The Final Solution of the Liar Paradox

[sigurdW]

Let us check the derivation of the liar paradox.

1 Sentence 1 is not true. (Assumption 1)
(LIAR SENTENCE)
2 Sentence 1 = "Sentence 1 is not true) (True by inspection of 1)
(LIAR IDENTITY)
3 "Sentence 1 is not true" is true if and only if sentence 1 is not true (by definition of truth)
4 Sentence 1 is true if and only if sentence 1 is not true (substitution from 2 to 3)(CONTRADICTION)

Instead of denying assumption 1 and end in paradox, we assume there is no sentence 1 ! (assumtion 2)
And see what happens:
2 sentence 1 = "Sentence 1 is not true"
(ONLY an ASSUMPTION)
3 (as before)
4 (as before)
Now we must conlude that:
5 It is NOT TRUE that Sentence 1 = "Sentence 1 is not true"

And we have proven that sentence 2 is LOGICALLY FALSE!
Question: Is not sentence 2 meaningless if there IS no sentence 1 ?
Answer : Fill in any sentence BUT sentence 1.
Reinstatement makes sentence 2 both logically false and true by inspection of 1. Which means that we make a logical error in introducing sentence 1 as it originally read.
Note that the solution demands a very minor restriction on the language in use in comparison with other solutions.

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

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

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

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

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

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

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

By conclusion 5 Paulus is shown to be a liar but he is not paradoxical since he is
using a Liar Identity instead of a Liar Sentence in his statement. (see Russells paradox)

(Conjecture) A real paradox has the following Logical Form:
1. Liar Sentence
2. Liar Identity

Put together the paradox is inevitable, but alone the LI can be restricted:
Mathematicians was quick to define the Russell Set to be no set but a Class!

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

Supposing there is such an x then we have:
1. x = "xZ" (assumption)

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

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

If, for any value of Z, the right side of 3 is not true,then the left side is not true! Which means that for some values of Z
x="xZ" is not true and x is not a self referential sentence!

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

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

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

Post Scriptum:
An Induction problem: Today and yesterday no visitor found any mistake done in my argument,what then will tomorrow bring?

 

[HallsofIvy]

Yes, what you are saying is true but it has been well known for years.
The only difference is that instead of asserting that "sentence 1" is a sentence and then conclude that "sentence 1 is not a sentence, it would be clearer if you used the word "proposition". In Logic, a proposition is a statement that is either true of false. "How are you?" is an example of a sentence that is not a proposition. Logic only deals with propositions.

 

[Stephen Tashi]

Let us check the derivation of the liar paradox.
In his letter to Titus apostle Paulus states something like the sentence 1 below.

It's amusing to note that some authorities (e.g. page 93 of https://www.amazon.com/dp/0062012614/?tag=pfamazon01-20 ) think that Paul's letter to Titus is a forgery, not actually written by the Apostol Paul.

 

[sigurdW]

In Logic, a proposition is a statement that is either true of false. "How are you?" is an example of a sentence that is not a proposition. Logic only deals with propositions.

How about this:
1 You publish in here a proper derivation of the Liar Paradox...

2 And i will (after successful inspection) show where and how the logical error is made...Ok?

 

[sigurdW]

It's amusing to note that some authorities (e.g. page 93 of https://www.amazon.com/dp/0062012614/?tag=pfamazon01-20 ) think that Paul's letter to Titus is a forgery, not actually written by the Apostol Paul.

Thats news!
But what is in my mind in the matter is not religion, its for instance set theory in the hands of Bertrand Russell.

In short: There are TWO ingredients in a REAL paradox:
1 A Liar Sentence
2 A Liar identity

Liar sentences are not easy to miss and litterature doesn't show many except in the Liar.
Liar identities are in abundance and (if alone) give rise to a milder form of paradox ...PseudoParadoxes?... easier to handle than Real ones. Note how things go back to normal when you define the Russel SET to be a CLASS.

 

[sigurdW]

Yes, what you are saying is true but it has been well known for years.
The only difference is that instead of asserting that "sentence 1" is a sentence and then conclude that "sentence 1 is not a sentence, it would be clearer if you used the word "proposition". In Logic, a proposition is a statement that is either true of false. "How are you?" is an example of a sentence that is not a proposition. Logic only deals with propositions.

1 I am not asserting that the term "Sentence 1" is a sentence
2 A quote sign is missing: that "sentence 1 is not a sentence
3 as for the rest of your statement: nothing is news
4 No! What I stated is not known before: Back your claim up!