Starting conjunctions and sentence fragments
We know that it is OK to start a sentence with a conjunction, but not OK to have incomplete sentences. This seems to me like a contradiction. How can a sentence that starts with a conjunction be a complete sentence? By its nature, a conjunction links to a previous sentence. Therefore, a sentence that starts with a conjunction does not form a complete thought, and is therefore incomplete.
Example guidelines:
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Start sentences with a conjunction (Grammarly):
There is nothing wrong with starting sentences with “and,” “but,” or other similar conjunctions. You may, however, encounter people who mistakenly believe that starting a sentence with a conjunction is an error, so consider your audience when deciding to structure your sentences this way.
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Avoid incomplete sentences (Grammarly):
No one escapes high school English without being penalized for writing the odd sentence fragment, but not everyone remembers what they are and how to fix them. Put simply, a sentence fragment is a clause that falls short of true sentencehood because it is missing one of three critical components: a subject, a verb, and a complete thought.
Fragments in speech vs Fragments in novels? does not quite answer the question, since it doesn't resolve why it is fine to start a sentence with a conjunction even if conjunctions create a fragment.
The missing part of the picture is the discourse within which a sentence stands.
Conjunctions are used in discourse to connect portions of sentences such as words or phrases, but they are also used to connect larger units of text such as sentences and paragraphs. And if you doubt that, this (current) sentence demonstrates such a use.
It can become unwieldy to push a conclusion, demonstration or highlight into the main sentence, as the paragraph above illustrates. However, placing the ideas into separate sentences might distance them in a way that the author finds unhelpful. At the same time, markers of conclusion such as thus, so, therefore, hence, etc can be too heavy - they place a greater emphasis on the trailing sentence than the author intends. Starting the trailing sentence with a conjunction such as and neatly solves both problems.
The warning about sentence fragments is a heuristic regarding clear communication - or more accurately, against unintelligible or unclear communication. For example, starting a discourse with "And now, ..." leaves the audience wondering what went before. You'd never cause such a problem if your sentences never start with conjunctions. But if the discourse has already covered related material, the sentence 'fragment' has a context within which it communicates clearly, and is therefore acceptable.
And that's the difference. :)