How to create language without self-contradiction?

[sigurdW]

OK, but if it's not a finite game, then it's not a valid choice for Player 1 to call out. Since Player 1 cannot call out Hypergame, Hypergame always terminates in a finite amount of time, and thus Hypergame is a valid choice for player 1 to call out!

If its infinite then its finite and vice versa...It is a paradox and i will probably solve it since it seems familiar...
The only paradox yet where I don't find my tecniques appliable is the Sorites Paradox.

Tomorrow it is! Good Night.

Edit: Now it is tomorrow... The plan is to finish off your subjects one at a time at a leisurely pace.

But I think wed better spend some time checking my solution of the Liar Paradox,
its good also for new comers who tend to read only the last few entries thereby missing important information.

1 Sentence 1 is not true. (assumption)
2 Sentence 1 = "Sentence 1 is not true." (Empirical truth from 1 by inspection)
3 Sentence 1 is true. (The negation of 1, by substitution from 2 to 1 and simplifying)

Here the core of the Liar Paradox is exposed!

Informally:If sentence 1 is true then it is not true, and if so then again its true and so on.
And since everything is either true or not true then sentence 1 is both true and not true!

This state of affairs contradicts the Law of contradiction and makes Classical Logic inconsistent! The Logicians abandoned Classical Logic and formulated Logics that excluded self referential sentences from the domain of their logic. Thereby excluding sentences like: I think this thought therefore I am !

Formally there is yet no contradiction arrived at, so let's add it:

4 Sentence 1 is not true and sentence 1 is true. (contradiction from 1 and 3)

Here the road to the paradox consist in denying the assumption expressed in sentence 1,

and that results in an affirmation instead of a denial...let us leave the road to defeat and

check the remaining alternative: Denying sentence 2!

 

[sigurdW]

On denying sentence 2! (Part One)

Let us listen to the opposition: But sigurdW, you yourself affirm that sentense 2 is true so IF you deny it you are contradicting yourself!

sigurdW: I claim that sentence 2 is an empirically true contradiction!

Thats to say: Sentence 2 is empirically true and logically not true and that is not to contradict myself!

Proof:

10 Sentence 1 = " Sentence 1 is not true." (ASSUMPTION!)
11 Sentence 1 is true if and only if "Sentence 1 is not true." is true (from 10)
12 Sentence 1 is true if and only if Sentence 1 is not true.(from 11)

Sentence 12 is a contradiction and the assumption in sentence 10 must be denied!

13 It is not true that Sentence 1 = " Sentence 1 is not true." (Logical Truth)
14 Sentence 1 = " Sentence 1 is not true." (Empirical Truth = sentence 2)

The extraordinary fact that an empirical truth and a logical truth contradicts each other must be explained...

 

[sigurdW]

Edit:

Today I make it simpler:

Definition:

y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

THESIS:No liar identity is Logically true.

Proof (Based on: (a=b) implies (Ta<-->Tb) )

1. Suppose x="x is not true" (assumption)

2. Then x is true if and only if "x is not true" is true (from 1)

3. And we get: x is true if and only if x is not true (from 2)

4. Sentence 3 contradicts the assumption. (QED)

The logical form of the foundation of the Paradox:

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

Some values for x makes the liar Identity Empirically true:

1. Sentence 1 is not true. (Liar Sentence)
2. Sentence 1 = " Sentence 1 is not true." (Liar Identity)

To get to the paradox one must produce " 3. Sentence 1 is true." from sentences 1 and 2.
But since sentence 2 is BOTH Empirically true and Logically false it can not be a well formed sentence!
Therefore no paradox can be derived from sentence 1,or any other liar sentence.


And here is how the thread originally started:

Alfred Tarski diagnosed the Liar Paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself).

To avoid self-contradiction, Tarski says it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

How to prove him wrong?

English is a semantically closed language so let's begin by stating the conditions for the Liar to arise:

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

Being careful I will not accept sentence 2 on its face value, perhaps its not true?

If it IS true then no harm is done if we declare it to be true...so this is what you should work with:

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

Now try to derive the Liar Paradox! (I predict you will fail to do so! Will you prove me wrong?)

 

[John Creighto]

How to prove him wrong?

English is a semantically closed language so let's begin by stating the conditions for the Liar to arise:

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

Being careful I will not accept sentence 2 on its face value, perhaps its not true?

If it IS true then no harm is done if we declare it to be true...so this is what you should work with:

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

Now try to derive the Liar Paradox! (I predict you will fail to do so! Will you prove me wrong?)

I find this thread hard to follow but I am returning to your first post in this thread to say that what you are trying to solve in the last post of this thread is not the liar's paradox. In sentence 2 you are using the symbol "=" in the normal scene but at the same time you are using it to mean "If and only If". I think the inconsistent use of the symbol "=" is confusing. Also I don't think what you start with in your final post in this thread:

x if and only if (x is not true)

is the liars paradox as self reference is completely removed.

 

[sigurdW]

I find this thread hard to follow but I am returning to your first post in this thread to say that what you are trying to solve in the last post of this thread is not the liar's paradox. In sentence 2 you are using the symbol "=" in the normal scene but at the same time you are using it to mean "If and only If". I think the inconsistent use of the symbol "=" is confusing.

I know the subject is difficult so I haven't been surprised that comments are few.

Suppose we have the identity "a=b" then from it we can get the equivalence "a is true if and only if b is true". The identity IMPLIES the equivalence but they are not identical. So you see I am not using the identity sign to mean anything but what it normally means!

Besides:Note that I am analysing what is supposed to be the beginning of a legitimate derivation of the liar paradox:

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

You must take care so you yourself doesn't solve the paradox by making the derivation of sentence 3 impossible.
(Thats my job: showing sentence 3 to be not derivable from sentences 1 and 2)

I thank you for your interest in this unbelievably (yes I am NOT joking.) difficult matter,
you are mistaken but you are an adventurous person honestly trying to check my argument.

Dont let my objection to your first attempt stop you from digging deeper into the matter :)

Also I don't think what you start with in your final post in this thread:

x if and only if (x is not true)

is the liars paradox as self reference is completely removed.

Im looking for this sentence in my post but I don't find it: "x if and only if (x is not true)"

Perhaps you can quote the post and make the objectional sentence (if its there) bold or something? The sentence "x if and only if (x is not true)" is indeed not wellformed and if I wrote it there's some correctioning (and self flagellation) that needs to be done.
Perhaps you read it while I still was editing the post? That would explain it. Cya ;)

 

[sigurdW]

I find this thread hard to follow

I take your comment very seriously! I am adressing a problem that is over two thousand years old and I should make every effort to present my solution in a clear manner aimed, not at the expert, but the general reader. So I will now start again from the beginning: What is the Liar Paradox?

I guess the dictionary will say something like: The Liar Paradox arises when you try to find out whether a sentence that says of itself that it is not true (Liar Sentence) is true or not.

Since, in my view there IS no Liar Paradox I define the objects believed to cause the paradox

My Definition:

y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

The most common way to introduce the LP is to start with a "Liar" definition:

A Liar Definition: Let the words "The liar" be a name of the sentence "The liar is not true."

Then it is assumed that:

1 The liar is not true. (Liar Sentence)

And from the definition is gotten:

2 The liar = "The liar is not true." (Liar Identity)

And from the above we deduce:

3 The liar is true.

A formal proof takes a few lines more but its not necessary,
the reader should be able now to foresee the disastrous result that follows!
Sentence 1 shows itself to be true if it is false, and false if it is true .
Thereby showing English together with Classic Logic to be inconsistent.

 

[sigurdW]

So what is my solution?

Whenever you look at a definition you should ask yourself if it is valid!

In this case you should ask: Isnt the definiens in the definiendum?

The General Liar Definition: x = "x is not true."

The proposer of the paradox will then say:
The paradox can be produced by other means... the definition is not necessary.
And there are other circular definitions accepted by scientists...
For example the definition of simultaneity in special relativity.

You should not,as a student of Logic, accept that answer... your cool reaction should be:

Oh well, let us assume the liar definition is valid then we get:

1 x = "x is not true"

Since (a=b) implies that (Ta <->Tb) then from sentence 1 we get:

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

The right side can be simplified and we get a contradiction:

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

Therefore the liar definition is NOT valid after all...

By what other means can the paradox be demonstrated did you say?

(But first let us rest, so the eventual reader might catch up, and raise objections!)

 

[John Creighto]

I know the subject is difficult so I haven't been surprised that comments are few.

Suppose we have the identity "a=b" then from it we can get the equivalence "a is true if and only if b is true". The identity IMPLIES the equivalence but they are not identical. So you see I am not using the identity sign to mean anything but what it normally means!

I know what you are doing. You are applying a truth function to each side of the equation. However, I don't think you get a=b. Instead you get A=A which isn't particularly useful.

I thank you for your interest in this unbelievably (yes I am NOT joking.) difficult matter,
you are mistaken but you are an adventurous person honestly trying to check my argument.

I actually wanted to start my own tread on this and their are things I want to say on this topic but before doing so we need to make the proceeding discussion much clearer.

Dont let my objection to your first attempt stop you from digging deeper into the matter :)

It will be easier if the target doesn't keep shifting. I see a lot of very similar posts and it is hard to know which to critique.

Im looking for this sentence in my post but I don't find it: "x if and only if (x is not true)"

My appologies. This was from sigurdW post (Post #38).

 

[sigurdW]

I know what you are doing. You are applying a truth function to each side of the equation. However, I don't think you get a=b. Instead you get A=A which isn't particularly useful.

Instead of defending proof 1 right now...
You should give your contra argument in better detail I am not at all sure why you think i get A=A.
Ill just prove the same thing differently, you can't use the same argument so what is your next contra?

Proof 2
From the Law of Identity we get:
1 x = x
By Double Negation we get:
2 It is not the case that x = "x is not true"

And here's another one for your third contra.

Proof 3
Suppose:
1 x = "x is not true"
Straight from the definition of truth we get:
2 "x is not true"is true if and only if x is not true
And now a contradiction is derivable:
3 x is true if and only if x is not true
Therefore:
4 Its not true that x = "x is not true"

And to make the fact finally obvious:

Proof 4
Suppose:
1 x = "x is not true"
Let x be "water is wet" then we get:
2 "water is wet" = ""water is wet" is not true = "water is not wet"
Now let x be "Water is not wet" then we get:
3 "water is not wet" = ""water is not wet"is not true" = "water is wet"
Neither a true sentence nor a false sentence makes sentence 1 true. Therefore sentence 1 must be a contradiction. QED

PS Make a truth table!

Please state your contra arguments so anyone (including me) can understand them.

 

[sigurdW]

Heres the Logical form of The Liar:

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

You should by now understand what the function of sentence 2 is?

It makes a statement identical to its own negation! Only a statement being both true and false can satisfy it.

And if sentence 1 gets selfreferential it CAN satisfy the Liar Identity.

So Logic forbids it.

Proof:

Suppose:
1 x = Zx
Then:
2 Zx = ZZx
And the logical conclusion (tautology) is:
3 (x = Zx) implies (Zx = ZZx)
Let:
4 Z = is not true
Then we get:
5 (x = x is not true) implies (x is not true = x is true)
and its a Logical Demand that:
6 Its not true that x = x is not true
So any time we construct a sentence that says of itself that it is not true then we defy the laws of logic.
And this EXPLAINS and SOLVES the Paradox of the Liar.

 

[John Creighto]

I want to return to posts #14 to # 16 with regards to prior’s solution. I’m not sure that prior solution is the correct solution. However, I think that by focusing on his solution we may better clarify the rules of logic which we are using. I believe this will help to clarify the posts which have been presented hitherto. I know there are different types of logic and when approaching such difficult paradoxes if we aren’t explicit about what rules of logic we are using then any complex derivations will be difficult to follow. Prior’s solution defines a rule of logic which isn’t universally agreed on. This rule is that any statement implicitly affirms its own truth.

I’m going to quote lugita15 (post #15) as a possible way to apply Prior's rule:

I'm saying that "This statement is false" is the same as saying "This statement is false and "This statement is false" is true."" Or to put in terms of P, P says "P is false", so it's implicitly saying "P is false and "P is false" is true", which is equivalent to saying "P is false and P is true", which is a contradiction.

As this seemed to produce some agreement. Now the criticism given by Wikipedia of Prior’s solution is as follows:

“ But the claim that every statement is really a conjunction in which the first conjunct says "this statement is true" seems to run afoul of standard rules of propositional logic, especially the rule, sometimes called Conjunction Elimination, that from a conjunction any of the conjuncts can be derived. Thus, from, "This statement is true and this statement is false", it follows that "this statement is false" and so we have, once again, a paradoxical (and non-conjunctive) statement. It seems then that Prior's attempt at resolution requires either a whole new propositional logic or else the postulation that the "and" in, "This statement is true and this statement is false", is a special type of conjunctive for which Conjunction Elimination does not apply. But then we need, at least, an expansion of standard propositional logic to account for this new kind of "and".[6]
...
6- Kirkham, Theories of Truth, chap. 9

http://en.wikipedia.org/wiki/Liar_paradox#Arthur_Prior
My response to the criticism which is cited from Kirkam, is that the word, "this", references the entire construct which is:
, "This statement is true and this statement is false"

And hence direct conjunction elimination is not possible. Now applying prior’s implicit assumption to sigurdW’s post #38:

we see that step two is superfluous since it is implied in step 1.

However, sigurdW is using propositional logic which deals more with reducing logical propositions then the assertion of truth. In contrast the laws of thought attempt to get more at the heart of what is true and false in the world.

We can certainly try to use propositional logic to prove a truth value of the liar paradox but I suspect that if possible, that it will be challenging to do so in a self consistent way which maintains the self reference.

and hence direct conjunction elimination is not possible. For purposes of propositional logic we could distinguish between: an "independent And" and a "dependent And" which is analogous how in statistics; we distinguish between independent and dependent random variables. Now applying prior’s implicit assumption to sigurdW’s post #38:

Step two is superfluous as it is implied in step 1.

Yet sigurdW is using propositional logic which deals more with reducing logical propositions then the assertion of truth. In contrast the laws of thought attempt to get more at the heart of what is true and false in the world.

We can certainly try to use propositional logic to prove a truth value of the liars paradox but I suspect that if possible, that it will be: challenging to do this in a self consistent way and at the same time maintain self reference -- since propositional logic distinguishes the atoms (things which we can assert as true or false) from the propositions.

 

[Hurkyl]

You still really don't get it.


First, you seem to have any concept of idea of "consistency". Or, more precisely, you seem to be unable to grasp what it would mean for something to be inconsistent.

Your entire point seems to be based entirely on an inability to comprehend that, in the presence of inconsistency, one can produce two valid arguments with contradictory conclusions, which has led you to the nonsensical rebuttal "there is no contradiction -- you can't do that because it would lead to a contradiction".

To resolve a pseudo-paradox, one must demonstrate that one (or both) of the arguments are flawed. To resolve a true paradox, one must actually abandon the inconsistent theory and create a new one in which is free from that contradiction.



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.

One could take the approach of rejecting formulas based on whether or not they lead to contradictions, but there are two serious flaws with this approach:

• An argument involves many formulas . One needs a rule to decide which of the many formulas are disallowed
• We can be faced with situations such as the possibility that "P and Q" might be a disallowed formula, even when "P" and "Q" are both allowed. Without rules to guarantee that one is allowed to combine formulas in various ways, it would be nearly impossible to reason at all

Third, I think it would be interesting to point out that in the logic of computation, there is no problem with there being a sentence P satisfying "P = P is not true". Here's an implementation in python:

def P():
    return not P()

however, your computer probably cannot do this computation: an equivalent implementation that will not overflow your stack is:

def P():
    while True:
        pass

 

[John Creighto]

sigurdW, I re-labeled your statements in the following:

Proof 3
Suppose:
s1: x = "x is not true"
Straight from the definition of truth we get:
s2: "x is not true"is true if and only if x is not true
And now a contradiction is derivable:
s3: x is true if and only if x is not true
Therefore:
s4: Its not true that x = "x is not true"

Now let's try to connect them:

SS1: s1&s2->s3

SS2: Not S3 -> Not S1 or Not S2We know that not S2 is false because S2 is true (by the law of identity). S3 appears to violate the law of identity so should be false. Consequently not S3 should be true. Thus Not S1 must be true and hence S1 must be false.

However, the conclusion we arrive at should be obvious so why did we assume the opposite in the first place? Also what does it have to do with the liar's paradox?

Now with regards to Hurkyl above post. I am not sure who he is responding to but I certainly admit I don't have a good grasp on this stuff but I'm not trying to resolve the liar's paradox. Rather I am only trying to see if any of the attempts to do so in this tread have made any sense. I think Hurkly's following comment sheds the most light on this:

"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.

 

[sigurdW]

I want to return to posts #14 to # 16 with regards to prior’s solution. I’m not sure that prior solution is the correct solution. However, I think that by focusing on his solution we may better clarify the rules of logic which we are using. I believe this will help to clarify the posts which have been presented hitherto. I know there are different types of logic and when approaching such difficult paradoxes if we aren’t explicit about what rules of logic we are using then any complex derivations will be difficult to follow. Prior’s solution defines a rule of logic which isn’t universally agreed on. This rule is that any statement implicitly affirms its own truth.

I accept that x implies "x is true", but I am not sure of what that may commit me to :)
Ive looked at the discussion between hurky and logita, and it seems that priors solution may resemble mine...Since we both seem to claim a sentence having two truth values is not well formed

My preferred resolution to the Liar Paradox is Prior's, summarized here. The idea is that the liar sentence, like all sentences, asserts its own truth. So a sentence that asserts both its truth and its falsity must be false.

But I don't get the conclusion that the Liar Sentence is false! It becomes a sentencefunction since its liar identity is not well formed!

Now applying prior’s implicit assumption to sigurdW’s post #38:

we see that step two is superfluous since it is implied in step 1.

You mean that the sentence (2. Then x is true if and only if "x is not true" is true ) is implied by priors assumption from (1. Suppose x="x is not true") ?
Or do you mean that (2. Sentence 1 = " Sentence 1 is not true.") is implied by(1. Sentence 1 is not true.)
My first impression is that I disagree in both cases.

#38:
y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

THESIS:No liar identity is Logically true.

Proof (Based on: (a=b) implies (Ta<-->Tb) )

1. Suppose x="x is not true" (assumption)

2. Then x is true if and only if "x is not true" is true (from 1)

Here Priors rule on 1 will give "x="x is not true" is true", but 2 is an equivalence!

3. And we get: x is true if and only if x is not true (from 2)

4. Sentence 3 contradicts the assumption. (QED)

The logical form of the foundation of the Paradox:

1. x is not true.
2. x = "x is not true".
And here Priors rule on 1 will give "x is not true" is true, but 2 is an identity!

Some values for x makes the liar Identity Empirically true:

1. Sentence 1 is not true. (Liar Sentence)
2. Sentence 1 = " Sentence 1 is not true." (Liar Identity)

To get to the paradox one must produce " 3. Sentence 1 is true." from sentences 1 and 2.
But since sentence 2 is BOTH Empirically true and Logically false it can not be a well formed sentence!
Therefore no paradox can be derived from sentence 1,or any other liar sentence.y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

THESIS:No liar identity is Logically true.

Proof (Based on: (a=b) implies (Ta<-->Tb) )

1. Suppose x="x is not true" (assumption)

2. Then x is true if and only if "x is not true" is true (from 1)

3. And we get: x is true if and only if x is not true (from 2)

4. Sentence 3 contradicts the assumption. (QED)

The logical form of the foundation of the Paradox:

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

Some values for x makes the liar Identity Empirically true:

1. Sentence 1 is not true. (Liar Sentence)
2. Sentence 1 = " Sentence 1 is not true." (Liar Identity)

To get to the paradox one must produce " 3. Sentence 1 is true." from sentences 1 and 2.
But since sentence 2 is BOTH Empirically true and Logically false it can not be a well formed sentence!
Therefore no paradox can be derived from sentence 1,or any other liar sentence.

Lets look into the details of my version of the Correspondence Theory of Truth:
Liar Identities are a special case of Referential Identities.

Definition:
y is a Referential Identity if and only if y is of the form: x is the object the "x" in the sentence "Zx" refers to.

Most referential identies are not sentences, say we have the sentence: The Sun is shining.Then the referential identity contains the words "The Sun" and the object that IS the Sun and there's a virtual equality sign joining them together.This makes the definition of truth work: The sentence "the Sun is shining." is true if and only if the Sun is shining.This is easier to understand if we only consider the set of self referential sentences...lets pick one for inspection: 1. Sentence 1 contains five wordsIts referential identity is a sentence!2. Sentence 1 = "Sentence 1 contains five words"And all we have to do is to count the words in the quote at the right side of the identity.

So I think any similarities to Priors theory are superficial. and youll have to convince me that Priors assumption is equivalent to my referential identities.

I like talking to you,I need to practise defence,so let's not be in any hurry,lets face the facts together :)

 

[sigurdW]

sigurdW, I re-labeled your statements in the following:

Originally Posted by sigurdW
Proof 3
Suppose:
s1: x = "x is not true"
Straight from the definition of truth we get:
s2: "x is not true"is true if and only if x is not true
And now a contradiction is derivable:
s3: x is true if and only if x is not true
Therefore:
s4: Its not true that x = "x is not true"
Now let's try to connect them:

SS1: s1&s2->s3

SS2: Not S3 -> Not S1 or Not S2


We know that not S2 is false because S2 is true (by the law of identity). S3 appears to violate the law of identity so should be false. Consequently not S3 should be true. Thus Not S1 must be true and hence S1 must be false.

However, the conclusion we arrive at should be obvious so why did we assume the opposite in the first place? Also what does it have to do with the liar's paradox?

Let me check if I understand you:
Do you accept that its not true that x = "x is not true"?
Its the cornerstone of my thinking:
y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then,and only then, x is a Liar Sentence defined by y.
So if y is not true then x is not a Liar sentence claiming itself to be not true!
And how then can there be a paradox?

1 Sentence 1 is not true (assumed Liar Sentence)
2 Sentence 1 = "Sentence 1 is not true" (logically false and empirically true Liar Identity)

An extraordinary fact is now coming up to the surface!
How CAN a sentence be logically false and empirically true??
Arent logical truths and falsehoods supposed not to ever collide with empirical reality? Logic was thought to be barren but it has brought forth a contradiction, Poincare said... Is this even worse?...Or is there a satisfactory explanation?

Now with regards to Hurkyl above post. I am not sure who he is responding to but I certainly admit I don't have a good grasp on this stuff but I'm not trying to resolve the liar's paradox. Rather I am only trying to see if any of the attempts to do so in this tread have made any sense. I think Hurkly's following comment sheds the most light on this:

"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.

Yes... I have a problem with hurkyl too, he doesn't seem to back up his cl...Whatever they are.
I think your statement in blue shows unbiased thinking.

 

[Hurkyl]

"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 $\Phi(P) := \neg P(P)$ 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 $\Phi(\Phi)$ anyways. ($\Phi$ is a second-order predicate, so we cannot evaluate $\Phi$ at $\Phi$)

In lambda calculus, all of the steps of the usual version of the Liar's paradox can be executed:
$$F := \lambda x. \mathrm{NOT}(x x)$$
$$S := F F$$
it's easy to see that S is a liar sentence:
$$S = FF = (\lambda x. \mathrm{NOT}(x x)) F = \mathrm{NOT}(F F) = \mathrm{NOT\ } S$$
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 let's 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.

 

[sigurdW]

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 $\Phi(P) := \neg P(P)$ 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 $\Phi(\Phi)$ anyways. ($\Phi$ is a second-order predicate, so we cannot evaluate $\Phi$ at $\Phi$)

In lambda calculus, all of the steps of the usual version of the Liar's paradox can be executed:
$$F := \lambda x. \mathrm{NOT}(x x)$$
$$S := F F$$
it's easy to see that S is a liar sentence:
$$S = FF = (\lambda x. \mathrm{NOT}(x x)) F = \mathrm{NOT}(F F) = \mathrm{NOT\ } S$$
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 let's 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 can't follow your thinking there but your description of the results is very interesting. But I won't object to what I don't 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 can't 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 can't 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 can't 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?

 

[John Creighto]

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 can't follow your thinking there but your description of the results is very interesting. But I won't object to what I don't 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.

 

[John Creighto]

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

 

[sigurdW]

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 haven't 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 don't formalize and I don't 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 don't 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 don't 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?

 

[John Creighto]

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]
Chinese remainder theorem.

 

[sigurdW]

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 don't 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 there's 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]
Chinese remainder theorem.

OK, now were getting somewhere. I am beginning to think you understand what I am saying...thats all I am interested in since everyone is entitled to ones own opiniion about things. Understanding? ,YES! Agreement? ... not necessarily :)

Well to tell you the truth: I don't 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 don't you (for its own sake) translate it into ordinary English? Can it, in your view, be done?

Why don't 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 don't 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.)

 

[Hurkyl]

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 $N = \{ x \mid x \notin x \}$ and from this we can construct the statement $N \in N$, which, by the definition of N, is equivalent to $N \notin N$.

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)

 

[John Creighto]

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 let's 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.

 

[sigurdW]

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”.

 

[John Creighto]

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.

 

[sigurdW]

"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 don't 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.

 

[nevere]

i'm a liar therefore I'm not true.

 

[sigurdW]

i'm a liar therefore I'm not true.

Seriously? I don't think you are a consistent liar :)

 

[Bill_McEnaney]

Aristotelian metaphysics to the rescue. Yay!
http://helsinki.academia.edu/Tuomas...Non-Contradiction_as_a_Metaphysical_Principle

 

[sigurdW]

Aristotelian metaphysics to the rescue. Yay!
http://helsinki.academia.edu/Tuomas...Non-Contradiction_as_a_Metaphysical_Principle

Hi! I've sort of forgotten this thread,
where I've 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!"?