What are the different beliefs of Socrates and the Sophists?

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The difference between Socrates and the Sophists is that Socrates believed that universal standards existed to guide individuals in matters such as justice and beauty, while the Sophists believed that it was powerful people's job to determine these points of knowledge themselves.

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The word "sophist" does not denote any particular beliefs. Though English words such as "sophistry" and "sophism" have become associated with dishonesty and specious reasoning, they originally meant no more than "someone who is paid to teach." The closest equivalent in modern English, therefore, is probably "professor" or "teacher"—and, just as we do not normally think that it is disreputable or disgraceful for a professor to accept payment for teaching students, there was originally no taboo in ancient Athens about the sophists accepting payment for teaching.

The fact that the sophists are often now disregarded as disreputable people is partly due to comic playwrights like Aristophanes, who lambasted the sophists and complained that they became rich by teaching absurd and useless subjects or that they taught immoral tricks to their students (as Socrates also complains in Plato's Gorgias). Aristophanes included Socrates among these sophists, but both Plato and Xenophon differentiate him sharply from the sophists—firstly on the basis that Socrates despised money and never took payment for teaching. Socrates himself criticized the sophists for selling their knowledge, which means that he too is responsible for the low esteem in which they are now held.

Socrates also had a fundamentally nihilistic approach to knowledge which would have prevented him from being a sophist even if he had wanted to accept payment. Since he believes that the wisest man in the world is the one who is aware that he knows nothing, most of his time is spent in dismantling the illusion of knowledge. It is doubtful whether many students would have paid to be shown that they knew nothing.

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Clarification: Socrates believed and made attempts to prove the existence of points of reference (universal standards) that should guide the individual with regards to matters such as justice and beauty.

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The Sophists believed that knowledge was power and acknowledged that those with the most power became the rulers of the world. According to them, education was a thus a source of knowledge that would later translate to power. Socrates believed that the acquisition of knowledge was important because it gave the individual an opportunity to convince others, bringing them closer to the truth.

The Sophists believed that matters of reality, morality and aesthetics were determined by the powerful. They believed that it was powerful, who set the precedence and determined right from wrong, good from bad. This was because no points of reference existed (universal standards), making it their responsibility to set these points of reference. Socrates believed and made attempts to prove that points of reference (universal standards) existed that should guide the individual with regards to matters such as justice and beauty.

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The primary difference between Socrates and the Sophists seems to lie in a disagreement on whether or not a truth (or knowledge) might be absolute.

Socrates (and Plato) pursued a line of rational thought intended to discover or determine real philosophical absolutes.

This belief in absolute truth(s) was epitomized in Plato's theory of the forms (which concerns the ideal/perfection of all things, including ideas). Through rational, logical pursuit a person might attain some insight as to the ideal, "true" form of any idea and thus be said to possess the truth. 

This is, of course, a simplification of Platonic ideals and of the Socratic method, but it serves as a counterpoint to the Sophists.

The Sophists argued that truth and morality were relative. They "claimed that the value of actions varied according to circumstances, that knowledge was necessarily imperfect, and that truth was relative" (eNotes). 

The political, moral and ethical ramifications of these distinctly opposed schools of thought should not be overlooked.  

"The Sophists’ ethical relativism was sharply attacked by a new philosophical movement, led by Socrates, reaffirming absolute values" (eNotes)

Using Socrates as a vehicle in his writing, Plato developed the notion of the philosopher-king, describing an entire mode of government and ethics based on his idea of who might be best equipped to understand "right and wrong" and to rule correctly.

While Socrates is famously hailed as saying that wisdom is essentially an awareness of how little one knows, his position vis-a-vis absolute truth suggests that he viewed ultimate wisdom as an attainment of an ideal knowledge. The Sophists, for their part, argued against the existence (even potentially) of such an ideal form of knowledge.

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In many ways, Socrates resembled the Sophists. The evidence from Aristophanes' Clouds suggests the Socrates was considered a sophist by many people in his lifetime. In Apology, as well, Socrates admits that he is often confused with the Sophists and tries to distinguish himself from them in two ways, first by pointing out that they charge fees for teaching and he doesn't, and second that they teach public speaking and he doesn't.

It's important to note that "sophist" does not represent an ideology, but rather sophists were simply 1) distinguished public speakers and 2) taught at a tertiary level for money. That does not imply any common beliefs beyond a commitment to education.

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