Parmenides inspired many philosophers to follow in his footsteps. The movement he founded is called the school of Elea, and its members are referred to as the Eleatics. The school of Elea was the first movement to treat pure reason as the sole criterion of truth. Logical consistency and internal theoretic coherence, rather than any sort of observational evidence, guided their entire search for knowledge. The main Eleatic positions were inherited from Parmenides: (1) there is no genesis or corruption; (2) there is no plurality out of unity; (3) there is no change; (4) it is impossible to speak or think of non-being.
Zeno of Elea was Parmenides' most eminent student and was also probably his lover. He was working at roughly the same time as Anaxagoras and Empedocles, and devoted his career to devising arguments in defense of the doctrine of the Parmenidean Real. In his famous paradoxes he attempted to shows that pluralism (i.e. the idea that there really is a plurality of existing things) runs into even greater absurdities than Parmenides' doctrine. His arguments use the method of reductio ad absurdum, in which he begins with the premise he wants to deny, and then shows that this premise leads to a logical contradiction. Zeno did not view these arguments as paradoxes, since he believed that the premises he was trying to undermine (for instance, the existence of motion) were false. Since we today believe that these premises are true, (i.e. we do believe that there is motion in the world, and we do believe that there is a plurality of existing things) we find his brilliant puzzles slightly disturbing.
Melissus of Samos was the last of the famous Eleatics, writing around 440 B.C. He argued for Parmenides' claims in his own original way, drawing on the distinction between "is" and "seems" and the metaphysical consequences of the former. If something "is" X, he claimed, then it must be X essentially, and so it can never not be X. So, for instance, if something is hot, and does not just seem hot, then it can never stop being hot. Since nothing retains properties indefinitely and through all circumstances, he argues, nothing really is, except the Parmenidean Real.
The line "is this a dagger which I see before me" is from Macbeth, not Hamlet. C'mon, Sparknotes! I expect better from you.
Is it possible that Parmenides was referring to the object orientation of our thoughts with his famous saying that "what is is and what is not is not"? Consider that when separating an object from its background, we can conceive of the object as something but can not conceive the background as a thing. The object is "what is", while the background is "what is not". This interpretation fits well with several ideas of the time, for example that opposites had a special position in our thought, that the universe is one (Zeno's paradoxes
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