According to Arlov Grammatically
Many instructors urge their students not to start their sentences with the word “but”, or any other coordination conjunction for that matter. This can be disheartening for students because we commonly speak with our sentences beginning with a conjunction. However, writing is not like speaking. There are two schools of thought on beginning a sentence with the word, “but”. According to Arlov, “Grammatically, it is correct to start a sentence with but or any other FANBOYS conjunction.
However, your instructors may discourage the practice for two good reasons” (343). She goes on to list the reasons.
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One good reason not to begin a sentence with the word but is because it is informal and casual. Academic writing is not usually casual or informal, and using “but” at the beginning of a sentence may make a more formal paper appear less than well thought out.
While it is OK for some sorts of writing such as narrative or dialog, a professional paper is seldom written with coordination conjunctions at the beginning of sentences. Also, using words like “but” is addictive! It is easy to use and overuse. Arlov advises using conjunctions to begin sentences sparingly so as not to make it a hard habit to break. The other side of the issue says using but is just like using any other word to begin a sentence, and therefore, there is nothing wrong with it.