We need to number pages of philosophical books.

-@TheMadFool

Makes sense.

I do believe that there are another player in this connection. I do agree that Mathematics is a science used to deduce conclusions and axioms which does not "need' philosophy. However, my objection is that philosophy also, is a discipline that does not necessarily need mathematics. I think that both mathematics and philosophy is a discipline in their own right.
The connection between them lies on third discipline that uses devices from both mathematics and philosophy, Bertrand Russell coined it as "Mathematical Philosophy" in his book 1919 book "Introduction to Mathematical Philosophy". Russell distinguishes "Mathematics", or "ordinary mathematics", from "Mathematical Philosophy". He said that both are dealing with the same objects but with a different mindset. (The details could be read from the book)
I also agree that philosophy borrows devices from mathematics, but the claim that philosophy necessarily needs mathematics is still broad, or it is maybe that the case it is not true at all. I also suggest that we should inquire first the nature of mathematics before proceeding to the claim that philosophy necessarily need mathematics through reason/reasoning.

Apparently, self-reference is inevitable, meaning, that there is nothing wrong with it. (proven as the diagonal lemma or fixed-point lemma)

It implies that there something wrong on how we do mathematics.

'This sentence' is a phrase (not a proposition) that refers to the sentence 'This sentence is false'. Thus, we would get a sentences like ''This sentence is false' is false' and so on. Let us take not that we are not assigning truth to phrase 'This sentence' but to the sentence it refers to: the sentence 'this sentence is not true'.

the problem here is that it is agreeable the every instance of usage of truth takes the form 'x is true', where 'x' is any truth-bearer such as sentence, proposition etc. Thus, 'x is true' somehow serves as a basic definition of truth. However, as we can see above, there is an instance which leads us to a contradiction.

Saul Kripke (1975) suggested a solution to the problem by introducing a third-value "undefined", still, it is still problematic (see "strengthened liar" or "revenge of the liar")

Alfred Tarski, on the other hand, worked on the assumption that self-reference (proved as the diagonal lemma) is inevitable and that truth is binary. He introduced the hierarchy of languages that is in order for us to talk about a language we should another language (a meta-language). The problem is Tarski aimed at introducing truth to formalized language, and claimed that we cannot introduce truth to ordinary language without resulting into a contradiction. (see 1944 Tarski)

I do believe that by saying "I am lying", what you really mean is "What I have just said is a lie". There is a context in that certain conversation that makes "I am lying" meaningful.

They said "no" :)

From what I know and read, a proposition is a statement or sentence that expresses an agreement or disagreement of its facts. Or to simply put it, a sentence that contains a truth value.

I think that the sentence already contains a truth value which is "not true". What I lack is a way to restate that sentence as its subject alone is problematic. This is still a possibility for me, there must be a way.

Maybe there some aspects of a proposition that I did miss, would you kindly tell me what those are?

I do agree. There is a complication alone on its subject. By looking at the earlier form of that paradox " I am lying". It refers to itself, thus becoming problematic.

One way of elucidating this type of self-referring sentence such as "What is this?" is by employing the context principle. On "I am lying", this type of sentences should not refer to itself for it to make sense.

"I am a female" > " I am lying" > " I am a male"
*a picture of a cat > "What is this?" > "A cat"

Thus, in statements such as "This sentence" it should refer to others and not itself. Does that satisfy it at all or not?

Attempting to clarify the ideas presented above:

I see that the sentence ""This sentence is not true" is true" as problematic as it contains 2 truth values.
*It is assumed - It is wrong to put a truth predicate on a sentence that already have a truth value in it.

The sentence "This sentence is not true" as problematic as it refer to itself, or to point out the core problem: the statement "This sentence".
- "This sentence is real" - "This sentence is square" - "This sentence is x"

*As stated above, if we could find a way to restate it to another way then the problem of contradiction would vanish. But there is still this problem of self-reference.

Also, even if I am right that "is false" is a truth-predicate and "This sentence" is stated rather differently. There is still this problem of self reference and lack of property.

On that quoted quote, It seems that I have not clarified that I am talking to statements in general and I do apologize.

On the other hand, it is wrong to put a truth value, or even assume, on a statement that already contain a truth-predicate. (I have stated ideas above that would bring light to what I am saying)

It is very difficult for me to reject the law of non-contradiction as it appeals to my intuition. There must be another way. If it is the case that dialetheism is the way to approach this kinds of sentences, I ll see into it that it would be etched into my mind the right way.

I do hope that you (refer to all) will be patient and understanding as I am a person who still lacks.

Thank you for clarifying my error. It seems that I am under illusion that "This sentence is not true" is a proposition as it already contains a truth-predicate (Its seems a bit weird and wrong. It is also weird to me, but I am trying to come of a better explanation)

The sentence contains a subj and pred. One thing that bothers me is the pred. "is not true". That must be a truth-predicate as it function as one.

Maybe there is a way to re-state this sentence...

Is the sentence ""This sentence is false" is true" could be restated as "This false sentence is true"?

I am quite new to this paradox and as I see it, It is well stated proposition that is clear and so on. It is maybe the case that this proposition or rather, this statement lacks the ground/s for it to be either true or false. But in the case of most or all of propositions, I think, that their mechanisms are very similar to models of mathematics or so, they are the same sort. They are models of reality as we imagine it (TLP 4.01), and so does not necessarily rely on fact/s.

It is maybe the case that statements should have grounds in order to assert whether that statement is true or false. But also, it is not wrong to assume its truth and falsity.

From what I see, as I have stated above, is it is a proposition thus it can be either true or false.It is also the case that it is truth-apt. But by looking at its components:

"This sentence is not true" is *true *assumed

Could we really put two truth-predicate in one proposition?

As I see it: "This sentence" is the subj, "is not true" is the predicate, and " is true" is the truth predicate.

Is it not that the predicate of the proposition counts as a truth predicate? If the predicate is not a truth predicate, then why do it function as a truth predicate?

Clarify me if I am wrong on some of my points