The two precise methods of philosophy recognized in philosophical inquiry are indeed protreptic and elenctic, also referred to as Socratic.
Protreptic involves engaging and encouraging individuals to reflect critically on their beliefs and assumptions. This method is about persuasion and often aims to lead students toward deeper understanding or even a change in viewpoint. It is marked by an approach that stimulates philosophical inquiry through questioning deeper principles or motivations underlying one's beliefs and way of life.
Elenctic, or the Socratic method, is characterized by a dialectical approach where the philosopher engages in dialogue, asking probing questions that reveal contradictions in the interlocutor’s beliefs. This method seeks to stimulate critical thinking, clarifying concepts, and drawing out underlying assumptions, ultimately aiming for clearer understanding or recognition of ignorance.
Together, these methods highlight the profound engagement with ideas and dialogue essential to philosophical practice, emphasizing the importance of questioning and reflection in the pursuit of wisdom and knowledge. This aligns well with the Socratic tradition, which prioritizes inquiry as a pathway to better understanding. The other methods listed, such as analytic and synthetic or deductive and inductive, refer more to logical and reasoning processes rather than to specific philosophical methodologies focused on interpersonal engagement and examination of belief systems