Can formal grammar (language) serve as the model for some logic and express the semantics of this logic? Different mathematical structures (e.g. relations etc.) may serve as the models for some logics and express the semantics of this logic. Can the formal grammar serve as the model for some logic? E.g. is it possible to express both the syntax of logic with one grammar and semantics for the same logic with other grammar?
Such semantic could clarify and formalize the translation among logics (with the definition of the preservation or transformation of the meanint).
This question arouses during acquaintance with https://www.grammaticalframework.org/ which is purely syntact translation among grammars and which could serve also as the engine for translation between logics.
Or maybe even the question can be - can the semantics of some logic be expressed by other logic? There is long standing tradition of Montagovian semantics of natural language. Abstract Categorial Grammar approach defines the formal grammar for the natural language (it is quite involved, of course) and it defines the semantics of the natural language via some more or less sophisticated mathematical logic. E.g. "boss thinks" became the logical expression "think(boss)", where "boss" is the instance of type "e", but "think" is the instance of type "e->t"
 A: 
Can the formal grammar serve as the model for some logic? E.g. is it possible to express both the syntax of logic with one grammar and semantics for the same logic with other grammar?

If I understand you correctly, then yes, a formal grammar can serve as the model for some logic. Usually a logic's syntax is defined up front using a formal grammar in BNF form, such as this pseudo formal grammar for propositional logic:
\begin{align}
Sentence &\to AtomicSentence \mid ComplexSentence\\
AtomicSentence &\to True \mid False \mid P \mid Q \mid R \mid ...\\
ComplexSentence &\to (Sentence) \mid Sentence\ Connective\ Sentence \mid \neg Sentence\\
Connective &\to\ \land\ \mid\ \lor\ \mid\ \Rightarrow\ \mid\ ⇔
\end{align}
If you want to specify the semantics of the logic, you can try using Operational Semantics or Matching Logic.
A: A famous result in automata theory says that every regular language could be decribed by a monadic second order formula with suitable predicated, and vice versa every such formula defines a regular language. Hence this logic (the semantic) is precisely captured by the regular languages. I do not know if this is what you are looking for, but here is an accessable reference by Straubing/Weil if you want to learn more about this.
