 0.

Not by chance, pushing the argument a bit farther, fuzzy logic is the mother, so to speak, or the entrance, to many-valued logics. Zadeh' s papers are clera enough about these aspects.

Strongly recommended:

"Vague language and corresponding models of inference and information processing is an important and challenging topic as witnessed by a number of recent monographs and collections of essays devoted to the topic.

This volume collects fifteen papers, the majority of which originated with talks presented at the conference "Logical Models of Reasoning with Vague Information (LoMoReVI)", September 14-17, 2009, in Čejkovice, that initiated a EUROCORES/LogICCC project with the same title.

At least two features set the current volume apart from other texts:

• first, the interdisciplinary nature of the topic is nicely reflected by the wide range of interests of the authors, who include philosophers, linguists, logicians, as well as mathematicians and computer scientists.
• Secondly, all the papers are accompanied by comments written by other authors and a few outside experts. These comments and corresponding replies by the authors document the very lively ongoing debate on adequate models of vague language."

http://www.amazon.com/dp/1848900376

A highly recommended introduction to FL to start with:

"This volume is an accessible introduction to the subject of many-valued and fuzzy logic suitable for use in relevant advanced undergraduate and graduate courses.

The text opens with a discussion of the philosophical issues that give rise to fuzzy logic - problems arising from vague language - and returns to those issues as logical systems are presented.

For historical and pedagogical reasons, three-valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic."

http://www.amazon.com/dp/0521707579

For the rather philosophically inclined ones:

"Our daily lives are full of vagueness or fuzziness. When we describe someone as "tall," for example, it is as though there is a particular height beyond which a person can be considered "tall."

In this stimulating book, Kees Van Deemter cuts across various disciplines--including artificial intelligence, logic, and computer science--to illuminate the nature and importance of vagueness. Van Deemter shows why vagueness is both unavoidable and useful, and he demonstrates how tempting--and how wrong--it often is to think in terms of black and white, instead of the richly graded spectrum of the world around us.

Vagueness, the author argues, allows us to focus on what matters, leaving out irrelevant details, and adding texture to what would otherwise be unintelligible facts. The embrace of vagueness, however, comes at a price, for when degrees of grey are accepted, concepts like truth, belief, and proof lose their power, and we are banished from that paradise in which truth and falsity are the only possibilities."

http://www.amazon.com/dp/0199545901

Most recent publication on FL for the rather mathematically inclined ones:

"This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work.

The book opens with a discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hájek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic.

Articles in the volume also focus on the treatment of vagueness, building connections between Hájek's favorite fuzzy logic and linguistic models of vagueness.

Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles."

http://www.amazon.com/dp/3319062328

All logic is fuzzy. The rules are not. The rules are necessarily ambiguous once we apply them. We have the premise, "All old men wear hats." Followed by "Juan is an old man." The conclusion is "Juan wears a hat." We establish the truth of each statement in order to reach the conclusion. We have not reached the inverse conclusion if "Juan wears a hat," "He is an old man." The rules of logic are necessary to ensure discipline in arguments and premises. The reality is that the actual state is fuzzy. "Most old men wear hats." "Juan is at age considered old." "Juan probably wears a hat."

The conclusion "Juan probably wears a hat," is more than simply fuzzy. The probably could be smaller than 0.01, yet, I have made a true statement. I have not made limits for probably. Ambiguity may be necessary for inclusion in statements, because we must have an inclusive set. The discipline of logic requires that the second premise further restricts the set and that we have a further restriction on accepting the conclusion.

Juan and Hanno

I rarely wear a hat. What is the hat status of the rest of our set?

As long as we are not precise, ambiguity will flood allo of the theroretical models. It stronlgy calls my attention the wide variety of fuzzy mathematical examples to "establish a rule". We will always require an outmost specific referent to have less possibilities of fuzzyness. Do not forget that saying in processes: garbage in leads to garbage out. No ofense at all to anyone.

There are at least five distinct versions of probability, but fuzziness isn't one of them. Even stronger claim: probability may be combined with fuzziness, that would not be possible at all, if fuzziness would be a variant of probability.

Please use "Attach:Publication" to refer to (any) publications uploaded on RG itself. Thanks.

@ Fausto Galetto :-: What do you mean with "fuzzy drawbacks"? as applications of FL under the name of Fuzzy Control and Robotics have had proven success in engineering (see early experiences in Japan) and Fuzzy Expert Systems are being intensely studied and applied up to this day, you can't apparently mean that there isn't a role for them in modern science and research.

Some people may be falsely applying it, but that applies to other theories and methods equally, e.g. logics or statistics "cf. the bestseller "How to lie with statistics?"). But I haven't heard of any accidents or people harmed by fuzzy engineering ("death by fuzziness").

So what kind of drawback are you thinking of, or, worse, did you experience? Please help us out! Never too late to learn another view.

On the other hand, reading the recent book "Petr Hájek on Mathematical Fuzzy Logic", you will be astonished about what has been achieved in so few years in this flourishing field (forget about Zadeh, he is a respected engineer, surely, but the real hero in this field should be Hájek and close colleagues in Czech Republic as well as Italy).