Set Theory, Rough Set Theory, Fuzzy Set Theory

[WWGD]

Hi. I hope this is not too far into philosophy.
Set Theory is commonly accepted as the foundations of Mathematics. Is it possible to develop
a different type of Mathematics by using Fuzzy sets or Rough sets instead?

 

[andrewkirk]

If we really want to look to foundations, then it is logic rather than set theory that is fundamental. Set theory is axiomatised (eg Zermelo Frankel, or Von Neumann Bernays Godel) using the language of logic and its theorems are developed by the application of logic.

Set theory is built on classical logic. There are different types of logic, such as intuitionist logic and minimal logic. Many-valued logic, in which a proposition can have values other than true or false, may be along the lines of what you are interested in. Then there is dialethian logic, in which a proposition is allowed to be both true and false - ie contradictions are allowed. Graham Priest is a well-known advocate of dialethian logic.

 

[WWGD]

Thank you, I was also thinking along the lines of the logic underlying fuzzy or rough sets. Do you know anything about this?

 

[andrewkirk]

I haven't gone into it myself (I'm a bit of a classical kind of guy) - rather I've only been involved in discussions of it with other people interested in logic.

I think the following book would cover much of what you are interested in:
https://books.google.com.au/books?id=rMXVbmAw3YwC&hl=en

It says it covers fuzzy logic which I would expect to be relevant to fuzzy sets (although it may not necessarily be).

Graham Priest often presents his motivation for exploring that area by means of the 'Liar Paradox': 'This sentence is false' , and the many variants of that. I don't agree with him that it is a paradox. I think there is a firm resolution. But that doesn't take away from the fact that the area he is motivated by it to explore is a rich and interesting one and as far as I can tell he's the leading expert on this area.

 

[Stephen Tashi]

Thank you, I was also thinking along the lines of the logic underlying fuzzy or rough sets. Do you know anything about this?

Fuzzy set theory is presented as definitions and assumptions that are stated using such ordinary mathematical concepts such as sets, real numbers, and functions. Results about fuzzy sets are proven using the same logic that you would use in algebra or calculus. Fuzzy set theory is built "on top" of whatever theory of sets that you wish to use for describing real numbers and functions.

That's the way texts on fuzzy sets are currently written. Perhaps someone will discover a different way of presenting things.

 

[TeethWhitener]

That's the way texts on fuzzy sets are currently written.

I've always seen fuzzy sets more as generalizations of standard set theory. Just like fuzzy logic, fuzzy sets rely on suspending the law of the excluded middle. So in standard set theory, we typically think of the statement
$$\forall x \forall S (x \in S \vee x \notin S)$$
as an axiom (inherited from first-order logic). But whereas in standard set theory the degree of membership of x in S is restricted to the values 0 and 1, the statement above isn't an axiom in fuzzy set theory because the degree of membership takes on a value in the continuum between 0 and 1. So in that sense, fuzzy set theory is a generalization of standard set theory. That being said, when you remove axioms from a system, you generally make the system weaker (fewer axioms means you can't prove as many statements to be theorems in the system). But just as you make the number of provably true propositions smaller, you also make the number of provably false propositions smaller, so in a sense, the system you're left with is "bigger" in that more statements can be true if you choose them to be. For instance, the (strong) negation of the axiom above
$$\forall x \forall S \neg (x \in S \vee x \notin S)$$
is always false in standard set theory, but in fuzzy set theory, it might be true, and if you wanted to, you could take it as an axiom, so that elements could never be full members of sets. (Technical note--The above is just an illustrative example; I haven't actually done the math, so it might turn out that this axiom makes the system inconsistent.) So to kind of answer WWGD's original question, I suppose you can put the machinery of, e.g., the ZF axioms on top of fuzzy set theory if you want, and probably regain much of math, but that's just because you'd never be looking at the part of fuzzy sets that distinguish them from standard sets and make studying them interesting.

 

[Stephen Tashi]

I've always seen fuzzy sets more as generalizations of standard set theory.

I suppose you can put the machinery of, e.g., the ZF axioms on top of fuzzy set theory if you want, and probably regain much of math,

The usual method is to build fuzzy set theory by employing ordinary set theory. For example, from the current Wikipedia article on "fuzzy set" we have:

A fuzzy set is a pair $(U,m)$ where $U$ is a set and $m: U \rightarrow [0,1]$.

So we have a reference to the ordinary concept of "set" used in defining a "fuzzy set".

 

[gill1109]

Fuzzy set theory was briefly a hype in the 60's. I think by now it is more or less completely forgotten. Nothing worthwhile or interesting came of it. Absolutely nothing.

People in AI have briefly played with it. I think by now almost everyone realizes that probability theory is the only useful and effective way to rigorously reason with uncertainty.

 

[Stephen Tashi]

Fuzzy set theory was briefly a hype in the 60's. I think by now it is more or less completely forgotten.

It's still an active and important subject in applied mathematics and statistics. For example, in control systems engineering there are electronic hardware devices called "fuzzy controllers".

 

[gill1109]

It's still an active and important subject in applied mathematics and statistics. For example, in control systems engineering there are electronic hardware devices called "fuzzy controllers".

https://en.wikipedia.org/wiki/Fuzzy_control_system ... yes they are analogue devices which take one or more continuous inputs and deliver a continuous output. Certainly they can be very useful. But you do not need fuzzy set theory either to design nor to study such devices.

 

[TeethWhitener]

The usual method is to build fuzzy set theory by employing ordinary set theory.

Sure. The usual method to do anything in foundational math is to employ ordinary set theory. But if you trash excluded middle at the outset, then you can consider sets where the following three statements hold:
$$\neg (x \in S)$$$$\neg (x \in S^c)$$$$x \in (S \cup S^c)$$
where S^c is the complement of S. I think this at least gets across the idea of partial membership in fuzzy set theory. Actually defining a membership function of x in S probably involves setting up some non-trivial topology that is likely beyond my ability, but I don't see why it couldn't be done.

 

[Stephen Tashi]

The usual method to do anything in foundational math is to employ ordinary set theory. But if you trash excluded middle at the outset, then you can consider sets where the following three statements hold:

Yes, you can consider "sets" .

I think you'll find most attempts to study mathematical structures that behave differently that ordinary sets and ordinary logic make heavy use of ordinary logic and ordinary sets. For example, theorems about such are "statements" - i.e. assertions that obey the law of the excluded middle.

To trash ordinary logic and sets, I think we'd have to use the scenario of an abstract "language" that has "derivation rules" that do not correspond to the principles of logical reasoning. And even if we did that, we'd end up trying to prove or disprove "statements" about the language and its rules in the ordinary sense of "statements" - meaning assertions that must be exactly one of the states: true or false.

 

[TeethWhitener]

I think we'd have to use the scenario of an abstract "language" that has "derivation rules" that do not correspond to the principles of logical reasoning.

At the risk of being too philosophical, what is logical reasoning besides an abstract language that has derivation rules? We might assign certain meaning to those rules, and we might be intuitionally wedded to some more than to others, but at its core, a logical system is just a list of axioms, some rules for how to put together wffs (grammar), and some other rules for how to get from one wff to the next (inference).

And even if we did that, we'd end up trying to prove or disprove "statements" about the language and its rules in the ordinary sense of "statements" - meaning assertions that must be exactly one of the states: true or false.

Or in a multivalent logic, we have the possibility that an assertion is always somewhere in between. I just contributed to a thread about the liar paradox not too long ago. One general method for tackling the liar paradox is to give it an interpretation in multivalent logic such that its truth value is identically 0.5.

 

[WWGD]

While maybe this is not strictly part of the Mathematics behind the theories, it seems like the associated interpretations, i.e. the Semantics of logical systems tied to Rough and Fuzzy set theories are very different from those tied to the standard Mathematical logic. Maybe this is part of what Teethwhitener said, if I understood correctly.

 

[Stephen Tashi]

At the risk of being too philosophical, what is logical reasoning besides an abstract language that has derivation rules? We might assign certain meaning to those rules, and we might be intuitionally wedded to some more than to others, but at its core, a logical system is just a list of axioms, some rules for how to put together wffs (grammar), and some other rules for how to get from one wff to the next (inference).

I agree that any formalization of logical reasoning so extensive that it could be implemented as a computer algorithm would be a formal language with derivation rules. Mathematics as practiced by human minds focuses on statements and uses the semantics that mathematical statements are exactly one of "true" or "false". Mathematics written by human beings only has informal rules of grammar uses informal derivations.

If human beings found a way to use multi-valued logical in some endeavor where the semantics were clear then they might "reason using multi-valued logic" instead of reasoning with the ordinary sort of logic.

 

[WWGD]

<Snip>

If human beings found a way to use multi-valued logical in some endeavor where the semantics were clear then they might "reason using multi-valued logic" instead of reasoning with the ordinary sort of logic.

Don't you think that Rough databases and Temporal databases are examples of this?

 

[Stephen Tashi]

Don't you think that Rough databases and Temporal databases are examples of this?

I don't know anything about them. Do they require a special type of logic?

 

[WWGD]

I don't know anything about them. Do they require a special type of logic?

I know very little about them, I was hoping you or someone else would know more. Let me read up some more.

 

[WWGD]

Thought someone may find this interesting, maybe (hopefully) answering @Stephen Tashi 's last post partially.
http://www.wseas.us/e-library/conferences/2012/CambridgeUK/AIKED/AIKED-37.pdf