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It is essential to establishing the balance between the deductive system and the semantics (see §5 below).
In Section 5, we turn to relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity.
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This establishes that the deductive system is rich enough to provide a deduction for every valid argument.
The deductive system for the new logic is formed by the rules of both logics.
Before turning to the deductive system and semantics, I mention a few features of the language, as developed so far.
As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics.
According to Lewis 1973, laws of nature are the true generalizations found in the deductive system that best describes our world (where "best" basically refers to the optimal trade-off between strength and simplicity; see the related entry for an introduction to, and further details on, this debate): laws of nature supervene on the four-dimensional arrangement of particulars and their properties.
In taking identity to be logical, we provide explicit treatment for it in the deductive system and in the model-theoretic semantics.
Early Chinese mathematics and science are likewise based on analogical reasoning and taxonomy, rather than on the model of the deductive system that dominates ancient Greek thought.
The relationships between a deductive system and a realistic ontology differ among the various cases that Russell lists as examples of logical constructions.
Stoic syllogistic has been formalized, and it has been shown that the Stoic deductive system shows strong similarities with relevance logical systems like those by McCall.
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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