Is the Liar's Paradox a Product of Conventional Mathematical Logic?

[Doron Shadmi]

The honest can't say :"I am a liar" , because he can't lie.

The liar can't say :"I am a liar", because he can't say the truth.

When someone in this boolean universe (where each person can be
honest XOR liar) uses the words "All people are liars"
he is included too.

It means that no one in this boolean universe can say those words,

Therefore, there is no paradox.



What do you think ?

 

[StephenPrivitera]

Isn't that the paradox? The paradox is that no one can truthfully say that he is a liar. You cannot tell the truth and lie at the same time. If someone says, "I am lying to you right now," then the statement he just made is false. But if the statement is false, then he is not lying and thus the statement is true. But if the statement is true...
The problem with this statement is that it has the unfortunate consequence p<=>~p, which of course is impossible.

 

[quartodeciman]

Let "liar" mean "consistent liar", and "antiliar" mean "consistent NON-liar".

If these categories (liars and antiliars) are instantiated, then members of these classes can't say this. But if they can, then the contradictions arise. In a logically consistent domain, this leads to the conclusion that these categories must be vacuous and all people lie sometimes and don't lie other times. That is interesting.

Also note that "lying" denotes more than just telling untruths. There are the dimensions of knowledge and intention to deceive to be considered. To lie, one must know the truth and must intend to deceive the hearer. Are there limits here?

 

[quantumdude]

Originally posted by Doron Shadmi
It means that no one in this boolean universe can say those words,

"I am a liar".

Nope, looks like I can say them.

Therefore, there is no paradox.

It is a paradox because it is a statement to which it is impossible to assign a truth value (that's what a paradox is). When I say, "I am lying" it means that it is the case that the statement I am making is false, which means I am not lying. Truth value: F

Which means it is the case that the statement I am making is true, which means that I am lying. Truth value: T

Which means that it is the case that the statement I am making is false, which means that I am not lying. Truth value: F
.
.
.
...ad infinitum

 

[Doron Shadmi]

Hi Tom,

When I say, "I am lying"

Not to be able to say it means: "When I sey" = {}

Therefore there is no paradox.

 

[Doron Shadmi]

Hi quartodeciman,

To lie, one must know the truth and must intend to deceive the hearer. Are there limits here?

To be a lair is first of all not to contradict your own property, so every layer knows exactly what is the truth.

The truth for a layer is like death.

Therefore, no one can hear a dead layer ( dead layer = {} ).

The same logic holds for the honest.

 

[Doron Shadmi]

Hi StephenPrivitera ,

But if the statement is false,...

Because the statement = {} , there is no paradox.

 

[quantumdude]

Originally posted by Doron Shadmi

Not to be able to say it means: "When I sey" = {}

Therefore there is no paradox.

I don't know what you mean by {}, but I think you are getting hung up on an irrelevant detail. The liar's paradox is not about liars, but about self reference.

Formal systems (such as logic) break down under self reference, or when statements refer directly to their own truth value. "Honesty" has nothing to do with it. You could just as well restate the paradox as follows:

This sentence is false.

So, it is the case that the above sentence is false. Truth value: F.
So, it is not the case that the above sentence is false. Truth value: T.
So, it is the case that the above sentence is false: Truth value: F.
.
.
.
...ad infinitum

The paradox is here to stay.

 

[Doron Shadmi]

Hi Tom,

{} = the empty set (see ZF set theory) = content does not exist = 0

Let X = This sentence

0 = false = {} = content does not exist
1 = truth = {X} = content exists

If X refers ot itself (directly or indirectly) as false, then X does not exist in any stage.

Therefore, there is no paradox.

The paradox arises because we force something that does not exist,
to act as if it exists.

And then on top of this act of forcing, we build our artificial paradox.

 

[quantumdude]

Originally posted by Doron Shadmi
{} = the empty set (see ZF set theory) = content does not exist = 0

Let X = This sentence

OK

0 = false = {} = content does not exist
1 = truth = {X} = content exists

If X refers ot itself (directly or indirectly) as false, then X does not exist in any stage.

How did you make this association? More specifically, what does the truth value have to do with existence of the statement?

The paradox arises because we force something that does not exist,
to act as if it exists.

And then on top of this act of forcing, we build our artificial paradox.

It looks to me more like the paradox is real and that your solution is artificial and forced. But maybe you can clear things up with your explanation of why "false" implies "does not exist".

 

[Doron Shadmi]

Hi Tom,

Part 1:
---------

How did you make this association? More specifically, what does the truth value have to do with existence of the statement?

not = ~

0 = false-condition = {} = content does not exist

1 = truth-condition = ~{} = content exists

Let X = some ~{}



------------------------------------------------------------------------------

Part 2:
---------

It looks to me more like the paradox is real and that your solution is artificial and forced. But maybe you can clear things up with your explanation of why "false" implies "does not exist".

{} = the empty set


not = ~


false-condition = {} = set's content ~exists = 0

truth-condition = ~{} = set's content exists = 1


false sentence exists = {sentence} = set's content exists = 1

truth sentence exists = {sentence} = set's content exists = 1



We must not mix between boolean-condition and {sentence}



truth-condition operation on sentences:
------------------------------------------------------------------------------
truth-condition on {sentence} implies ~{} = set's content exists = 1
------------------------------------------------------------------------------


false-condition operation on sentences:
------------------------------------------------------------------------------
false-condition on {sentence} implies {} = set's content ~exists = 0
------------------------------------------------------------------------------




false-condition operation on conditions:
------------------------------------------------------------------------------
false-condition on false-condition implies ~{} = set's content exists = 1

false-condition on truth-condition implies {} = set's content ~exists = 0
------------------------------------------------------------------------------


truth-condition operation on conditions:
------------------------------------------------------------------------------
truth-condition on false-condition implies {} = set's content ~exists = 0

truth-condition on truth-condition implies ~{} = set's content exists = 1
------------------------------------------------------------------------------


The paradox arises because of the mixing between conditions and sentences.


A sentence is a statement, not a condition.


The one who can be operated (on itself and/or on a sentence), is the boolean-condition.


So, we must not mix between an operator and anything that is not an operator (like some statement).


Let X = "This sentence is a false" = {sentence} = ~{} = content exists

X is not a boolean-condition but it tells us what boolean-condition
should be operated on it:

false-condition on X = set's content ~exist, therefore there is no paradox.


------------------------------------------------------------------------------
To be {truth sentence} or to be {false sentence}, that is not the question.


To be(=~{}), or not to be(={}), that is the question.

------------------------------------------------------------------------------

 

[quantumdude]

Originally posted by Doron Shadmi
0 = false-condition = {} = content does not exist

1 = truth-condition = ~{} = content exists

You are simply begging my question here. I asked you on what basis you associate "falsity" with "nonexistence". All you did here is re-assert the relationship.

The way I see it, you are making 2 mistakes here.

1.) First, you are associating the number 0 with the empty set. There is no justification for that.

2.) Second, you seem to be confusing identificaiton with predication. That is, you are taking the mistake in 1.) and compounding it by identifying "this sentence" with "falsity", rather than taking "the sentence" as the subject of the predicate "false".

Just think about it. The sentence "this sentence is false" must exist, because the idea of it can be formed in your mind. That alone is sufficient to qualify as existence for an abstract object of logic.

 

[Doron Shadmi]

1.) First, you are associating the number 0 with the empty set. There is no justification for that.

The boolean system is based on X^2 = X

Therefore X=1 or X=0.


X = 1 = The universal set = {{}} = set's content exists = truth-condition

X = 0 = The empty set = {} = set's content does not exist = false-condition


So, in a boolean system there are only two states:

{} = no members = 0 = false-condition

~{} = all members = 1 = truth-condition





Sorry Tom,


But you did not read the second part of my answer.

Please go back and raed the second part of it.

After you read it, then and only then you will understand my answer.


Yours,

Doron

 

[quantumdude]

Originally posted by Doron Shadmi
The boolean system is based on X^2 = X

Therefore X=1 or X=0.

Yes, I know.

X = 1 = The universal set = {{}} = set's content exists = truth-condition

X = 0 = The empty set = {} = set's content does not exist = false-condition

So, in a boolean system there are only two states:

{} = no members = 0 = false-condition

~{} = all members = 1 = truth-condition

Once again, you are begging the question as to why truth value should be associated with existence.

Sorry Tom,


But you did not read the second part of my answer.

Please go back and raed the second part of it.

After you read it, then and only then you will understand my answer.

I did read it, and I stand by my objections.

 

[Doron Shadmi]

Hi Tom,

In a boolean system any element logically belongs to {}(=0) XOR ~{}(=1)


For example:

Let X = "2 is in {3,4}" is false, therefore X logically belongs to {} = emply set = 0.

Let X = "2 is in {2,7}" is truth, therefore X logically belongs to ~{} = {2,7} = non-empty set = 1.


As you can see, in both cases X is some existing sentence.


But we are not talking about the existence of a sentence as some thought or as some string of notations.

We are talking about the logical existence of some element in a boolean logical system.

Therefore, any element that refers to ITSELF (directly or indirectly) as logically false ("This X is false"), logically belongs to {} = empty set.

The paradox is examined in the boolean logical space.

Because its generator logically send itself to {}, there is no logical base for the paradox.

Therefore, there is no paradox in the logical boolean space, but in some nonlogical imagination of the thinker.

 

[Hurkyl]

Let X = "2 is in {3,4}" is false, therefore X belongs to {} = false = emply set = 0.

The defining property of the empty set is that nothing is an element of the empty set...

 

[Doron Shadmi]

Hi Hurkyl,

Thank you, I mean logically belong ...


I'll fix it.

 

[Hurkyl]

Well, it doesn't really logically belong to the empty set either (whatever that means)... logic doesn't know what sets or numbers are; those are additional structures imposed by mathematical theories.


In mathematical logic, "true" and "false" are not fundamental ideas; deducibility is the fundamental idea. However, one can define things called a truth assignments which is a logical function from well-formed sentences to either true or false.


So, in set theory, for any truth assignment T, we have:

T( T(2 is in {3, 4})=true )=false

(I don't recall with 100% confidence, but I think truth assignments are fair game to be used like this)


But the point is that truth assignments map things to the objects "true" and "false", and logic does not state that these two objects have any connection with 0, 1, {}, {2, 7}, or whatever ~{} is supposed to mean.

 

[Doron Shadmi]

But the point is that truth assignments map things to the objects "true" and "false", and logic does not state that these two objects have any connection with 0, 1, {}, {2, 7}, or whatever ~{} is supposed to mean.

~ = not

In a logical boolean space any state holds(=~{}} or does not hold(={}}

 

[Hurkyl]

The problem is that {} is not a truth value, only "true" and "false" are truth values (at least in conventional mathematical logic). You can't negate it, xor it with anything, or say that sentences have some fundamental relationship with it.

 

[Doron Shadmi]

You are right, the conventional mathematical logic does not based on the simplest possible concepts of to be(=~{}) XOR not to be(={}).

Therefore we have the liar's paradox.