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Owen Holden Feb23-05 10:02 AM

Resolving the Barber Paradox and the Russell's Paradox
 
#1
There is a man in a town that shaves all and only those that do not shave themselves.
That is, this barber shaves x if and only if x does not shave x, for all x.

Does the barber shave himself?


No, he cannot shave himself!

Where x and y are existent individuals and S is the relation 'shaves'...
For any x [Ay(xSy <-> ~(ySy)) -> (xSx <-> ~(xSx))], is the paradox.

The assumption that there is an x such that: Ay(xSy <-> ~(ySy)),
leads to the contradiction (xSx <-> ~(xSx)), therefore,
There is no x such that: Ay(xSy <-> ~(ySy)).

ie. Ax~Ay(xSy <-> ~(ySy)), ie. ~ExAy(xSy <-> ~(ySy)) is a theorem.

An x such that Ay(xSy <-> ~(ySy)) cannot exist in classical logic.
The x such that Ay(xSy <-> ~(ySy)) cannot exist either.

There is no existent individual that satisfies the description [an x such
that: Ay(xSy <-> ~(ySy)].
The 'barber' does not exist. He can't shave himself nor can he be shaven.

All primary predications of 'the barber' are false.
We cannot say what it is, but, we can say what it is not.
The barber is not among the members of: those that do shave themselves,
or, those that do not shave themselves.

The barber cannot be: a man, a woman, a robot, etc.
There is no entity that satisfies the description of 'the barber'.



Also..
There is no existent individual class that satisfies the description [an x
such that Ay(y e x <-> ~(y e y))].
The 'Russell Class' does not exist.

~ExAy(yRx <-> ~(yRy)) and ~ExAy(xRy <-> ~(yRy)) are theorems for all relations R.

Including:
1. ~ExAy(x(Shaves)y <-> ~(y(Shaves)y)).
2. ~ExAy(x=y <-> ~(y=y)).
3. ~ExAy(y e x <-> ~(y e y)).
etc.

The answer to Russell's question "Is the class of those classes that are
not members of themselves, a member of itself?" is No, because it does not
exist.
It is neither a member of any class nor is anything a member of it!

(Contrary to NBG, von Neumann-Bernays-Godel set theory, which claims that the Russell class is a proper class.)


The barber paradox, and the Russell Paradox, are resolved without the need for a theory of types or a special category of 'proper classes'.

Bartholomew Feb24-05 01:22 PM

I am not an expert on this issue by a long shot, but I think the problem is that by the original definition of sets, not all sets are logically consistent. People generally take it for granted that if they define some object according to an established template, it should be logically consistent with itself.

anti_crank Feb24-05 04:48 PM

Quote:

Quote by Owen Holden
The barber paradox, and the Russell Paradox, are resolved without the need for a theory of types or a special category of 'proper classes'.

While this is true, it does not contain a novel solution to Russell's paradox. The Russell paradox was based on the now-discredited axiom that every predicate defines a set. The answer that the Russell set does not exist, along with the concession that some predicates do not define sets at all, has been the solution eventually chosen by mathematics and logic after the discovery of the paradox. From that point on, it may or may not be of interest whether more general objects like types or classes can be defined in a consistent manner, depending on whom one asks. The more important effect has been to modify the axioms of sets, since one must now answer the question of which predicates are to be allowed to form sets such that the resulting theory is strong enough to be useful but does not allow the expression of the paradox.

Philocrat Feb26-05 08:12 AM

Quote:

Quote by Owen Holden
#1
There is a man in a town that shaves all and only those that do not shave themselves.
That is, this barber shaves x if and only if x does not shave x, for all x.

Does the barber shave himself?


No, he cannot shave himself!

Where x and y are existent individuals and S is the relation 'shaves'...
For any x [Ay(xSy <-> ~(ySy)) -> (xSx <-> ~(xSx))], is the paradox.

The assumption that there is an x such that: Ay(xSy <-> ~(ySy)),
leads to the contradiction (xSx <-> ~(xSx)), therefore,
There is no x such that: Ay(xSy <-> ~(ySy)).

ie. Ax~Ay(xSy <-> ~(ySy)), ie. ~ExAy(xSy <-> ~(ySy)) is a theorem.

An x such that Ay(xSy <-> ~(ySy)) cannot exist in classical logic.
The x such that Ay(xSy <-> ~(ySy)) cannot exist either.

There is no existent individual that satisfies the description [an x such
that: Ay(xSy <-> ~(ySy)].
The 'barber' does not exist. He can't shave himself nor can he be shaven.

All primary predications of 'the barber' are false.
We cannot say what it is, but, we can say what it is not.
The barber is not among the members of: those that do shave themselves,
or, those that do not shave themselves.

The barber cannot be: a man, a woman, a robot, etc.
There is no entity that satisfies the description of 'the barber'.



Also..
There is no existent individual class that satisfies the description [an x
such that Ay(y e x <-> ~(y e y))].
The 'Russell Class' does not exist.

~ExAy(yRx <-> ~(yRy)) and ~ExAy(xRy <-> ~(yRy)) are theorems for all relations R.

Including:
1. ~ExAy(x(Shaves)y <-> ~(y(Shaves)y)).
2. ~ExAy(x=y <-> ~(y=y)).
3. ~ExAy(y e x <-> ~(y e y)).
etc.

The answer to Russell's question "Is the class of those classes that are
not members of themselves, a member of itself?" is No, because it does not
exist.
It is neither a member of any class nor is anything a member of it!

(Contrary to NBG, von Neumann-Bernays-Godel set theory, which claims that the Russell class is a proper class.)


The barber paradox, and the Russell Paradox, are resolved without the need for a theory of types or a special category of 'proper classes'.

If The barber does not wish to shave himeslef, then he must so declare and not mislead his audience with the fiction that there is a paradox. If he is LOGICALLY and QUANTITATIVE confused, he must so declare. He must never hide behind the fiction of Paradox, for there is none.

If he speaks, writes, or instructs, he must clearly and distinctively say or indicate whether he wishes to include himself in the Set of bearded and shavable people or not. A Universal Set or a Universe of Discourse always contains a precise number of members or things. If a Cretan wants to include himself in the set of all the Cretans that are Liars, then he must take care and construct his sentence in a manner that quantitatively and logically avoids 'Self-Referential Errors'. Exclussionary Laws exist both in Formal Logic and in NL (Natural Langauge) for 'self-debugging' when speaking, writing or instructing. If you cannot self-debug, why speak, write or instruct in the first place?
--------
THINK NATURE.....STAY GREEN! MAY THE 'BOOK OF NATURE' SERVE YOU WELL AND BRING YOU ALL THAT IS GOOD!

Owen Holden Feb26-05 12:13 PM

Quote:

Quote by Philocrat
If The barber does not wish to shave himeslef, then he must so declare and not mislead his audience with the fiction that there is a paradox. If he is LOGICALLY and QUANTITATIVE confused, he must so declare. He must never hide behind the fiction of Paradox, for there is none.

The described barber cannot wish. It does not exist!

Philocrat Feb27-05 03:51 AM

Quote:

Quote by Owen Holden
The described barber cannot wish. It does not exist!

If so, much the better.....either way (exist or not exist) there is no paradox! If you have patience, I may turn up with some Devices later, as long as such devices are NL-Specific and relevant!

---------------------------------
Think Nature....Stay Green! Above all, never harm or destroy that which you cannot create! May the 'Book of Nature' serve you well and bring you all that is Good!

Chronos Mar30-05 02:00 AM

Logical paradoxes are internally inconsistent. First principles are usually violated. The mathematical equivalent of singularities.

Pensador Mar30-05 09:48 AM

A computer programmer once showed me two sets which included each other. Set A was a member of set B, and set B was a member of set A. I was dumbfounded by what I thought to be a logical impossibility, yet he explained to me how that was possible and I found the solution quite simple, though perhaps not intuitive.

It left me thinking of how anonymous people all over the world are faced with those "paradoxes" of philosophy and, not having heard of them, go ahead and solve them in clever ways. I was under the impression that Russel's set does exist and the problem was finding how the paradox was solved. Perhaps we should hire a few computer programmers?

As to the barber paradox, I'd venture to say it's related to the principle behind oscillators. The current state is unstable and forces a different state to emerge, but the new state is as unstable as the previous one and forces its return. In the case of the barber, he has to shave himself when he doesn't, and that act forces him not to shave himself, which brings back the previous scenario, and so on. The key point here is that time solves the paradox, though only temporarily (sort of redundant but that's precisely the point)

phoenixthoth Mar30-05 01:03 PM

In ternary logic, the conjunction of a statement and it's negation is not false necessarily.

Philocrat Mar31-05 02:08 PM

Quote:

Quote by Pensador
A computer programmer once showed me two sets which included each other. Set A was a member of set B, and set B was a member of set A. I was dumbfounded by what I thought to be a logical impossibility, yet he explained to me how that was possible and I found the solution quite simple, though perhaps not intuitive.

It left me thinking of how anonymous people all over the world are faced with those "paradoxes" of philosophy and, not having heard of them, go ahead and solve them in clever ways. I was under the impression that Russel's set does exist and the problem was finding how the paradox was solved. Perhaps we should hire a few computer programmers?

Programmers are undisputedly ahead of the game. And this is reason that I sometimes look at most philosophical statements (and even some scientific statements as well) and I just smile knowing fully well that some of the logics that these disciplines are wrestling with are already secretely resolved by programmers. For example, most of the so-called Paradoxes have already been successfully captured in the 'DATA STRUCTURES' specifications in computer operating systems, Databases, and in most of the so-called High Level Languages, espcecially those with advanced and more up-to-date Object-Orientation capabilities. In fact, in my own opinion programmers are no longer doing programming, rather they spend most of their time specifying communications channells between theortical self-sustained objects. Infact, some of these self-sustained theoretical objects are multiple computational instruction sets within instruction sets. Some of them are so sophisticated that they can theoretically and in actual fact self-refer in that they can fuctionally activate or call instances of themselves while in action.

The question of resolving the cretan or barber paradox is a non-starter because they can be programmatically specified.



Quote:

As to the barber paradox, I'd venture to say it's related to the principle behind oscillators. The current state is unstable and forces a different state to emerge, but the new state is as unstable as the previous one and forces its return. In the case of the barber, he has to shave himself when he doesn't, and that act forces him not to shave himself, which brings back the previous scenario, and so on. The key point here is that time solves the paradox, though only temporarily (sort of redundant but that's precisely the point)
In our Natural Langauge (NL), there are also declaratory constructs and devices for exclusively avoiding all these so-called paradoxes. The formalists just want to pointlessy prove rigour and elegance when there is no need for this.

Philocrat Mar31-05 02:16 PM

And also, the claim that 'CONSCIENCE' (let alone CONSCIOUSNESS) cannot be captured in computer programs is a fundamental mistake. It can be done, except that we need to define it first incase it has evolutionary value in the overall scheme of things in the universe.

Pensador Mar31-05 03:29 PM

Quote:

Quote by Philocrat
In our Natural Langauge (NL), there are also declaratory constructs and devices for exclusively avoiding all these so-called paradoxes.

That is interesting. Could you provide an example?

I do find it interesting that ordinary people are never confronted with paradoxes, which certainly implies our language is sophisticated enough to avoid them. I just don't know exactly how we do that, which is why I asked you for an example.

Philocrat Mar31-05 09:29 PM

Quote:

Quote by Pensador
That is interesting. Could you provide an example?

I do find it interesting that ordinary people are never confronted with paradoxes, which certainly implies our language is sophisticated enough to avoid them. I just don't know exactly how we do that, which is why I asked you for an example.

The constructs are already embedded in NL, and any competent native speaker of it can almost unconsciously 'self-debug' and generaaly disambiguate sentential constructs in normal conversations. If you were having conversation with someone or a group of your friends and the following types of sentences kept on cropping up:

(1) John is tall

(2) America is well fed

(3) All politicians are liars

(4) All Rain forest people are carnibals

how would you disambiguate them? Even if you yourself failed to do so, would your friends miss the opportunity to do so? Conversation is not only a domain for communicating just for the sake of doing so, and for justifying all the constituent terms of all the propositions (sentences, questions, exclamations, metaphors, etc), and ground the intermediate truths and final truth of it, but also, and most importantly, for disambiguating metaphysically vexing terms, forms and constructs. For example, even if you were the maker of propositions (3) and (4) and you were in actual fact a politician or a rain forest person, how would you resolve the paradoxes in these two sentences, if any in the first place? Of course for the formalists logical paradoxes are like drugs......they are good for daydreaming!

Do not worry about being wrong.....we are not in a classroom here. I just want you to take at least a mere guess of how the above sentences would be conversationally disambiguated in NL with all their quantificational and logical forms or constructs fully intact. Just take a guess!

Philocrat Mar31-05 09:47 PM

If people claim and insist, as they constantly do, that our NL is vague and riddled with paradoxes, then like I have suggested already NL itself needs to be disambiguated and clearly taught at all levels of our education. But to think that we can take Logic and Mathematics out of NL, purify and teach them to only the ellitist few is a fundamental error. Everything that we need to speak, write, think and instruct clearly is already fully contained in NL. At worst it only needs to be debugged and clearly but fully implemented universally.

Pensador Apr1-05 08:27 AM

Quote:

Quote by Philocrat
If you were having conversation with someone or a group of your friends and the following types of sentences kept on cropping up:

(1) John is tall

(2) America is well fed

(3) All politicians are liars

(4) All Rain forest people are carnibals

how would you disambiguate them?

(1) and (2) are ambiguous because they lack context. More information will make it clear what the sentences mean, and whether they are true or false. As to (3) or (4), anyone who knows a politician who is not a liar, or a rain forest person who is not a cannibal, can know that the sentence is false; on the other hand, no one knows the truth about "all politicians" or "all rain forest people", so whoever utters the sentence cannot possibly be making a valid statement.

Quote:

Of course for the formalists logical paradoxes are like drugs......they are good for daydreaming!
It is my impression that philosophers love to take sentences no one will ever utter, and then spend an eternity trying to figure out what people mean when they say what they never say.

Quote:

Do not worry about being wrong.....we are not in a classroom here. I just want you to take at least a mere guess of how the above sentences would be conversationally disambiguated in NL with all their quantificational and logical forms or constructs fully intact. Just take a guess!
I enjoyed the exercise, and I'm looking forward to your comments.

phoenixthoth Apr1-05 05:29 PM

Quote:

(1) and (2) are ambiguous because they lack context. More information will make it clear what the sentences mean, and whether they are true or false. As to (3) or (4), anyone who knows a politician who is not a liar, or a rain forest person who is not a cannibal, can know that the sentence is false; on the other hand, no one knows the truth about "all politicians" or "all rain forest people", so whoever utters the sentence cannot possibly be making a valid statement.
This is why it's clear that our brains use a form of fuzzy logic. Is John tall? Well, can you answer that as true or false in any context? Suppose the context is the set of all Americans and John is the tallest American. Then is John is tall true or false? Suppose the context is the set of all pigmies on an island and John is the tallest pigmy islander. Then is John is tall true or false? Could it be that truth depends on interpretation? ;)

As Philocrat observed, we can work out these puzzles in our head without having to bog our biochemical "cpus" with such questions. There is a whole spectrum of tallness which many would agree on. If John is 7'6", everyone would call him tall right? But what if we compare John to a redwood tree that is 1000 years old? John isn't so tall now. This doesn't bother us as we can charge forward and assign a graded truth value to the statement after setting up criteria.

(2) America is well fed. Again, as well is a matter of opinion, just as tall is, the truth value of this statement can't be evaluated. It's not true and it's not false. If we set a context, like not well fed means anything less than 1600 calories per person on average and well fed means anything over 2200 calories, then we can assign (arbitrarily--from the get go) an intermediate truth value to (2) which depends on our criteria.

(3) All politicians are liars . This may be true. Only if every politician satisfies the definition of liar. This could be true without anyone having verified it. Something can be true even if no one has proved it. I can argue that there are infinitely many true and false statements; so there are more than we have written down. So, something can be true even if no one has proved it.

(4) All Rain forest people are carnibals . Same thing as the last one. Note that if there is one instance of this not being true, then it makes (4) false.

Oh... How would you disambiguate them? I missed that! Sorry.

(1) John is over 6'3".
(2) America is in the top 20% in terms of calories per person
(3) Every politician I have encountered is a liar, and I have encounter a great many of them
(4) similar to (3)

By disambiguate do you mean turn into crispy logic? What's wrong with fuzzy?

Philocrat Apr2-05 10:37 PM

Quote:

Quote by Pensador
(1) and (2) are ambiguous because they lack context. More information will make it clear what the sentences mean, and whether they are true or false. As to (3) or (4), anyone who knows a politician who is not a liar, or a rain forest person who is not a cannibal, can know that the sentence is false; on the other hand, no one knows the truth about "all politicians" or "all rain forest people", so whoever utters the sentence cannot possibly be making a valid statement.

Absolutely! Spot on! Yes, (1) and (2) do lack context and that's why they are vague. But the fundamental question here is "what context are we referring to here?". My own answer to this is that such context is a Metaphysical one. We need to crack these sentences open with a METAPHYSICAL HAMMER to reveal their numerical contents with which to speak, write, think and instruct properly about the subject matter that we at hand. We cannot just continue to claim that these sentences are paradoxical or are naturally loaded with paradoxes without making some effort to metaphysically disambiguate them. In fact, (1) is a sentence that is 'metpahysically pregnant' with two or more numerical values John's height and the height of anyone standing next to John or within the same space and time locality as John. In Modal Logic we would then say that in a possible world where there is only one person in existence, in this case John, that there is no way of knowing whether John is tall or not without someone else with which to compare his height. So in a world where there is only John, the term 'Tall' is devoid of semantic and truth values. It is semantically meaningless and epistemologically truth-valueless (neither true nor false). So, in NL you would say that "John is tall" always implies "John is taller' than x". Another way to appreciate the implication of this interpretaion is to always think of every proposition as a 'Conclusion of a Deductive Argument'. Doing this allows you to see every proposition as a fully argued statement of fact.

Let me shock you even more by dropping the bombshell that propositions (2), (3) and (4) have the same formal structures. They share the same logical and quantitative formal structures. They too need to be metaphysically cracked open with a metaphysical hammer. I will post the full analysis of these three sentences later.

Quote:

It is my impression that philosophers love to take sentences no one will ever utter, and then spend an eternity trying to figure out what people mean when they say what they never say.
I just couldn't agree with you more on this. Infact, it is not only philosophers alone that get bugged down on irrelevant, scientists are more or less the same. It is very sad that we are turning our higher institutions into Paradox Hunting grounds, where the drive for academic success now nearly entirely depends on how many paradoxes each of us can find in the world and in the very language with which we describe it. We seek rigour and complexity where there is none and often where pure simplicity and honesty would do. We dump down academic excellence in pursuit of irrelevances. We cry wolf, pump up sensationalism and hype up everything. It is a viscious circle and up till this day I am still wondering how we are going to get out of it.

NOTE: Now, note that the fundamental issue at stake here is not about the truth-value of a given proposition being either true or false, but rather it is about the truth-value of such proposition resulting in a paradox. For example, in the proposition (3), the issue is not about whether it is true or false that all politicians tell lies. Rather, the issue is about whether, in self-referential context where the maker of the proposition were to actually be a politician, the resulting truth-value of it (true or false) manifests into a paradox. Although some people might not agree, any proposition that gives rise to self-refrential errors or paradox of any kind is almost always due to 'Category Mistake' as it is sometimes called in Metaphysics. When this happens, you need to get as many metaphysical tools as you can lay your hands on to disambiguate them at the grassroot level.

Philocrat Apr3-05 12:44 AM

Quote:

Quote by phoenixthoth
This is why it's clear that our brains use a form of fuzzy logic. Is John tall? Well, can you answer that as true or false in any context? Suppose the context is the set of all Americans and John is the tallest American. Then is John is tall true or false? Suppose the context is the set of all pigmies on an island and John is the tallest pigmy islander. Then is John is tall true or false? Could it be that truth depends on interpretation? ;)

Ok, Let's test this concept with a little bit of Modal Logic or the Possible World Logic or Possible World Semantics as sometimes called. Given a world where there is only John in existence, the metaphysical implication of this scenario is that the constituent term "Tall" is devoid of Semantic value (meaning) and the truth-value (true or false). That is, it is semantically meaningless and epistemologically truth-valueless. But as soon as John is placed in a world where there is more than one person, the semantic value and the truth-value of the term 'tall' automatically come alive, because there would be at least one person in the vincity where John is physically installed with which to compare John's height with. The comparative numerical values of John's height with someone else's height are the things which give both the semantic value and the truth-value to the constituent term 'tall'. And once you crack it open with a metaphysical hammer to reveal its numerical content, then you can make clear and logically precise statement about John. Now, supposing we moved to a world populated with many people with same height, what do you reckon would be the semantic and epistemological fate of the term? Well, it is obvious: it would share the same semantic and epistemological fate as the world with only one person for there would be no way of knowing who is tall or not.

Semantically and epistemolgically, the term "tall" is metaphysically pregnant with two fundamental numerical values that are relevant to a competent native speaker of NL for saying something concrete about John relative to anyone that geometrically extends from him. The proposition "John is tall" logically and quantitaively implies:

There is an A with Numerical Property x;
There is a B with Numerical Property y,
Such that x > y


Which in NL simply implies

"A is taller than B" or "A is numerically greater than B"

In fact, there are countless ways to say this in NL. Here I am only trying to sort things out at the metaphysical level, it is therefore up to the formalists to properly symbolise them in any branch of logic or mathematics that they want. That is entirely up to them.

Quote:

As Philocrat observed, we can work out these puzzles in our head without having to bog our biochemical "cpus" with such questions. There is a whole spectrum of tallness which many would agree on. If John is 7'6", everyone would call him tall right? But what if we compare John to a redwood tree that is 1000 years old? John isn't so tall now. This doesn't bother us as we can charge forward and assign a graded truth value to the statement after setting up criteria.
Well, this is where METAPHYSICAL CATEGORISATION comes in. The world is metaphysically, semantically and epistemologically messy. I totally agree with you. There is no dispute about that. But because there is a fundamental need for the human race to intellectually progress, we just cannot afford to sit back and not make any attempt to categorise things in the world into manageable fundamental types. We have to do this. For example, you do not have to compare a human being with a tree in order to conceptualise and derive the semantic value and the truth-value of this metaphysically vexing term "tall'. You can derive these values from a single class of variants such as a class of human beings with height variants, even where such derived values are mere approximations. In the real world we use such numerical ceoncepts as averages within a given class of variants. That is ok, since the issue of deriving absolute semantic and truth values is a completely different matter. The issue at stake here is that off deriving the right concepts that avoid paradoxes in our routine description of the world using NL, and not that of deriving absolute truth values from the process.

Quote:

(2) America is well fed. Again, as well is a matter of opinion, just as tall is, the truth value of this statement can't be evaluated. It's not true and it's not false. If we set a context, like not well fed means anything less than 1600 calories per person on average and well fed means anything over 2200 calories, then we can assign (arbitrarily--from the get go) an intermediate truth value to (2) which depends on our criteria.
Here I am only interested in cracking open this sentence in order to reveal its proper numercal contents with which to properly describe the underlying subject matter in a manner that avoids the so-called hidden paradoxes, which till this day I vehementaly kick against. Of course, I do agree with you that defining the concept of 'well-fed' is a completely different matter, let alone deriving the absolute truth-values of the whole proposition. We will come back the epistemological issues proper later. But here I am only interested in the metaphysical mechanism that is apparent in the proposition the we can reveal with which to do colateral damage to any paradox or paradoxes that may be hidden in it.

Quote:

(3) All politicians are liars . This may be true. Only if every politician satisfies the definition of liar. This could be true without anyone having verified it. Something can be true even if no one has proved it. I can argue that there are infinitely many true and false statements; so there are more than we have written down. So, something can be true even if no one has proved it.
Equally, here I am not concerned with truth values. My interest is in what we need to avoid paradoxes from derivable truth values.

Quote:

(4) All Rain forest people are carnibals . Same thing as the last one. Note that if there is one instance of this not being true, then it makes (4) false.
Same as above. We will deal with truth-values on their own later. Fingers crossed.

Quote:

Oh... How would you disambiguate them? I missed that! Sorry.

(1) John is over 6'3".
(2) America is in the top 20% in terms of calories per person
(3) Every politician I have encountered is a liar, and I have encounter a great many of them
(4) similar to (3)

By disambiguate do you mean turn into crispy logic? What's wrong with fuzzy?
I love Fuzzy ......because it tells it as it is. It is by far the best reflection of the worl! In Philosophy, Modal Logic is used to examine a range of possibilities in the world......to speak of the world as you see or find it. The 'yes' or 'no' camps (the advocates of the 'LOGIC OF STRICT IMPLICATION') are the ones causing the problems. Of course, if you add Modal Logic or Fuzzy Logic to all other types of Logic, the notion of 'TRUTH-VALUES' will include not only 'True' or 'False' but also other types of truth values such 'Necessary', 'Possibly', 'Contingent', 6.3" etc. All that matters here is that you can say something about the world that may result in any of thses types of truth-value without falling into a paradox in the process.


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