How does Aristotle relate virtue to happiness in Nicomachean Ethics?  

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In the Nicomachean Ethics, Aristotle claims that the achievement of happiness is a universal ambition and is the highest good because it is an end in itself—not a means to some other end. He then says that happiness is dependent on virtue.

The Greek word Aristotle uses is arete, and it could be translated as "excellence." Aristotle's definition of arete is somewhat vague. He claims it is a disposition not an activity. The virtuous person takes pleasure in doing virtuous acts, hence the link between virtue and happiness. There is no value, however (and presumably no happiness either), in performing virtuous acts if one is not essentially virtuous.

Virtue, for Aristotle, means following the golden mean. Extremes of all kinds are to be condemned (such as rashness and cowardice, parsimony and excessive waste, etc.). The most important virtue of all is justice.

The moral virtues, however, must be contemplated by the intellectual virtues: reasoning, intuition, wisdom, and knowledge. Without these, it is impossible to choose the most virtuous course of action.

While Aristotle spends much more time in the Nicomachean Ethics discussing virtue (and other subjects, such as friendship) than he allots to happiness, he still claims happiness as the highest of all goods and virtue as a means to reach it.

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