This is another example of how to use the phrase: "Something is all but Something" ... I have seen some examples of using this phrase as "almost completely", so, that would mean the solution is very easy, but I have also heard this "but limits the totality of the all", so that would mean the solution is very hard ... I don't understand how to use that expression
As others have said, "all but X" means "nearly X" and can be used in phrases like "all but empty," "all but completed," etc.
I think the other answers are little hung up on the meaning of trivial. Many words have one or more general meanings when used in common speech but a specific meaning when used in a specific field or profession; this is an example of jargon.
Trivial is one such word. When used in the context of mathematics, it means
a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). [...] The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove.
In everyday use trivial means "having little importance or value." But as used here it means "requiring very little explanation; self-evident." Thus all but trivial means almost obvious, barely requiring effort to prove.
Collins says that
You use all but to say that something is almost the case.
- The concrete wall that used to divide this city has now all but gone.
- He has been all but forgotten.
So your sentence "The solution is all but trivial", means that the solution is almost/nearly trivial (shouldn't be hard to find).
More context would be needed for further explanations.
When you say that "but limits the totality of the all", you are speaking about a different use of the expression all but:
All but a particular person or thing means everyone or everything except that person or thing.
- The general was an unattractive man to all but his most ardent admirers.
- The plant will stand all but the worst winters out of doors.
So it is when all but is followed by a noun that it means except for and "but limits the totality of the all" as you say. In your sentence all but is used with an adjective, so it cannot limit totality.
The solution is all but trivial
would normally mean that the solution is so easy that it could almsot, but not quite, be considered trivial or self-evident. I could imagine this phrase in a math text, say, as a starting point before the text goes on to discuss a harder, but somehow related, problem.
I cannot imagine how this phrase or a similar phrase might be used together with "but limits the totality of the all" -- a phrase which seems rather obscure and of a different tone. But given an actual paragraph where both are used, I might better understand what has the OP confused.
The construction "all but" is very contextual, relying on the surrounding communication to be correctly interpreted. However, standing alone, the phrase "all but trivial" would generally (at least in American English) be taken as "it is anything but trivial" - meaning it is NOT trivial.
On the other hand, as an example, one could use "all but complete" to indicate either that completion is simple, or that completion is nearly impossible. It would depend on the context of the exchange. Context here could include such things as body language, or the general direction of the preceding conversation (or writing, for written communications).
This is not a construction I would recommend an English Language Learner using, due to the possibility of misunderstanding, and the ease of constructing alternatives. And, if one were to receive this in a communication, a request for clarification would not be out of order.
Meaning of “Solution is all but trivial”
The answer seems to be in the context of all but However I suggest in this case it is not. The fact we have **"All" before "But" changes the meaning, it Becomes the opposite of what we would normally expect the sentence to mean. It now has the meaning except.
“Solution is all except trivial”
but preposition, conjunction meaning ‘except’ Cambridge English Dictionary
But means ‘except’ when it is used after words such as all, everything/nothing, everyone/no one, everybody/nobody:
The fact we have **"All" before "But" sometimes does not change the meaning, it remains what we would normally expect the sentence to mean. It now has the meaning almost.
“Solution is almost trivial”
All but meaning ‘almost completely’ Cambridge English Dictionary I had all but finished the essay when the computer crashed and I lost it all.
all but: Collins Dictionary almost; nearly; example all but dead.
The answer lies in the meaning of Trivial, “Solution is almost having little value or importance", makes no sense. We have already establish the object has almost (little) value or importance by the use of the word trivial. Therefore it can be concluded that in this instance the meaning is
“Solution is all except trivial”
trivial: Cambridge English Dictionary adjective;having little value or importance: