How to solve the Liar Paradox

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  • #51
Hurkyl
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"Secondly, the problem the liar's paradox brings to light is that various rules for forming formulas conflict with logical semantics. If one is not to alter logic, one must instead alter the grammar by which formulas are constructed."

Perhaps this is what the previous posters were trying to do but in doing so then there is no paradox -- and hence there is nothing to prove. However, if we are left with nothing to prove then all attempts to do so are superfluous.

There is nothing to do in the various forms of logic used today. For example, first-order logic solved the issue by simply disallowing predicates to operate on predicates entirely. The grammar only allows one to evaluate predicates at variable symbols. P(Q), for example, is simply not in the language of well-formed formulas, if P and Q are both predicate symbols.

One can look for other ways to slip self reference into the logic: this is essentially what a Gödel numbering is, and the liar's paradox becomes becomes Tarski's theorem on the undefinability of truth. (Gödel's first incompleteness theorem is the same idea, but referring to provability rather than truth)

This continues with higher-order logics. e.g. second-order logic introduces second-order predicates that are allowed to operate upon first-order predicates and variables, but not second-order predciates. Both steps of the usual formal version of the liar's paradox fail:

  • We can't define a predicate [itex]\Phi(P) := \neg P(P)[/itex] because P(P) isn't a well-formed formula. (P is a first-order predicate, so we cannot evaluate P at P)
  • Even if we could, we can't consider [itex]\Phi(\Phi)[/itex] anyways. ([itex]\Phi[/itex] is a second-order predicate, so we cannot evaluate [itex]\Phi[/itex] at [itex]\Phi[/itex])


In lambda calculus, all of the steps of the usual version of the Liar's paradox can be executed:
[tex]F := \lambda x. \mathrm{NOT}(x x)[/tex]
[tex]S := F F[/tex]
it's easy to see that S is a liar sentence:
[tex]S = FF = (\lambda x. \mathrm{NOT}(x x)) F = \mathrm{NOT}(F F) = \mathrm{NOT\ } S[/tex]
It's also easy to see the right hand sides are both lambda expressions so one cannot weasel out of a paradox by claiming that either F or S is not well-formed. So we are stuck with a lambda expression S with the property that S is neither TRUE nor FALSE.

Fortunately, there are plenty of other things S can be, so there is no paradox.

Note that an older form of Lambda calculus suffered from the Kleene-Rosser paradox. Stanford's pages state that Curry considered the paradox as analogous to Russel's paradox and the Liar's paradox.


In the theory of computation, the recursion theorem lets us write down a liar Turing machine directly, by the program:
  • Let P be my own source code.
  • Simulate the execution of P.
  • If P returns True, then return False.
  • return True
But again, no paradox: this is simply a Turing machine that never halts.


In various modern forms of logic, the Liar's paradox simply isn't paradoxical. Or more precisely, no way is known to construct an inconsistency of logic using the idea of the Liar's paradox. Instead, the idea simply becomes a useful proof by contradiction technique, e.g. to prove in ZFC that the class of all sets is a proper class, or in the theory of computation to demonstrate the halting problem is not computable.

The Liar's paradox only remains a threat of inconsistency when one is trying to devise new logics, trying to understand the semantics of natural languages, or other similar sorts of situations.
 
  • #52
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Hi! Here you write clearly...no backing up is needed, and I essentially agree with you.
There is nothing to do in the various forms of logic used today. For example, first-order logic solved the issue by simply disallowing predicates to operate on predicates entirely. The grammar only allows one to evaluate predicates at variable symbols. P(Q), for example, is simply not in the language of well-formed formulas, if P and Q are both predicate symbols.

One can look for other ways to slip self reference into the logic: this is essentially what a Gödel numbering is, and the liar's paradox becomes becomes Tarski's theorem on the undefinability of truth. (Gödel's first incompleteness theorem is the same idea, but referring to provability rather than truth)


This continues with higher-order logics. e.g. second-order logic introduces second-order predicates that are allowed to operate upon first-order predicates and variables, but not second-order predciates. Both steps of the usual formal version of the liar's paradox fail:

  • We can't define a predicate [itex]\Phi(P) := \neg P(P)[/itex] because P(P) isn't a well-formed formula. (P is a first-order predicate, so we cannot evaluate P at P)
  • Even if we could, we can't consider [itex]\Phi(\Phi)[/itex] anyways. ([itex]\Phi[/itex] is a second-order predicate, so we cannot evaluate [itex]\Phi[/itex] at [itex]\Phi[/itex])


In lambda calculus, all of the steps of the usual version of the Liar's paradox can be executed:
[tex]F := \lambda x. \mathrm{NOT}(x x)[/tex]
[tex]S := F F[/tex]
it's easy to see that S is a liar sentence:
[tex]S = FF = (\lambda x. \mathrm{NOT}(x x)) F = \mathrm{NOT}(F F) = \mathrm{NOT\ } S[/tex]
It's also easy to see the right hand sides are both lambda expressions so one cannot weasel out of a paradox by claiming that either F or S is not well-formed. So we are stuck with a lambda expression S with the property that S is neither TRUE nor FALSE.

Fortunately, there are plenty of other things S can be, so there is no paradox.

Note that an older form of Lambda calculus suffered from the Kleene-Rosser paradox. Stanford's pages state that Curry considered the paradox as analogous to Russel's paradox and the Liar's paradox.


In the theory of computation, the recursion theorem lets us write down a liar Turing machine directly, by the program:
  • Let P be my own source code.
  • Simulate the execution of P.
  • If P returns True, then return False.
  • return True
But again, no paradox: this is simply a Turing machine that never halts.


In various modern forms of logic, the Liar's paradox simply isn't paradoxical. Or more precisely, no way is known to construct an inconsistency of logic using the idea of the Liar's paradox. Instead, the idea simply becomes a useful proof by contradiction technique, e.g. to prove in ZFC that the class of all sets is a proper class, or in the theory of computation to demonstrate the halting problem is not computable.

The Liar's paradox only remains a threat of inconsistency when one is trying to devise new logics, trying to understand the semantics of natural languages, or other similar sorts of situations.
Im trying really hard to find something objectionable in your post and all I can come up with is your sentence in red and even there I mostly agree with it.(Since I have no Idea what Lambda calculus is I cant follow your thinking there but your description of the results is very interesting. But I wont object to what I dont understand even I itch to do just that!)

Our disagreement (if there is one) has to do with:
1 What logic is used when we think? Originally Classic Logic was considered as the Natural Laws of Thought, a claim not made by Modern Logic.
2 What is the Anatomy of Paradoxes? You mention a distinction between Real Paradox and Pseudo Paradox...On what criterion is the distinction made? If the logic used does not permit self referent sentences then how can Paradoxes be analysed,classified and eventually solved?
3 Isnt it supposed that all the results and formulas in any formalization can be translated into English? That there is nothing that cant be said equally true in English as in any formalized language? If so then my results originally intended only to apply to English together with Classic Logic might be of concern to formalized languages, and in particular the observation that: (x=Zx) implies (-Zx if and only if -ZZx).From which,for example, it may be deduced that there is no English sentence saying of itself that it cant be proven.

Suppose:
1 x = "x is not provable"
Then:
2 x is provable if and only if "x is not provable" is provable
And if all provable sentences are true then:
3 x is provable if and only if x is not provable
And therefore:
4 There is no sentence x saying that it is not provable.

So if there are undecidable sentences then they cant be translated into English,
but how then can we understand (thinking in English),
looking in from outside of the Goedel system,
that the Goedel sentence is true?
 
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  • #53
Hi! Here you write clearly...no backing up is needed, and I essentially agree with you.
Im trying really hard to find something objectionable in your post and all I can come up with is your sentence in red and even there I mostly agree with it.(Since I have no Idea what Lambda calculus is I cant follow your thinking there but your description of the results is very interesting. But I wont object to what I dont understand even I itch to do just that!)
This will help:

F:=λx.NOT(xx)

is equivalent to

F(x) = Not (x(x))

and

S:=FF

is equivalent to the composition of F and F.

It is simply another way to right functions.
 
  • #55
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sigurdW, I noticed that in the Stanford Encyclopedia, the liar's paradox was used to derive a contradiction (see section 2.3.3). You may be interested in seeing how it compares to what you've done so far.



http://plato.stanford.edu/entries/liar-paradox/#LiaSho

Interested I am but the formalized language used is not familiar to me...

I am somewhat excentric, I stay off formalization and that is not the norm.

There seem to be a logicians Tower of Babel somewhere. So I even look at the equality sign with some suspicion but so far I have decided to keep and use it. Howzit going? I notice you havent yet declared truth of x=x and the untruth of x="x is not true"!

So are we finding common ground or not? I uphold the three Laws of Logic:
1 Law of Identity
2 Law of contradiction
3 Law of excluded Middle

But I dont formalize and I dont think I treat them as Axioms. (They are,in my view: Prescriptions.)
Together with this we need to understand Negation and Truth.
IF we understand negation...and dont tell me you dont... then truth must be understood by ourselves also. (Notice that negation is customarily defined by "truth" tables.)

Theres many ways of negating sentences so I only note that a negation of x is "x is not true"
(One minor technical point should be mentioned because stupid philosophers will attack otherwise, and that is that I/we dont put quotationmarks on "x" if it is a sentence. Whenever x is replaced with the sentence meant then marks go on if necessary.)

Keep in mind the "trivial" observation that it is not true that x = "x is not true"
and look at the logical form used to derive the liar paradox:

1 x is not true
2 x = "x is not true"

Its a system of sentence functions and it is mistakenly believed to have solutions!
Put x = "Sentence 1" and we get:

1 Sentence 1 is not true
2 Sentence 1 = "Sentence 1 is not true"

Are we communicating? Do you understand WHY no paradox can be derived here?

The Existence of the Liar Sentence makes the Liar Identity "true",
and that in turn makes the Liar sentence Legitimate!
Which in turn makes the Liar identity "Legitimate"!?
A beautiful criminal pair arent they?
 
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  • #56
Interested I am but the formalized language used is not familiar to me...

I am somewhat excentric, I stay off formalization and that is not the norm.

It should be apparent because there aren't that many symbols in logic but to me it is apparent that:

¬ means Not
⊢ means implies
∨ means or
∧ means and
⊣⊢ means logical equality (if and only if)


Notice that the last symbol is two implication signs pointed in opposite directions.

They also seem to use word versions of these symbols but with a lower precedence. In other we evaluate ⊢ before we evaluate "then".


So are we finding common ground or not? I uphold the three Laws of Logic:
1 Law of Identity
2 Law of contradiction
3 Law of excluded Middle
I accept these as standard rules of logic yet I can't accept any of them as absolute.

Keep in mind the "trivial" observation that it is not true that x = "x is not true"
and look at the logical form used to derive the liar paradox:
This appears to be the liar identity but even though "if and only if means" means equality in terms of logical equivalence I have trouble accepting this is the same as thing as true equality.


1 Sentence 1 is not true
2 Sentence 1 = "Sentence 1 is not true"

I like this way of labeling better then you have previously done so because I think it is more clear. An even more clear way to write it would be:

S1: S1 is not true
S2: S1 = "S1 is not true"

but I don't think this is valid. I think that a more proper way to write sentence S1 would be:

S{1,1}: S{0,1} == S{0,1} is not true

and then the second sentence becomes redundant.

I'm using Colin as a label, == for logical equivalence, and I want to reserve = for true equality. Now here is the point, the truth value of S{0,1} is not the same thing as the truth value of S{1,1}. Said another way: The question of whether S{0,1} has some some Boolean truth value is a completely different question then if the relationship is true.

I think the liars paradox just captures an equivalence relationship which is inconsistent. We can chose to call the liar identity as false. More generally, we can define predicates on equivalence relations. For instance, we can define predicates on binary equivalence relations such as logical equivalence relationships.

We can go further. Why should we only use one variable on each side of the equation? Why not instead use use many? There are many systems of equations which are inconsistent so we may expect that there should also be many systems of binary equations which are inconsistent. Perhaps, logical equivalence is just a special case of the http://en.wikipedia.org/wiki/Chinese_remainder_theorem][/PLAIN] [Broken]
Chinese remainder theorem
.
 
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  • #57
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It should be apparent because there aren't that many symbols in logic but to me it is apparent that:

¬ means Not
⊢ means implies
∨ means or
∧ means and
⊣⊢ means logical equality (if and only if)


Notice that the last symbol is two implication signs pointed in opposite directions.

They also seem to use word versions of these symbols but with a lower precedence. In other we evaluate ⊢ before we evaluate "then".



I accept these as standard rules of logic yet I can't accept any of them as absolute.
Since there is nothing that is absolute you should definitely not accept them or anything else as absolute.


This appears to be the liar identity but even though "if and only if means" means equality in terms of logical equivalence I have trouble accepting this is the same as thing as true equality.
And...ahem... I have trouble accepting there is such a thing as "true equality" unless perhaps if you by "equality" mean "inequality". But I dont think it will work. so...nah! "True equality" means the same as "A supposed equality that is not an inequality".

I like this way of labeling better then you have previously done so because I think it is more clear. An even more clear way to write it would be:

S1: S1 is not true
S2: S1 = "S1 is not true"

but I don't think this is valid. I think that a more proper way to write sentence S1 would be:

S{1,1}: S{0,1} == S{0,1} is not true

and then the second sentence becomes redundant.
And youve made it difficult for me to interprete the argument...Good work! (Sarcasm.)

Sigh...I think I should remind you that successful communication is possible if and only if theres a shared Media ,a shared (and notation of) Logic and a shared manner of Conduct.

I'm using Colin as a label, == for logical equivalence, and I want to reserve = for true equality. Now here is the point, the truth value of S{0,1} is not the same thing as the truth value of S{1,1}. Said another way: The question of whether S{0,1} has some some Boolean truth value is a completely different question then if the relationship is true.

I think the liars paradox just captures an equivalence relationship which is inconsistent. We can chose to call the liar identity as false. More generally, we can define predicates on equivalence relations. For instance, we can define predicates on binary equivalence relations such as logical equivalence relationships.

We can go further. Why should we only use one variable on each side of the equation? Why not instead use use many? There are many systems of equations which are inconsistent so we may expect that there should also be many systems of binary equations which are inconsistent. Perhaps, logical equivalence is just a special case of the http://en.wikipedia.org/wiki/Chinese_remainder_theorem][/PLAIN] [Broken]
Chinese remainder theorem
.
OK, now were getting somewhere. I am beginning to think you understand what Im saying...thats all Im interested in since everyone is entitled to ones own opiniion about things. Understanding? ,YES! Agreement? ... not necessarily :)

Well to tell you the truth: I dont understand you in general (sometimes I do understand you so I think our communicative relation may improve)... but its of minor importance since you are probably one of a kind... My own view is, as far as I know, unique! It would be a shame if it died with me because I failed to explain it. So have you found some proposal the same or similar to mine? The translation formula is it complete? ...wasnt there rectangles in the damned thing as well?
That was what finally made me close the site in irritation. Why dont you (for its own sake) translate it into ordinary English? Can it, in your view, be done?

Why dont you accept this very question? Do you accept English or not? If you deny something using ordinary English need I then not accept it as a proper denial of something!? And if YOU claim something, could I then shrug my shoulders and say... Say: Since your not using, an unknown to you, way of forming statements then your "so called statements" really dont state what they may seem to state? Are you denying the possibility of translation between systems intended to express and communicate statements?

I think my concept of "Referential Identity" can be helpful here: Take the following sentence: "This sentence hasnt been read by anyone else but sigurdV" Its obviously true...to me at this moment... but it will be obviously not true to you! How to explain? Well... the two instances does not have identical referential identities so its two different statements expressed by thev same sentence. (My "hunch" now is that this matters when it comes to Quantification and Substitution into Modal Matters.)
 
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  • #58
Hurkyl
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Science Advisor
Gold Member
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2 What is the Anatomy of Paradoxes? You mention a distinction between Real Paradox and Pseudo Paradox...On what criterion is the distinction made? If the logic used does not permit self referent sentences then how can Paradoxes be analysed,classified and eventually solved?
A real paradox is when a theory contradicts itself. Russel's paradox was a real paradox of Cantor's set theory: the axiom of (unrestricted) comprehension guarantees that there is a set [itex]N = \{ x \mid x \notin x \}[/itex] and from this we can construct the statement [itex]N \in N[/itex], which, by the definition of N, is equivalent to [itex]N \notin N[/itex].

A pseudoparadox is when there isn't a contradiction in the theory, but the argument is a fallacy or is merely counter-intuitive. The twin paradox is a pseudoparadox of special relativity: the argument involves applying a formula for time dilation in a situation that doesn't satisfy the formula's hypotheses.

Sometimes, which a paradox is is relative. e.g. Skolem's paradox is a pseudoparadox of model theory. However, it is genuinely paradoxical to a number of opinions on the philosophy of mathematics.

(actually, in my opinion it is not genuinely paradoxical, because I'm quite convinced that the people who believe it to be genuinely paradoxical simply do not understand it -- I believe that if they did understand it, they would find the result to be merely disappointing rather than actually contradictory to their philosophy)
 
  • #59

I think my concept of "Referential Identity" can be helpful here: Take the following sentence: "This sentence hasnt been read by anyone else but sigurdV" Its obviously true...to me at this moment... but it will be obviously not true to you! How to explain? Well... the two instances does not have identical referential identities so its two different statements expressed by thev same sentence. (My "hunch" now is that this matters when it comes to Quantification and Substitution into Modal Matters.)

The part of the liar’s paradox that I find difficult is the part where we go from the sentence:

“This sentence is false”

To the accepted abstract representation:

X=(X is false)

To comprise a logic you need two of three things. Two of these three are syntax and semantics. The axioms of logic you gave are for the syntax of logic although there are metaphysical versions of these axioms which you could use as semantics.

However, if we blindly do this, it lets us introduce new rules of syntax through our semantics which can lead to inconsistencies such as we see with the liars paradox. Clasical logic is “post complete” which means we cannot add any more(non-derivable) axioms to it without making it trivial.

One form of logic which discusses semantics is modal logic:
http://en.wikipedia.org/wiki/Modal_logic#Semantics

In modal logic semantics they use an operator that looks like this |= to stand for entails. Wikipedia says the following:

“If there is a world w such that w ⊨ P, then P is true at w. A model is thus an ordered triple, <G, R, ⊨>.”
It would seem to me that the “is” operator of classical logic would also define semantics but cannot be applied blindly.

Now, with regards to your time reference in "This sentence hasn’t been read by anyone else but sigurdV", according to the Wikipedia page, Prior extended modal logic in 1957 to create temporal logic and in 1976 Vaughan Pratt introduced dynamic logic. I found some video lectures which might be helpful on these topics:

Dynamic logic:
http://videolectures.net/ssll09_schmitt_dlog/

Modal logic:
http://videolectures.net/ssll09_gore_iml/

Non Classical Logic:
http://videolectures.net/ssll09_mares_ncl/

The lectures seem to be geared somewhat towards computer science students.
 
  • #60
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The part of the liar’s paradox that I find difficult is the part where we go from the sentence:

“This sentence is false”

To the accepted abstract representation:

X=(X is false)
It gave me some headache too: its better to not use exactly that representation, use the following instead:

"This" in the sentence "This sentence is false” refers to "This sentence is false”

It means the same thing! Has the same effect. Its an alternative formulation of the liar identity defining “This sentence is false”.
 
  • #61
It gave me some headache too: its better to not use exactly that representation, use the following instead:

"This" in the sentence "This sentence is false” refers to "This sentence is false”

It means the same thing! Has the same effect. Its an alternative formulation of the liar identity defining “This sentence is false”.

This seems to get us closer to the abstract form yet it is further a way from the concrete form. We could be more explicit and write:

"This" in the sentence "This sentence is false” is logically equivalent to "This sentence is false”.

but then the question is why do this. Perhaps this historical example given in Wikipedia is a better way to state the paradox:

Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?"

Yet one might answer Eubulides by saying that the man is lying by omission.
 
  • #62
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"This" in the sentence "This sentence is false” is logically equivalent to "This sentence is false”.
You could say that both sentences has both true and false as truth value... but my point is that if the liar identity has two truth values then THE LIAR "SENTENCE" IS NO SENTENCE AT ALL! It has no true and only true referential identity! It is a sentence function! So they are not logically equivalent after all. I think this is the most overlooked part of my theory. Again: Liar sentences are not sentences, they are sentence functions because they lack proper referential identities! This claim is new: Its not the same claim as saying they are meaningless, they simply dont have a defined subject, they have only a predicate that needs a subject to make a statement...

Look at it backwards: begin with the sentence function "x is not true": if x = "x is not true " then the sentence "x = "x is not true"" is both logically false and empirically true so x is not allowed to take that value! And if we force it to take the value we brake the laws of logic by creating sentences that are both true and false.
 
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  • #63
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i'm a liar therefore i'm not true.
 
  • #64
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i'm a liar therefore i'm not true.
Seriously? I dont think you are a consistent liar :)
 
  • #66
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Hi! Ive sorta forgotten this thread,
where Ive met lots of interesting ppl. I WILL read what is said in http://helsinki.academia.edu/Tuomas...Non-Contradiction_as_a_Metaphysical_Principle
I love Finnish Philosophy "Perkkele! Ei Ymmere"... But not right away... Ill study some beer first.

SO? My theory is like Zenons arrow still alive & swimming?

Cheers to you all ;)

sigurdV


edit: ‘the same attribute cannot at the same time belong and not belong to the same subject in the same respect’(Aristotle 1984: 1005b19-20).

There are References/names?/descriptions? : "xn" (n=1,2,3, ...n)
And Qualities/predicates?. "Qn"
Is negation a quality?

Lets simplify!
Let there be no language:


We are the audience. We look at a fire. Around it we see stones.
Sitting, there are two players supposed to invent language.

My instant guess is that there are three beginning words "this" . "yes" and "no".
But it might be necessary to create other "words" like "aha!" first...

At the moment no word has been created.

Act 1: A player looks at what? and...(Well ahem...) does WHAT?
Yes, friends in the audience, what must happen?
Will a player stick a finger into the fire and say: "Ouch!"?
 
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