Yes, when writing in APA style there isn't much restriction on sentence formatting such as starting one with "And" or "But." Restrictions typically apply more to formatting the title page, reference page, and the heading at the top of the page. However, for scholarly writing, it is typically avoided to start a sentence with "And." Starting with "but" has less taboo surrounding it. If you would like to have your paper reviewed for spelling, grammar, and/or rhetoric, contact the Academic Support Center.
