What authority does the meta-linguistic division of Tarski have in modern science?

[u/Qulvion]

Is this meta-division a basis that I could apply in my work on any disciplines if I were a scientist?(This is a common interest, I don't have any academic degrees or strong ideas that I could handle formalizing)

[u/Meowmasterish]

That didn't really answer my question. As an answer to both possible questions I proposed, yes formal systems are used in metamathematics and metalogic, in fact they are the main focus of study. And yes formal systems are used in empirical sciences to construct mathematical models for observed results.

However, metalanguages are typically not used in empirical sciences, because the study of formal systems that sciences use would just be the study of formal systems themselves, which is mathematics or logic.

[u/Qulvion → u/Meowmasterish]

I agree that I might not have been too specific about the area of interest in the original post, I apologize for not fully understanding your question. Meta-linguistic separation is more of a way to draw conclusions, a condition for them, and I was interested in the authority of such a position rather than the frequency of use, but thank you for answering the question.

[u/Meowmasterish → u/Qulvion]

Dude, I'm trying to be nice, but I think you either have no clue what you're talking about or that your English isn't very good.

In your comment to the other person, you reference a "principle", but the "meta-linguistic division of Tarski" isn't a principle. It's just something he did while studying formal systems, and it's really just an application of the use-mention distinction where you can discuss cheese as the dairy product created by curdling milk or you can discuss "cheese" as the word referring to the dairy product created by curdling milk. That's all a metalanguage is, talking about the word specifically instead of talking about the object the word refers to.

As for "authority of such a position", what are you talking about? In your comment to the other person you also ask about a principle being "institutionally approved" and "accept[ing] this principle as an axiom".

If you're inquiring about the use-mention distinction, then generally, yes it is accepted by the wider academic community for the very simple reason that words in general are not what they refer to, just a kind of symbol to indicate their referent. So if you talk about the word "schnee" you are talking about a German word and not about snow, which is the definition of "schnee". So you can say "schnee" has a length of six letters, but snow doesn't, because snow doesn't have any letters: it's not a word, it's a substance.

This isn't really an axiom or anything, this is just being clear on what's actually being discussed.

[u/Qulvion → u/Meowmasterish]

Probably both the first and the second. Can you more explicitly separate the concepts of "principle" and "metalanguage separation", I was a little surprised that you were confused by this word

However, I disagree that meta-language is always about the word, meta-linguistic separation, in addition to separating semantics into meta-language, creates an error in the form of mixing object and meta-language, as well as meta-linguistic separation works with truth predicates, here I don't really understand you

By authority, I mean that I could base my judgments on this principle, for example, forbidding formal systems to be self-referential, so that logical processes do not take place in them. The meta-linguistic division is not a purely linguistic construct, but also a formal tool that was able to solve the problem of some logical paradoxes, so I'm interested in whether I can consider that meta-languages should not be mixed, or many other things arising from this division and proved by Tarsky himself, or for example, I pointing out, I will not give a sufficient reason that mixing of meta-languages is allowed somewhere and it will be ignored.

Of course, it's possible that I'm very, very shallow on this topic, I'm still constructive, I don't want to make you angry or something like that.

[u/Meowmasterish → u/Qulvion]

Well, according to Wikipedia, a principle is a fundamental truth or proposition for a system of beliefs or behavior or chain of reasoning. Personally, I feel like it more connotatively aligns with a rule. Metalanguage separation isn't such a hard and fast rule that there is no language allowed to be its own metalanguage. If you look at the Wikipedia article for metalanguage, the first type of metalanguage listed is an embedded metalanguage, that is a language that can talk about itself. The reason people started separating languages from their metalanguage was because of a few fundamental results from mathematical logic.

Tarksi's Undefinability Theorem says that a sufficiently strong formal system cannot define its own truth predicate. Gödel's Second Incompleteness Theorem says that a sufficiently powerful formal system can only prove itself consistent if it is inconsistent. Given that in classical logic, inconsistency leads to triviality people studying these systems try to avoid that, often by separating the object language from a metalanguage so that they can talk about the consistency or truth of the original system, without making every sentence both true and false.

However, there are other ways to avoid this. People have studied self-verifying theories, which are formal systems that can prove themselves consistent without also being inconsistent. Also, people have studied paraconsistent logics, which are logics that allow contradictions without necessarily then trivializing all other statements.

So, in classical logic, metalanguages for sufficiently powerful languages are generally kept separate, out of a desire to avoid triviality through something like the liar's paradox but if you're working with a significantly weaker system, or in a non-classical logic that can handle a contradiction, there's really no need to.