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"Knowledge" is the usual translation of the Greek word episteme. It is a mistake to think of this knowledge, however, as a body of information that can be communicated simply. Rather, episteme also includes the types of skill more frequently referred to by words like craft, profession, art, or science. The kind of knowledge being discussed by Socrates is, then, more like the knowledge of a language, or the ability to play music. It contains elements of theory that can be laid out by verbal rules, but it also contains a practical element.
We can uncover some sense of what is entailed in this by tracing the roots of the word "ethics," which is the central form of knowledge being argued over by Socrates and Protagoras. The Greek root of ethics, ethos, denotes a person's habits, or the inner character of their behavior. One's ethics are embodied in all of one's habits. To be truly known, virtue must infiltrate and shape all aspects of one's life. Once knowledge is understood in this way, Socrates's claim that it is impossible to commit an evil action in the knowledge that it is evil starts to make more sense. However, the question of how such knowledge can be learnt is not explicitly answered within the dialogue, although elements of the form of the Socratic dialogue provide clues.
The term "Sophist" comes from the Greek word sophistes, which means a wise person, or an expert. Largely thanks to the work of Plato and Aristotle, however, sophistry has now become synonymous with specious reasoning (arguments that appear to be logical but that are, in fact, false). As such, sophistry connotes an irresponsible attitude towards philosophy. Plato's attacks on sophistry are based on both these factors: he is critical of the Sophists for accepting payment for their services, and for teaching how to win arguments regardless of the truth of what is being argued.
To some degree like modern-day lawyers, sophists were considered to be tongues for hire, more interested in making money and impressing with their rhetorical pyrotechnics than in uncovering the truth of the matter at hand. More recently, the controversy between sophistry and the Platonic thinkers has been interpreted as playing out a conflict between democratic social movements and conservative aristocratic forces. In this interpretation, Plato is suspicious that sophistry teaches argumentative skills to anyone capable of paying, and therefore expands too greatly the circle of those who can think critically. However, his class-based interpretation of the Sophists in the Platonic dialogues remains controversial. In Protagoras, the Sophists are Hippias, Prodicus, and Protagoras himself.
Syllogisms are forms of argument in which conclusions inevitably follow from two or more premises, and in which no unnecessary premises are included. For example, the two statements "all humans are mammals" and "all mammals have scales," when taken together, must yield "all humans have scales." Whether the argument is true or not, the logic of the form guarantees that the conclusion follows from the premises. Because of this, syllogistic logic requires very close examination of premises prior to their being accepted.
Protagoras is centrally concerned with the concept of virtue: what is virtue, and how is it acquired? The Greek term arete is usually translated by "virtue," but it can also mean excellence, good disposition, or even courage. Thus, to understand "virtue" here as referring only to morality is a mistake. "Virtue" can also refer to the good qualities of things, objects and forms of behavior that are not immediately included in the notion of moral virtue. This is important to remember when considering the difference between Socrates's and Protagoras's ideas about how virtue can be taught. If virtue can be considered as a set of verbal doctrines, or moral rules, than it is easy to see how it could form the basis of a system of education. If, on the other hand, virtue cannot be so easily extracted from social forms and ways of living, but is nonetheless a single and unitary quality, it becomes far more difficult to see how it can be taught.
One way of understanding Protagoras is that it is a debate based on these two different concepts of virtue. By the end, Socrates has the upper hand—virtue is a form of knowledge, as he has argued—but what that knowledge (episteme) is and how it can be acquired are questions that remain to be investigated.