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In other cases the rules of an institution are not explicitly formalized.
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A statement about the completeness of a relation can also explicitly be formalized as a sentence of first-order logic.
Frege formalized his system more explicitly, but his system was not precisely a set theory in the modern sense: the most that can be said is that his system is inconsistent, for basically the reason given here, and a full account of the differences between Frege's system and our "naive set theory" is beside the point (though historically certainly interesting).
The metadata in XML files can be formalized as well-defined common data elements whose definition and usage semantics are explicitly described in an accessible, unique document.
A generative grammar, in the sense in which Noam Chomsky used the term, is a rule system formalized with mathematical precision that generates, without need of any information that is not represented explicitly in the system, the grammatical sentences of the language that it describes, or characterizes, and assigns to each sentence a structural description, or grammatical analysis.
His treaty was never formalized.
She formalized her marriage as an "arrangement".
"It formalized a hobby," he said.
He formalized the theory of matrices.
It formalized the approval in February.
"We've got everything formalized now".
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com