|School of law
Fajia (Chinese: 法家; pinyin: fǎ jiā), often translated as Legalism, is one of Sima Qian's six classical schools of thought in Chinese philosophy, forming a school of thought in the Han dynasty. It represents several branches of thought from early thinkers such as Guan Zhong, Li Kui, Shen Buhai, Shang Yang, and Han Fei, whose reforms and ideas contributed greatly to the establishment of the bureaucratic Chinese empire. The Qin to Tang were more characterized by the tradition.
Though the origins of the Chinese administrative system cannot be traced to any one person, prime minister Shen Buhai may have had more influence than any other in the construction of the merit system, and could be considered its founder. His philosophical successor Han Fei, regarded as their finest writer, wrote the most acclaimed of their texts, the Han Feizi. Sun Tzu's Art of War incorporates both a Daoistic philosophy of inaction and impartiality, and a Legalist system of punishment and rewards, recalling Han Fei's use of the concepts of power (勢, shì) and technique (術, shù).
Concerned largely with administrative and sociopolitical innovation, Shang Yang was a leading reformer of his time. His numerous reforms transformed the peripheral Qin state into a militarily powerful and strongly centralized kingdom. Much of Legalism was the development of the ideas underlying his reforms, which led the Qin to ultimate conquest of the other states of China in 221 BCE. With an administrative influence for the Qin dynasty, succeeding emperors often followed the templates set by Han Fei, Shen Buhai and Shang Yang.
Although the old term Legalism has still seen some conventional usage in recent years, such as in Adventures in Chinese Realism, academia has avoided it for reasons which date back to Sinologist Herrlee G. Creel's early work on the subject. With Li Kui a predecessor of Shang Yang, Shang Yang is the most 'Legalist' of them. Shang Yang most commonly has fa (standards) as law, while Han Fei's predecessor Shen Buhai uses fa (standards) in the administration, which Creel translated as method.
Reward and punishment was a major emphasis for Shang Yang and Han Fei, and Han Fei can even be said to have more belief in the requirement of it's permanence. But he was only an advocate. Together with Shen Buhai and another of Han Fei's figures, the scholar Shen Dao, Han Fei was still primarily administrative. Han Fei and Shen Dao make some use of fa akin to law, and some use of reward and punishment, but generally use fa similarly to Shen Buhai: as an administrative technique. With a quotation from Han Fei as example:
An enlightened ruler employs fa to pick his men; he does not select them himself. He employs fa to weigh their merit; he does not fathom it himself. Thus ability cannot be obscured nor failure prettified. If those who are [falsely] glorified cannot advance, and likewise those who are maligned cannot be set back, then there will be clear distinctions between lord and subject, and order will be easily [attained]. Thus the ruler can only use fa.
Sima Qian himself seemed to be clearly aware that fa had senses which included both law and administrative method. Dividing up his forebears into different elements, Han Fei differentiates Shen Buhai's fa (standards) as administrative Technique; he had disorganized law. While Creel and some others have simply used Fajia, and some none at all, the Stanford Encyclopedia of Philosophy has introduced them as the fa tradition and fa thinkers.
The early reforms of Zichan, from the central Zheng state, concerned the Jin that law would destroy the social order and even the state. However, the Warring States period mainly characterized the Qin as barbarian; Xun Kuang knows about Shen Dao and Shen Buhai, but does not appear to know about Shang Yang, leaving reception of him for the period's late figure: Han Fei. But while the Qin did not abandon all of Shang Yang's reforms, they did abandon his heavy punishments and suppression of commerce before the founding of the Qin dynasty.
More like a house of thought, Sima Tan defined the school in terms of office divisions and responsibilities. He does not name anyone under the schools; the figures are named under it in the Book of Han, one hundred years later. Despite later associations, Sima Qian divided Shang Yang from Shen Buhai and Han Fei, taking them as more Daoistic thinkers, like Laozi and Zhuang Zhou. He may not have actually had them in mind for it. The Book of Han similarly lists the Guanzi as Daoist before it is later taken as Legalist.
Han Fei however presented Shang Yang and Shen Buhai as the opposite components of his own doctrine, and pseudo-Li Si, per Sima Qian, makes the same discussion, associating Shen Buhai only with administrative control of the bureaucracy, Shang Yang only with law, and Han Fei with both. Associations of Shen Buhai, Shang Yang and Han Fei become common in the early Han dynasty, and although the figures were influential, they were probably never an organized or self-aware movement in the sense of the Mohists and Confucians; they are essentially Han Fei's philosophical lineage. The Book of Han would list six men apart from Han Fei and his: Shang Yang, Shen Buhai, and Shen Dao, though it does not give them biographies.
Although reducing capital punishment, correlating facts and claims, the broader mileu of Shen Buhai's administrative method would be paired with Shang Yang, obscured as criminal law, and incorporated into the Han dynasty penal law offices. Han dynasty figures Liu Xiang and Liu Xin would assign the school a fictional origin in an ancient department of criminal justice, together with departments for the other schools, and the Book of Han would receive them as something more akin to Legalists. 
The scholar Shen Dao, discussed in chapter 40 of the Han Feizi, was taken as having a group by early scholarship, and is included in the 'typical canon' modernly as part of Han Fei's lineage. He was early remembered for his secondary subject of shi or "situational authority", of which he is spoken in Chapter 40 of the Han Feizi and incorporated into The Art of War. He only uses the term twice in the his fragments, and Xun Kuang calls him "beclouded with fa". Sima Qian lists him with other figures of the Jixia academy, like Xun Kuang and Mencius. Although evidentially known by some in his time, and included in the Book of Han, he has no record of any notable activity, and is only mentioned in the shiji in a stub with the claim that he had studied Sima Qian's own Huang-Lao ideology. Creel found no following for him comparable to Shang Yang or Shen Buhai in either the Warring States period or Han dynasty.
Han Fei's three predecessors are glossed in early scholarship under the elements of fa, shu, and shi. But shi is a minority, while Shu ("Technique") is Han Fei's distinctive for Shen Buhai's fa.