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Distinguishing the semantics of a language from its syntax means (at least) distinguishing the meaning of the expressions (what is being represented) from the grammatical structure and formation rules (the means of formal representation).

Distinguishing object and meta language means distinguishing a language that is being talked about (the object language) from the language being used to do so (the meta language).

Both distinctions are claimed to have emerged in the very late 1920's, in particular with Gödel's incompleteness papers and Tarski's undefinability theorem.

My question is:

Is it possible to have drawn one of these distinctions with acuity, whilst ignoring the other, or (more likely) what aspect of the historical development led to their simultaneous recognition? In other words, what sorts of relations hold between the two distinctions that would compel an early researcher to recognize both, if he or she were to recognize one?

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  • $\begingroup$ Obviously, the distinction syntax/semantics was a "natural" one, already available with the study natural language. Not so for the distinction between object- and meta-language. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Dec 23, 2018 at 19:13
  • $\begingroup$ The term meta-mathematics was coined by Hilbert in the 20s, while the similar term meta-logic was introduced in 1930 by Lukasiewicz and Tarski : "In the course of the years 1920-30 investigations were carried out in Warsaw belonging to that part of metamathematics—or better metalogic—which has as its field of study the simplest deductive discipline, namely the sentential calculus." $\endgroup$
    – Mauro ALLEGRANZA
    Commented Dec 23, 2018 at 19:22

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Meta/object $\to$ syntax/semantics:

Once you have the distinction between a meta-language and a formal language, you certainly have the idea of a language. And the very idea of a language (as opposed to just a set of sequences) suggests having meaning assigned to the terms of the language.

Syntax/semantics $\not\to$ meta/object:

I believe you can have syntax and semantics without a meta-language as long as the language is rather basic. For instance, consider a language of Boolean algebras using just connectives like $\neg$ and $\wedge$. It seems that considering various models, i.e., various Boolean algebras, does not force you to think that your meta-language is also a mathematical object.

However, once a language $\mathcal L$ that purports to cover all mathematical reasoning, like a formalization of set theory in first-order logic, is considered, it seems natural when studying $\mathcal L$ mathematically to also considered whether you are using another mathematical formal language $\mathcal L'$ in doing so.

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  • $\begingroup$ What about the contrapositive of the first direction? Would failure to distinguish syntax/semantics make it impossible to draw a coherent meta/object language distinction? $\endgroup$
    – Mallik
    Commented Dec 5, 2018 at 22:57
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    $\begingroup$ @Mallik well, the contrapositive of any statement $\varphi$ is equivalent to $\varphi$, right? $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Dec 6, 2018 at 22:40

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