Each language has its own complex systems of word, phrase, and sentence construction, the guiding principles of which are often summarized in grammar descriptions for the consumption of linguists or language learners. However, manual creation of such descriptions is a fraught process, as creating descriptions which describe the language in "its own terms" without bias or error requires both a deep understanding of the language at hand and linguistics as a whole. We propose an automatic framework AutoLEX that aims to ease linguists' discovery and extraction of concise descriptions of linguistic phenomena. Specifically, we apply this framework to extract descriptions for three phenomena: morphological agreement, case marking, and word order, across several languages. We evaluate the descriptions with the help of language experts and propose a method for automated evaluation when human evaluation is infeasible.