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caumaan
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I don't really know a lot about computers, so maybe I should just say this...
What is fuzzy logic and what are some of its applications?
What is fuzzy logic and what are some of its applications?
Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth -- truth values between "completely true" and "completely false".
Cauuman: What is fuzzy logic and what are some of its applications?
Fuzzy logic is used directly in very few applications. The Sony PalmTop apparently uses a fuzzy logic decision tree algorithm to perform handwritten (well, computer lightpen) Kanji character recognition.
Originally posted by Hurkyl
If you treat all the logical operators as if they were operating on binary logical values, then of course you're going to derive a contradiction if the values aren't restricted to be 0-1.
Fuzzy logic is one way to avoid hard decision making. For example, 5'9" is the average height for a male (or it was at one time), and 6'3" is tall. What do you call a person who is 5'11"? Average height? Tall? Does it even make sense to have a cutoff? Fuzzy logic let's you assign a number between 0 and 1 for both "average" and "tall", thus avoiding making this hard decision.
An extension of two-valued logic such that statements need not be true or false, but may have a degree of truth between 0 and 1.
Originally posted by phoenixthoth
hurkyl, that's exactly the example i was going to give.
if every statement is either true or false, then tell me whether the following statements are true or false, ok?
1. i am tall (i'm 5'11'')
2. i am very tall
3. i am somewhat tall
4. i am roughly of average height
5. i am beautiful
6. i am one of the most beautiful people in the world
7. i always lie
8. this statement is false
9. the barber who shaves every man who doesn't shave himself shaves himself
10. the barber who shaves every man who doesn't shave himself does not shave himself
11. the set of all sets that are not members of itself is a member of itself.
12. the set of all sets that are not members of itself is not a member of itself.
13. this statement is true but not provable.
14. this statement is true but you can prove that it is false.
or did you just mean that all well formed formula in binary logic are either true or false?
this is the third thread with basically the same topic now.
Originally posted by Canute
It doesn't matter whether Goedel's sentence was a sentence or a statement in this context. He did not suggest that some statements are neither true nor false. His proof was of the fact that any system of truths and falsities (formal axiomatic systems of a certain level of complexity etc.) must contain statements which are undecidable within the system.
As this is true for all such systems then there will always be statements which are undecidable. However this does not mean that they are neither true nor false in reality, just that they cannot be decided without extending the system.
But some people take this to mean that all such systems cannot produce truth, and infer from this that there are statements which are neither true not false in reality , just as non-dual philosophers assert.
As Lao Tsu said " Words that are strictly true seem to be paradoxical" (Tao Teh Ching). This relates directly to Goedel, in that the more true (provable) words become, the more inconsistent the system used to prove them appears. This relates also to something called the 'Quine-Duhem thesis', which states (by implication) that all axiomatically derived truths are only relatively true, and cannot ever be shown to be completely true.
Drawing conclusions from all this is tricky however, and academics argue incessantly about the real meaning of these proofs.
I think you're missing the significance of Goedel's theorems. Certainly not many people share your view that he was mistaken. Most people, even today, think he should have got a Nobel prize. Perhaps you know better.Originally posted by StarThrower
I've already completely analyzed the Godel sentence, and isolated an error in its analysis, as I now see it, there is nothing to argue about. It certainly does matter whether or not the Godel sentence is a statement, because there is no reason to use binary logic to try and process sentences which don't denote statements, no reason at all.
I can post my old work which isolates Godel's reasoning error if you wish.
Fuzzy logic is total nonsense.
Any statement is either true or false. There is no such thing as a statement being 1/3 true and 2/3 false, and that is exactly what fuzzy logic claims is possible.
Basically, of all possible logics that a reasoning agent can use, only one will be free from contradiction, and that is binary logic.
The following statement is true:
For any statement x, either X is true or X is false, and not (X is true and X is false.
It doesn't matter whether Goedel's sentence was a sentence or a statement in this context. He did not suggest that some statements are neither true nor false. His proof was of the fact that any system of truths and falsities (formal axiomatic systems of a certain level of complexity etc.) must contain statements which are undecidable within the system.
Very well put imo.Originally posted by Bob3141592
If I understand Goedel (now that's a fuzzy statement!) he wasn't making a statement about the nature of absolute truth. He was making a statement about the nature of formal systems. Lots of people seem to miss this point. A formal system is one in which there a finite number of explicitly defined symbols, and a finite number of explicitely stated axioms, and where there are a finite number of explicitely stated operations on those symbols. The symbols themselves do not contain "meaning," and it is not necessary nor even desirable to have any interpretation of the symbols. A formal system can be reduced to a set of algorithmic actions on the symbols or the derived statements of the system.
This is where I begin to disagree. I think I see what you're saying. In a sense you're right, there is no such thing as a formal system, since all such systems must be incomplete or inconsistent and therefore not strictly formal. But we treat systems as formal, in other words we act as if we deal with formal systems. We assume, for instance, that proofs can be derived from formal systems despite their lack of complete formality. This is ok on a superficial level, (2 + 2 does truly equal 4), but it goes horribly wrong when we use axiomatic systems to attempt to prove truths about reality, for it can't be done.Very little we deal with is really a formal system. I'm not sure that mathematics itself is a formal sstem, since I don't know for a fact that set theory can be completely axiomized.
I'd say that this is the heart of the issue. The non-dual view of reality is that it is built on something (/nothing) of which nothing completely true or false can be asserted. In this sense the totality of reality is seen as being beyond description by any formal system of truths and falsities. By this view Goedel's theorems are true as a natural consequence of the structure of reality, for all formally axiomatic systems used to describe it must leave out the most important part of it, and must therefore not be incomplete or inconsistent, for if fully developed they will be found to be wrong, only partially true.And reality itself may not be a formal system. We can't make it a formal system unless we can both axiomize all of the principles by which reality operates, which is a dubious proposition, and define without any ambiguity all of the elements on which reality is built.
I agree with that, but feel that the theorems are vitally significant to philosophy, so the dangers have to be faced. I like Roger Penrose's approach, but I know many people think he abused Goedel.So while Goedel's Proof is very important logically, it's commonly overextended and used in domains where it really doesn't apply. Philosophically, it's trecherous to use without abusing it.
Originally posted by Canute
This is where I begin to disagree. I think I see what you're saying. In a sense you're right, there is no such thing as a formal system, since all such systems must be incomplete or inconsistent and therefore not strictly formal. But we treat systems as formal, in other words we act as if we deal with formal systems. We assume, for instance, that proofs can be derived from formal systems despite their lack of complete formality. This is ok on a superficial level, (2 + 2 does truly equal 4), but it goes horribly wrong when we use axiomatic systems to attempt to prove truths about reality, for it can't be done.
As nearly all our reasoning is formally axiomatic, and certainly all our systems of proof, I would say we do deal all the time with what we treat as being formal axiomatic systems, but that, as you say, we are kidding ourselves for no such thing exists, and these systems contain no non-contingent proofs or truths. (I'd be interested to know whether you agree with this point or not)
I'd say that this is the heart of the issue. The non-dual view of reality is that it is built on something (/nothing) of which nothing completely true or false can be asserted. In this sense the totality of reality is seen as being beyond description by any formal system of truths and falsities.
... but feel that the theorems are vitally significant to philosophy, so the dangers have to be faced. I like Roger Penrose's approach, but I know many people think he abused Goedel.
That wasn't quite what I said. No "nontrivial" formal system can be both complete and consistent (nontrivial in this case means powerful enough to do multiplication, but I'll use it in a differenct sense later). That's what Godel proved.
Yes, I was trying out an idea about systems to see if I got away with it or not. I didn't.Originally posted by master_coda
I just feel that I should point out that for something to be a formal system, it does not have been be consistent and complete.
And even if consistency and completeness were a requirement for a system to be formal, that does not imply that there are no formal systems. There certainly are systems that are both consistent and complete.
Originally posted by Canute
Yes, I was trying out an idea about systems to see if I got away with it or not. I didn't.
I was wondering whether a system could be accurately called 'formal' if it was inconsistent or incomplete. It's a odd way of looking at it I suppose. Such as system is formal in intent, but it cannot be proved to be entirely formal in fact. There will always be some doubt, with any system subject to Goedel's limits, as to whether the system is actually formal or not. After all if we know that there has to be a contradiction in it somewhere then we can define it as not being strictly formal.
Does that make any sense?
I agree with this. See also my reply to 'masta coda' above.Originally posted by Bob3141592
That wasn't quite what I said. No "nontrivial" formal system can be both complete and consistent (nontrivial in this case means powerful enough to do multiplication, but I'll use it in a differenct sense later). That's what Godel proved. The systems are formal. And if they're formal, they're limited by the constraint of Godel's proof. If, however, they're not formal, meaning they don't meet at least one of the essential criteria of a formal system, maybe they can be both complete and consistent. Is reality a formal system? If it isn't, then it can be both complete and consistent. It may not be completely comprehensible to us in that case, but in and of itself it can be complete and consistent.
Interesting. I've convinced myself that it cannot be. The trouble is that if it cannot be, then there's no way of proving it cannot be. If there is a way that reality can be described by a non-trivial formal axiomatic system then I wonder what the fundamental axiom is, or was.Note that I'm not saying that it is here. That would be a statement of faith that I'm not prepared to make at this point (but if you ask me later, I'll say that it is, I hope).
I don't disagree with what you said about chess and so on, but I take it to mean something different.I can't agree with that, because I don't think all of our reasoning is formally axiomatic at all. For example, I play chess, and I play moderately well. Most people think of chess as a formal system, that it involves nothing but it's own special form of reasoning. The rules about... etc
No, no, not religious. But you mentioned reality and metaphysics became unavoidable.Now that sounds almost like a religious statement (is it acceptable to include religious notions in the discussions in these folders?).
Often it is yes. But quite often it isn't. There are some quite good non-religious reasons for believing in a 'higher plane' of some sort.And if anything, it is religious beliefs that some, even many people believe to be a higher reality than the physical world.
I'll just agree.Myself, I'm rather areligious, since I think no concept of a non-trivial God can be consistent (in other words, makes any sense at all: and in this case, non-trivial means any active God, as opposed to the Deistic notion of a Creator who wills the universe into existence and then separates himself from it--that's what I call a trivial God, like the trivial solution to the harmonic equation). But we don't have to go there.
No offense, but as an idea that doesn't seem any more consistent than the idea of God. Can one have more of nothing? I'm not sure.I take a more poetic view (poetry is another way around the restraints of a formal system, since the symbols it invokes have meaning but aren't unambiguously defined). I say the universe is nothing, just nothing that's unevenly distributed. Some places in the universe contains less nothing than other places, and some contain more nothing, so overall it balances out. And that's the truth, at least in a sense. Right?
Hmm. Sorry to write so much but this topic fascinates me.I agree, the existence of Godel's theorem is significant even outside of the domain where it applies. It's useful in aguments by simile and by analogy, which are dangerous ways to philosophize themselves. But we have to use something to get a handle on things, and since we can't really define the symbols of our notions of the world to another, we have to do something. So trecherous or not, we forge ahead. [/B]
Yeah, I think there's two ways of looking at it. What you say is right but I still wonder whether a system known a priori to be either incomplete or inconsistent can properly be called formal. But I'm only being pedantic.Originally posted by master_coda
I think I see what you are saying. But problems that happen "within" the system (such as inconsistancy and undecidability) don't actually affect formality, since a systems formality comes from how it is defined.
The truth-values of statements within the system don't really have anything to do with formality.
Originally posted by selfAdjoint
As I keep posting, maybe it will take one of these days, Tarski proved that geometry is complete, and Hilbert and others proved it is consistent. And you can do multiplication in geometry (application of areas, or Eudoxian proportions). But the proofs don't use set theory, so they escape the flaws that lead to incompleteness.
I'm with Bob on this. Are you sure of your facts here?Originally posted by selfAdjoint
As I keep posting, maybe it will take one of these days, Tarski proved that geometry is complete, and Hilbert and others proved it is consistent. And you can do multiplication in geometry (application of areas, or Eudoxian proportions). But the proofs don't use set theory, so they escape the flaws that lead to incompleteness. [/B]
Originally posted by Canute
I'm with Bob on this. Are you sure of your facts here?
What do you mean by complete? Do you mean that the system proves its own axioms? Surely that would make the system trivial (unable to refer outside itself, and therefore tautological).
Originally posted by Hurkyl
For the sake of accuracy, it means that given a statement "G", either "G" or "not G" is deducible from the axioms. (A very fine distinction)
My point was that if the axioms of geometry can be proved from within the formal system of theorems to which they give rise then that system is trivial in that it is tautological and cannot refer outside of itself. Such a system therefore is not subject to the contraints of the incompleteness theorems, and therefore Tarski doesn't seem relevant here.Originally posted by selfAdjoint
He took the whole modern set of axioms for geometry. This is an axiom system just as set theory has a system of axioms. Goedel proved the axioms of set theory, and anything that depended on them, to be incomplete. This was a proof in the meta theory, where the set theory axioms are content.
Tarski constructed a proof at the same level as Goedel's, in the meta theory of geometry, that showed that Geometry was complete. Tarski's proof is just as valid as Goedel's and is well known in the foundations community. Google on Tarski, complete.
My point was that if the axioms of geometry can be proved from within the formal system of theorems to which they give rise then that system is trivial in that it is tautological and cannot refer outside of itself.
Originally posted by Canute
My point was that if the axioms of geometry can be proved from within the formal system of theorems to which they give rise then that system is trivial in that it is tautological and cannot refer outside of itself. Such a system therefore is not subject to the contraints of the incompleteness theorems, and therefore Tarski doesn't seem relevant here.
I'm happy for you to explain why I'm wrong about this, if I am. It's something I've been taking for granted, but I'm no mathematician.
Originally posted by Canute, on my suggestion that reality might be expressible by a formal system
Interesting. I've convinced myself that it cannot be. The trouble is that if it cannot be, then there's no way of proving it cannot be. If there is a way that reality can be described by a non-trivial formal axiomatic system then I wonder what the fundamental axiom is, or was.[/B]
No offense, but as an idea that doesn't seem any more consistent than the idea of God. Can one have more of nothing? I'm not sure.