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A grammar can never be a complete description of an actual language but must always be a reduction of it.

Probably. IIRC one of the objectives of Linguistics was to form a theory that would account for all, and only, grammatical utterances. I don't think it's been done yet.

Since grammars do not attempt to give an exhaustive analysis of lexis, phonology, ideolects. register, style and a long etc that is obvious.

If by "a grammar" one means "a book learners can use to study an L2", there's no way it can be exhaustive. What's "an actual language" anyway? A variety of English?

Since grammars do not attempt to give an exhaustive analysis of lexis, phonology, ideolects. register, style and a long etc that is obvious.

To whom? That a grammar is all one needs in order to know the how the language works seems to be a view held by many, students, teachers and other such commentators, in classrooms around the world and on fora all over the Internet.

That a grammar is all one needs in order to know the how the laguage works seems to be a view held by many, students, teachers and other such commentators, in classrooms around the world and on fora all over the Internet

A grammar is all one needs to get basic principles in a language and communicate at a rudimentary level. When I learn a new language, grammar is the first thing I seek to make the fastest progress as possible at first. After that, I seek real people who speak the language. Grammar can't replace experience and conversation, through which tone of voice, inflection, accent, thinking patterns, etc., come into play.

A grammar is all one needs to get basic principles in a language and communicate at a rudimentary level.

Would that be equal to saying "A grammar can only give one the basic principles that allow one to communicate at a rudimentary level", IYO?

no. It also depends on if you mean a grammar for a second-language learner or for a native speaker. A good textbook for language learners though probably does a good job at balancing basic rules with real-life exercises or conversations: it doesn't just cover grammar. This is the kind of textbook I was thinking of. If it did just cover dry grammar with no example sentences, it might not even give you enough to communicate with others. I think you really need to see the language in context, either by reading or listening.

Hmm. I'm interested to know what differences in meaning you see between these two statements:

A grammar is all one needs to get basic principles in a language and communicate at a rudimentary level.

A grammar can only give one the basic principles that allow one to communicate at a rudimentary level.

Well, first, I meant a good textbook instead of grammar. Does that change things? I think a quick look through a textbook, or pure grammar, is all you need to get by in a language (for the first time), probably having a very heavy accent and your own thinking patterns inundating the foreign language. Of course to perfect the language and see it in it's authenticity, you need to see real text in the language, like advertisements, novels, newspapers — and talk to real people.
For the second statement, a good textbook need not limit itself just to grammar principles. It could probably provide real advertisements or reading contexts from a real newspaper, etc. One may not be a perfect speaker of a language regarding accent or other things, but if you go through a textbook series and spend a lot of time with a textbook, you certainly can get more than just rudimentary communication.
To sum up, I think for rudimentary communication, a textbook is all that's necessary, but it can also provide more than just that, even though it can't perfect your ability in the language.