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# The Final Solution Of The Liar

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2 Then come back here :)

(You dont need to read more.)

Let us check the derivation of the liar paradox.

1 Sentence 1 is not true. (Assumption 1)

(LIAR SENTENCE)

2 Sentence 1 = "Sentence 1 is not true) (True by inspection of 1)

(LIAR IDENTITY)

3 "Sentence 1 is not true" is true if and only if sentence 1 is not true (by definition of truth)

4 Sentence 1 is true if and only if sentence 1 is not true (substitution from 2 to 3)

Instead of denying assumption 1 and end in paradox, we assume there is no sentence 1 ! (assumtion 2)

And see what happens:

2 sentence 1 = "Sentence 1 is not true"

(ONLY an ASSUMPTION)

3 (as before)

4 (as before)

Now we must conlude that:

5 It is NOT TRUE that Sentence 1 = "Sentence 1 is not true"

And we have proven that sentence 2 is LOGICALLY FALSE!

Question: Is not sentence 2 meaningless if there IS no sentence 1 ?

Answer : Fill in any sentence BUT sentence 1.

Reinstatement makes sentence 2 both logically false and true by inspection of 1. Which means that we make a logical error in introducing sentence 1 as it originally read.

Note that the solution demands a very minor restriction on the language in use in comparison with other solutions.

Aristotle said that it is false to say of what is that it is not... But to assume of what is that it is not, is not the same as to say of what is that it is not! Still it would be nice if we could do without contrafactual assumptions...

In his letter to Titus apostle Paulus states something like the sentence 1 below.

1. There is a sentence, x , such that x = "x is not true" and this is true.

2. a = "a is not true " (x=a)

3. a is true if and only if " a is not true " is true (xZ=aZ)

4 a is true if and only if a is not true (contradiction)

5 It is not true that there is a sentence,x, such that x = " x is not true "

By conclusion 5 Paulus is shown to be a liar but he is not paradoxical since he is

using a Liar Identity instead of a Liar Sentence in his statement. (see Russells paradox)

(Conjecture) A paradox has the following Logical Form:

1. Liar Sentence

2. Liar Identity

Put together the paradox is inevitable, but alone the Liar Identity can be restricted:

Mathematicians was quick to define the Russell Set to be no set but a Class!

(Definition) The sentence, x , is a "selfreferential sentence" if and only there is a predicate, Z , such that x = "xZ"

Supposing there is such an x then we have:

1. x = "xZ" (assumption)

2. xZ = ""xZ"Z" (from 1)

3. x="xZ" implies "xZ" =""xZ"Z" (conclusion)(Logical Truth)

If, for any value of Z, the right side of 3 is not true,then the left side is not true! Which means that for some values of Z

x="xZ" is not true and x is not a self referential sentence!

Such values are,for example: Z="is not true" and Z="is not provable"

Form the set of all such values of Z that are not permitted to form self referential sentences and you know what self referential sentences are not permitted by logic.

You have been very patient, thank you for your attention: SigurdV

Post Scriptum 1: No refuting error has been found, then whats new is:

1 All derivations of the Liar Paradox contain a logical error.(The Liar identity being logically false but empirically or conceptually true!)

2 Earlier solutions also excludes Liar Sentences from appearing in derivations, but no explaining of the paradox is done.

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...

Form the set of all such values of Z that are not permitted to form self referential sentences and you know what self referential sentences are not permitted by logic.

You have been very patient, thank you for your attention: SigurdV

i only dabble in this stuff, but my most recent such dab deals explicitly with the apparent horrors of self-reference. while i read this whole book, i gave it to someone else to read & so i can't reference it any further than this bit below that i quoted in another thread. i thought i posted a dougy dandy of a self-referencial statement that would make russel squirm, but i can't seem to find it. anyway, here's the quote; the blue title is a link to the wiki page on i am a strange loop.

I Am A Strange Loop by Douglas R. Hofstadter: Chapter 4: Loops, Goals, and Loopholes: Intellectuals Who Dread Feedback Loops pg. 63

...What remained with me, however, was the realization that some highly educated and otherwise sensible people are irrationally allergic to the idea of self-reference, or of structures or systems that fold back on themselves.

I suspect that such people's allergy stems, in the final analysis, from a deep-seated fear of paradox or of the universe exploding (metaphorically), something like the panic that the television sales clerk evinced when I threatened to point the video camera at the TV screen. The contrast between my lifelong savoring of such loops and the allergic recoiling from them on the part of such people as Bertrand Russell, B. F. Skinner, this education professor, and the TV salesperson taught me a lifelong lesson in the "theory of types"--namely, that there are indeed "two types" of people in the world. ...

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Im a dabbler as well... Writing whatever I was writing above I had a hard time editing out irrelevancies such as:(1) Who am I? (2) What am I doing? (3) Why am I doing it here? (4)Where IS here?

Oh! I forgot... WELCOME!

Now, taking a brake, I orient myself towards real things... I hereby hasten to assure you that I, probably a figment of my imagination, is as of now residing in some literary frame, eagerly collecting the multitude of specimens needed for my coming bestseller: The Proper Place for Frames not in their Proper Places.

Sorry! I was trying to make a joke:) Here are the correct answers:

(1) Sigurd Vojnov, Swedish Citizen,

(2) succeeded ( ...let us wait and see...) perhaps for the first time in recorded history in actually solving an example of the Liar Paradox! (Other Liars should be vulnerable to the same treatment.)

(3) I did it in here because I felt it to be a Proper Place.

(4) Trying to find out, let us check out the view from top of Mount Einstein.

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(1) Sigurd Vojnov, Swedish Citizen,

(2) succeeded for the first time in recorded history, in actually solving an example of the Liar Paradox! (Other examples should be vulnerable to the same treatment.)

Not to rain on your parade, Sigurd, but you better wait until published in something peer reviewed before making claim #2! I think we agree it’s better to wait to be lauded by others, rather than lauding yourself.

At a glance, I’m inclined to think that the various “solutions” to the liar paradox made by set theorists from Cantor’s time to Gödel's incompleteness theorems (from around 1897 to exactly 1931) predate, and likely duplicate, your general approach, which appears to me to be to forbid certain self referential statements. This approach is effectively the same as the one taken by Zermelo Fraenkel in 1922, which these days is know as ZFC, for “Zermelo and Fraenkel’s axiom of Choice”.

As Turtle’s Hofstadter quote alludes to, self-reference was a source of intense worry from the beginning of what we now know as set theory (The late 19th century). Interest in banishing it waned substantially after Gödel, because hope of generating a consistent formal system including at least the rules of arithmetic, was compellingly proven impossible.

Speaking of Hofstadter, I’ve long liked one of his formulations of the liar paradox, which goes something like this:

Achilles (In his youth, Doug used the characters Achilles and Tortoise a lot) has a purchased a perfect record player.

Turtle – I mean Tortoise ;) – brings Achilles a present of a special record titled “Sounds including the sound that destroys perfect record players, and only perfect record players” – but doesn’t tell Achilles what it is :evil:.

Achilles plays it, destroying his treasured record player.

But, since a perfect record player can play any record perfectly from beginning to end, and Achilles’s broke part way through playing Tortoise’s gift, it wasn’t perfect, but if it wasn’t perfect, the perfect-and-only-perfect-record-player-destroying sound shouldn’t have destroyed it.

Fun stuff! :)

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Goedel, Escher, Bach: an Eternal Golden Braid

Is hereby recommended! (Alas,its too thin.) ;)

Intellectual news has a tendency passing Sweden by...

Next Ill check if the bookstore contains some Strange Loops.

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Hi Again Craig :)

Of course! You are absolutely right!

I SHOULD wait...

But how long? It is as of now NINE YEARS since I started trying to get attention... I operated my heart ten tears ago, I was born in 1945, so what are my chances of being ALIVE when my lifework is checked by ...whoever else (besides my two Distinguished Visitors) is checking new results !?

I am now perhaps somewhat tooooooooooooo impatient so I will edit claim 2 a little... perhaps next time edit out myself as a person: I need no attention. All wanted is assurance that I can go on to examine consequenses of the solution...

(For instance: Alfred Tarski claims that Natural Languages are inconsistent since The Liar Paradox can be derived in them... I should answer: Sir! Show me a derivation of the Liar Paradox and I will show where and how the logical error was made!)

...without worrying about eventual errors in it.

(Meanwhile i am happy to have something else besides paradoxes to think of, i KNOW i have some book written by Einstein somewhere but in the meantime Rindlers work on Special Relativity will have to do.)

Next: It is true that my solution is one among many and theres no guarantee that it is unique...still... using an assumption to remove the liar sentence from the derivation

and then showing it cant be put back again surely is a surprising move that should have been heard of, if it was done before. (Unless there is something wrong with it.)

Note: By adding a restriction axiom, R ,to ZF set theory in order to exclude trouble making entities ,T... Z gave my thinking a push... The axiom being independent, there also should be a consistent theory ZF+(-R) (provided ZF+R is consistent)... Therefore T:s still exists somewhere and some other way must be found to treat the disease.(By me now diagnosed as in general being caused by LS:s and/or LI:s)

Also: I find it economical. One set of solutions excludes all self referential sentences, another raises ladders of language levels towards infinity... I only suggest we obey the laws of logic!

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So lets exemplify :)

1 x = x. (Law of Identity)

2 "x is true" = x (Definition of Truth)

Question: Is sentence 3 now permissible as a definition of negation of x?

3 "x is not true" = x

Answer: No! Since "x is not true" is the ONLY object certainly not x! (My Negative Definition of "Negation of x")

3 "x is not true" = y = A negation of x, if and only if x is not y. (My positive Definition of Negation of x)

Now look at the beginning of a Traditional Derivation of The Liar:

( "Liar" Definition) Let "The Liar" be a name of "The Liar is not true.".

1 The Liar is not true

2 The Liar = "The Liar is not true"

Here,clearly, sentence 2 is a violation of the negative definition of negation!

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• 1 month later...

I'm sorry to say so Sigurd, but I agree that you appear to have done essentially what others had already done: i. e. to put the blame on self reference. This was exactly what prompted Tarski to construct his metalanguages proposal, in order to avoid self reference being possible, yet allowing to make statements about the truth value of statements. :shrug:

Apart from this, I'm not one of those who agrees with banning self reference. Here's my approach to the liar's paradox. I see the liar statement as corresponding to an equation without solutions.

[math]X=\neg X[/math]

There is no boolean value that solves this equation, unlike the similar algebraic equation:

[math]x=-x[/math]

which has the solution of 0, or the analogous case for the reciprocal which has 1 as a solution, whereas the one for boolean negation has no solution.

In your approach, sentence 2 is somewhat like the above equation for negation, rather than being a statement (similarly, you specify that 1 is a statement and 2 is "an identity"). You state that we must conlcude your sentence 5, apparently by reductio ad absurdum, but this only means it is not an identity (it could still have solutions as an equation, without being an identity).

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Hi Qfwfq! :)

Thank You for Your Insightful Posting on the matters of my thread.

(It is just what i have been waiting for.)

I'm sorry to say so Sigurd, but I agree that you appear to have done essentially what others had already done: i. e. to put the blame on self reference. This was exactly what prompted Tarski to construct his metalanguages proposal, in order to avoid self reference being possible, yet allowing to make statements about the truth value of statements. :shrug:

Apart from this, I'm not one of those who agrees with banning self reference. Here's my approach to the liar's paradox. I see the liar statement as corresponding to an equation without solutions.

[math]X=\neg X[/math]

There is no boolean value that solves this equation, unlike the similar algebraic equation:

[math]x=-x[/math]

which has the solution of 0, or the analogous case for the reciprocal which has 1 as a solution, whereas the one for boolean negation has no solution.

In your approach, sentence 2 is somewhat like the above equation for negation, rather than being a statement (similarly, you specify that 1 is a statement and 2 is "an identity"). You state that we must conlcude your sentence 5, apparently by reductio ad absurdum, but this only means it is not an identity (it could still have solutions as an equation, without being an identity).

There is a territory ( How the concepts "liar sentence" and "liar identity" relate to each other and how they produce

the liar paradox.) that must be travelled to arrive at a solution.

The territory is not new, but it was not properly mapped: the concept "Liar Identity" did not exist and had to be defined by me. (Or is that little step on the way of progress also old news?)

What is my ESSENTIAL claim is that A L L derivations of paradox are flawed!

Again: There IS no Logically Correct Derivation of the Liar Paradox!!

The above statement is not new?! (Or true?)

Then, please , tell me who said (and proved it) before! (Not true? Then show me a correct derivation!)

PS there are some remarks i would like to do on your text some other time, but here and now I only consider the question: What is the difference between my solution and ALL other solutions... (Why did I call it "Final"? (etc))

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I'm sorry to say so Sigurd, but I agree that you appear to have done essentially what others had already done: i. e. to put the blame on self reference. This was exactly what prompted Tarski to construct his metalanguages proposal, in order to avoid self reference being possible, yet allowing to make statements about the truth value of statements. :shrug:

If my work is only duplicating already known truth I could live with that :)

Its difficult to avoid. Especially if you are (as I am) an isolated worker.

But before contemplating defeat I will resist...

By pointing out that self reference is not (as your post seems to say) a single cause of paradox.

There are exactly THREE things involved:

1 The definition of truth

2 The definition of negation

3 The definition of self reference

What must be done, and is already known, is that by doctoring the definitions carefully paradoxes will not arise.

BUT! It is only the Definition of Negation that is in need of repair. So the critics of my solution really should say:

Its not new, we already knew the definition of negation had to be fixed! (for detail see post#7)

I further claim that my solution has an explanatory value not encountered earlier.

And at last: You mentioned Alfred Tarski who (besides other things) claimed that natural languages are inconsistent because the liar paradox can be derived in them...

His and my views collide: The Liar Paradox is Underivable! (Unless you break the laws and definitions of logic.)

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The above statement is not new?! (Or true?)
Unfortunately Sigurd, I am not a fluent speaker of Swedish and it is rather hard to follow your English presentation, so it isn't easy for me to be sure of understanding your questions and giving the right answer to them. That is why I said that you appear to have done what has been done; it isn't easy to be sure of what you mean, exactly. That's why I specified what I meant, putting the blame on self reference. You could easily look up the history of this.

However, concentrating on your posts more in detail, I found the further matter about calling something an identity when it isn't really such.

If my work is only duplicating already known truth I could live with that :)

Its difficult to avoid. Especially if you are (as I am) an isolated worker.

You might find folks at Uppsala who can help you in your own language, therefore with less ambiguity in meaning. I'm not sure they are brimming with logicians but there would be less of a language problem.

There are exactly THREE things involved:

1 The definition of truth

2 The definition of negation

3 The definition of self reference

What must be done, and is already known, is that by doctoring the definitions carefully paradoxes will not arise.

BUT! It is only the Definition of Negation that is in need of repair. So the critics of my solution really should say:

Its not new, we already knew the definition of negation had to be fixed! (for detail see post#7)

I further claim that my solution has an explanatory value not encountered earlier.

And at last: You mentioned Alfred Tarski who (besides other things) claimed that natural languages are inconsistent because the liar paradox can be derived in them...

His and my views collide: The Liar Paradox is Underivable! (Unless you break the laws and definitions of logic.)

I find this a bit confused. I don't see why negation would need to be doctored, in any strictly boolean formalization of logic. What Tarski was doing is, in a sense, toward the same goal that you attempt to reach: your constructions are striving to be more precise than natural language. Your sentence 2 is not what we'd call natural language. It shows that the truth value of the liar sentence must satisfy the equality I wrote yesterday:

[math]X=\neg X[/math]

Why do I call it an equation and not an identity? Simply because an equality is an identity if it holds for any value(s) of its variable(s), with no exception. If it holds for only some combinations it isn't an identity, it is an equation with solutions, whereas if it holds for none of all possible combinations then it is an equation with no solutions. For one single boolean variable, it is trivial to check; the above equality is not an identity, it is an equation with no solutions. For either of the two possible values of [imath]X[/imath] it reduces to: "true = false" which simply means the equality doesn't hold.

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Hi Qfwfq!

And I see no error in your approach.

I think we will soon understand each other :)

Doctoring Negation.

(i) Sentence 1 exemplifies the Liar Identity, a statement function in natural language:

1 x = "x is not true". (Assumption)

(ii) Derived from the definition of Truth, this Definition of Negation contradicts the liar identity:

2 -x = "x is not true"

(iii) The Liar Paradox is underivable. (Conclusion)

What is said above has not been said elsewhere. (Conjecture)

The solution is minimal and then (perhaps) final.

PS "Identity"I think i call any expression of the form " x=y", its more or less a routine ... so your objektion there is correct. There might be cases where the distinktion is important. But,hello, how precise must one in general be?

The topic we discuss is not easy ... I dont think the language has very much to do with understanding what we say about logic, reality and paradoxes.

PPS Right, I should approach Uppsala. Come here ye swedish philosophers ;)

PPPS My original intention in this forum was to initiate discussion of foundations...Here and Elsewhere :)

On Time and Mind

On Space and Life

On Truth (The "true" topic of this tread.)

PPPPS Is: x = "x" an Equation identity or a definition, or perhaps all at once?

(to be edited)

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• 2 weeks later...

I have a philosophical background, which means you are free to make fun of my butcherings of scientific theories. ;)

First, let's get a little clearer understanding of the Liar's Paradox.

In bivalent symbolic logic, you only have two options with a semantic variable: T and F. If a logical proposition can only return T, it's a tautology (e.g. A = A). If it can only return F, it's a contradiction (e.g. A = ~A). Both tautologies and contradictions are well-formed statements as long as they can hold stable truth-values. If we can't settle on a set of truth-values for the operations, then the statement is "inconsistent." In classical logic, we use a "truth table" to track the results of a logical formula, like so:

So, here's the compact version:

P
=
P
is false

The problem here is not that P is a contradiction; it's that P cannot hold a stable truth value, and is thus inconsistent.

If you assert that "P is true," then that means P is actually false, so the truth-value flips to false. However, if we say that "P is false," then we have proven it to be true, and the truth value is forced to flip again. This process would repeat indefinitely as the truth-values "oscillate," and this means that P is inconsistent.

SigurdV's initial solution, if I'm reading it right, is apparently to create a property of "self-reference." Unfortunately, it is trivial to see why this doesn't work, as follows.

x = Z(x)

This is a statement of identity, which means that every time we have x, we can substitute Z(x). This triggers an infinite regress:

x = Z(x) = Z(Z(x) = Z(Z(Z(x)) = Z(Z(Z(Z(x))) = .....

The idea that "the odds are true" doesn't work, because logically the "odds" and "evens" are supposed to hold the same truth-value. Thus, this ends up being inconsistent, because you're equating T and F, or we can say that as above, the truth-value constantly oscillates between T and F (which is a mark of inconsistency).

This is not a solution to the Liar's Paradox. In fact, it's the very problem Gödel identified at the start of his proof of the Incompleteness Theorem.

His second solution appears to be some form of trivalence: e.g. truth-values of T, F, and "neither." This doesn't work either, because it's just an assignment that P is inconsistent, which means it cannot be allowed within a consistent axiomatic formal system. It's like saying "I solved the Liar's Paradox by saying it can't be solved." See the problem? :D

The closest thing I've seen to a "solution" was the use of fuzzy logic, and I'm not sure this genuinely resolves the issue either. In bivalent logic, we have T and F. In fuzzy logic, we assign gradations of truth-values ranging from 0 to 1 (or 0 to 100), which captures the common-sensical notion of a "partial truth."

For example, if Z = "the glass is full," and we see that the glass is 25% full, then Z will be false but this doesn't capture the fact that it's partially full. We'd have to test another proposition (Z' = "the glass is 25% full") instead. In fuzzy logic, the truth-value of Z is .25.

So, let's look at:

P
=
P
is false

The consistent fuzzy logic answer to this is .5. This is not especially easy to translate into natural language, since we might say that .5 means "neither true nor false" and/or "both true and false."

The reason I don't think this works is because we'd just turn to another variant:

Q
=
Q
is inconsistent

I'm not entirely sure how you'd express this in symbolic logic, but the natural-language version should suffice. If we claim that Q is true (1), or false (0), or is neither/both true/false (.5), then as long as that truth-value is stable, we've proven that Q is in fact consistent. However, it cannot be consistent, because Q is inconsistent. Thus, no matter what truth-value we assign to Q, it is inconsistent. Fuzzy logic may allow a handful of self-referential statements.

In short, I'm fairly confident that Gödel's Incompleteness Theorem holds, and there is no "solution" to the Liar Paradox in the context of any axiomatic formal system. If you state that it is inconsistent and/or not a well-formed statement, then you're stating that the Liar's Paradox cannot be solved. If you allow it in your system, then per the Principle of Explosion your entire formal system turns into gobbledygook, because you can simultaneously prove that any statement or formula as true, false, neither true nor false, both true and false, and any combination thereof.

So, there is no way to solve the Liar's Paradox. Since formal systems like ZFC seem to be just fine without them, the only "solution" is to recognize that certain types of statements can't be expressed in axiomatic formal systems.

Does that clarify things, without repeating too much of other people's posts? :D

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Hi All, and especially Fluxus :)

I like the the first look of Fluxus post, but i will now proceed without mentioning it further

since i will first write what was on my mind when coming here. (Next post will perhaps examine it.)

In English we are allowed to form sentences using any subject and any predicate...

My wiew is that the laws and definitions of Classic Logic forbids this!

I think that wiew is new and defendable, and thats why im in here writing:

Take "This" and "is not true." and put them together into "This is not true.".

The sentence formed is ordinarily used together with a pointing out what "This" is referred to. The Liar Paradox occurs if "This" refers to "This is not true."!

Lets have the derivation mode:

1 This is not true. (Liar sentence if 2 is true)

2 "This" in "This is not true" refers to "This is not true." (Liar identity: This = "This is not true.")

Abstraction gives:

1 xZ (Liar sentence if 2 is true and Z = "is not true)

2 x = "xZ" (Liar identity if Z = "is not true" (or any related predicate))

The identity is a necessary precondition for the sentence in general!

Notice how we decide the truth value of ordinary sentences like "The sky is blue":

The subject "The sky" refers to " " and is indeed blue...

Obviously the sky cant be put between the quotation signs, but there within is the place where it should be put if it were possible.

But! With selfreferential sentences the subject of the sentence CAN be put there!

Making a good model of the Correspondence Theory of Truth.

So... if we can prove that NO liar identity is logically true then we have proved that any liar sentence is a law breaker!

(Contrary to natural law, laws of logic can be broken.)

Well then, that is easy, suppose that there is an x such that:

1 x = "x is not true"

Then we get:

2 x is true if and only if "x is not true" is true

3 x is true if and only if x is not true!

Therfore there is no x satisfying the liar identity!

This is the special theory, there is also a general theory:

Suppose that:

1 x = "xZ"

Then:

2 xZ = "xZ"Z

And the conclusion is:

3 (x = "xZ") if and only if (xZ = "xZ"Z)

Now we have a logical truth we can use for each an every predicate to check if logic permits its use in a self referential sentence!

To see what happens if Z = "is not true" is left as an exercice, here we check the related Z = "has no proof"

3 (x = "x has no proof") if and only if (x has no proof = "x has no proof" has no proof)

4 if "x has no proof" has no proof then x must be true and have a proof!

5 (x = "x has no proof) if and only if (x has no proof = x has a proof)

6 The right side of the equivalence (in 5) is false therfore the left side is also false!

Conclusion: The predicate " has no proof" may not be used in a selfreferential sentence.

Rather surprising... right?

PS The only way a liar identity ever can be true is to make x = a sentence identical with its own negation.. but the laws of classical logic forbids that!

So... a liar sentence is a sign that logic has been broken by someone.

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Hi Fluxus! And thanx!

1 You have stated the wiew im opposing... well done!

2 I thought Wittgenstein was first in using truth tables, maybe im wrong there.

3 note: You start with " P = P is false " ... What i call a liar identity

4 Dont a truth table on the identity show that both sides cant have the same truth value and that the identity therfore cant be satisfied?(Unless P IS identical with its negation thereby breaking ALL THREE laws of logic.)

5 " x = "xZ" " implies an infinitude of sentences and an infinite identity sequence/statement containing them all.

I have no problem with that... I accept, for instance an infinite sentence that lists its alfabethical constituents.

Once we know how it starts we can fathom the rest of it.

(And wonder if its construction is similar to self awareness.)

6 Fuzzy logic is for me untried territory.

7 And: Gödels proof is flawed! (Or theres a flaw in my theory.)

8 I didnt want to state that in here before...one thing at a time ;)

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I 'm pretty sure truth-tables predate Wittgenstein by a considerable period of time.

Predicate logic was superseded by propositional logic a litte over 100 years ago (by Frege). "P = P is not true" is in propositional logic, nor is "true" regarded as a predicate.

The Liar Paradox only "breaks" logic if you allow it into your formal system. Again, it is inconsistent; therefore, per Gödel, you have a choice between excluding it (in which case your formal system is incomplete) or including it (in which case your formal system is inconsistent.) Allowing an inconsistent statement into your axiomatic formal system will basically trash it; look up the Principle of Explosion to see why. Thus, as far as I know ZFC chooses to rule out these type of inconsistencies, and isn't too concerned with being incomplete.

The Liar Paradox is fairly well understood by now. I'm not an expert on it but not much surprises me about it at this point. There are many iterations, as well as paradoxes which use very similar (if not the same) mechanism. E.g. Russell's Paradox is "x is the set of all sets that do not belong to themselves." If x is not in x, then we have shown that x ought to be in x. But if x thus belongs in x, then it cannot belong in x.

You can't work with an infinite regress. As I pointed out, if you try to assign a truth-value to one stage of the regress, you add another level and the truth-value flips.

With all due respect, you're going to need some very serious chops in logic and mathematics to even consider refuting Gödel's Incompleteness Theorem.

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• 2 weeks later...

Hi Fluxus!

The Liar Paradox is not well understood yet :(

In , for example , The Revision Theory of Truth by Anil Gupta and Nuel Belnap one finds the following:

Let us introduce into English the name "The Liar" according to the following stipulation:

(The Liar) The Liar is not true.

(End of quote.)

Stipulation, Definition, call above what you want... I call it a "Liar Identity", necessary for paradox but ALL such identities are false!

Proof: Suppose that x = "x is not true" , then

x is true if and only if "x is not true" is true, which reduces to

x is true if and only if x is not true, and therefore

there is no x such that x = "x is not true"

The book was printed in 1993 and the authors were unaware that their definition was incorrect ... as the proof above shows ... I dont think the situation has improved since then.

It is with pleasure I declare that: There Is No Logically Correct Derivation Of The Liar Paradox.

Show me your derivation and I will show you where, and what, your mistake is!

Ok.. lets next compare Russells Paradox: The difference is that the Liar Paradox has both Liar Sentence and Liar Identity ...

Russells Paradox has no Liar Sentence, only a Liar Identity.

Thats to say: the set of all sets that do not contain themselves as an element in themselves is never shown ...only mentioned! ( As I just did :) )

And in distinction to the Liar Paradox theres no problem in assuring there is no such set ...since no counter example can be given.

And lastly: Any careful and intelligent reader can see how I will proceed on G ...

I have already stated all that is needed. He uses an unexpressed (and false) assumption in his proof ;)

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