Aristotle saw tragedy as centering around a tragic hero, a basically noble person (well, he said "man") with a tragic flaw--hamartia. This flaw ususally took the form of hubris--excessive pride, so excessive that the person with it considered himself equal to the gods and thus, with no sense of his own ability to make mistakes, made some really terrible ones. Aristotle's tragedy involved a change (reversal) of fortune, which could go from bad to good just as well as from good to bad. He did, however, consider to good-to-bad reversal, with an unhappy ending, more artistic. But that unhappy ending did not require the hero to die.
Shakespeare incorporated many of Aristotle's ideas into his concept of tragedy, but he was also influenced by the medieval concept, according to which tragedy simply involved a fall from a high and fortunate position. Medieval tragedy was narrative, not dramatic, and usually consisted of a series of tales about people who suffered downfalls, often undeservedly. (Chaucer's Monk's Tale is a good example.) Shakespeare incorporated the idea of the necessary unhappy ending into all his true tragedies.
I would have to say that Shakespeare's tragedies show more influence of Aristotle than of the medieval idea, but they have the characteristics of both where the two overlap. His reversals go from good to bad, with the protagonists in happy circumstances at the beginning and successful up until the climax, when their fortunes begin to decline. His heroes always die, and in some way they always bring their downfalls upon themselves.
Aristotle would probably not have approved of some of the features of Shakespeare's tragedy, such as his subplots. Aristotle believed in unity of plot, and the plots of Greek tragedies are usually much more streamlined than those of Shakespeare. (However, although it happens that most Greek tragedies take place in the course of one day, with no change of scene, Aristotle actually said nothing about unity of time...
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