Pinker of Harvard says this about starting a sentence with a conjunction:
“You can’t begin a sentence with a conjunction.”
Teachers instruct young students that it is incorrect to begin a sentence with a conjunction (and, because, but, or, so, also) because it helps keep them from writing in fragments, Pinker writes, but it’s a lie that adults don’t need to follow.
Avoid writing an ugly “megasentence” full of connected independent clauses, and feel free to start a sentence with a conjunction.
starting a sentence with a conjunction
What the good professor does not say, and I will, is that in an utterance such as 'But she was so tired, she did not finish painting.', the but is presuppositional.
1) She painted like a fiend all day. **But she was so tired, she did not finish painting.** [see refining below. The but clause refines the idea of why she did not finish.]
2) She painted like a fiend all day but she was so tired she did not finish painting.**
3) She painted all day like a fiend. But she was so tired [that] she didn't finish the painting. without a comma.
The but in 1) can be called a "presupposition trigger".
A presupposition trigger is a construction or item that signals the existence of a presupposition in an utterance. (I say: also in a previous utterance)
In the OP's example the presupposition precedes the utterance. And as a coordinating conjunction what it coordinates is not given in the sentence, it precedes it.
So, I think I would argue here that 1) is really a variation of 2), and I'd call 1) a defective independent clause in its materialisation in 1) and a normal independent clause in 2) and 3).
- But she was so tired she didn't finish the painting. without a comma.
It is "defective" because most initial-position buts are not dependent clause-clause like. They are fully independent clauses. If you look at the paper I quote below, I think that is a fair statement.
Putting but in the initial position is called initial-position but (SIB). And one linguist calls this cancellation or refining (depending on the clause). The quote below is from an academic paper about academic writing but its descriptions of SIBs can be applied to non-academic contexts such as this one.
[...] then SIB facilitates argument development by its canceling or refining the previous argument. It is beyond the scope of this paper to explore the many ways in which cancellation operates in terms of argument development, but I give examples of some of the main ways [it does].
- After a few days Mary regains her ability to talk. But her speech has lost the usual inflection and tone, making a machine-like and dead impression. (Medicine,
Health Care and Philosophy) [...]
cancellation and refining
(Please note: all bolding is mine)