How to solve the Liar Paradox

lugita15

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!

sigurdW

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 dont 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 lets 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:

John Creighto

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 know the subject is difficult so I havent 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 doesnt 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 :)

Im looking for this sentence in my post but I dont 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 theres 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 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 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 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.
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.

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

sigurdW

Instead of defending proof 1 right now...
You should give your contra argument in better detail Im not at all sure why you think i get A=A.
Ill just prove the same thing differently, you cant 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 heres 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:

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

John Creighto

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

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 accept that x implies "x is true", but Im 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

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

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.

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


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 lets not be in any hurry,lets face the facts together :)

sigurdW

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?


Yes... I have a problem with hurkyl too, he doesnt seem to back up his cl...Whatever they are.
I think your statement in blue shows unbiased thinking.

Hurkyl

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.