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Ethics Mill's - Utilitarianism

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I'm reading John Stuart Mill (Utilitarianism) second edition.

Is Mill's solution to the problem posed by the desire for virtue for its own sake successful? Why or Why not?

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This solution explores whether Mill's solution to the problem posed by the desire for virtue for its own sake is successful, including why or why not.

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RESPONSE:

1. Is Mill's solution to the problem posed by the desire for virtue for its own sake successful? Why or Why not pp 36-38.

It seems to address the problem posed by the desire for virtue for its own sake, but was it successful OR does it lack logic? For example, Mill argues that,

(a) an utilitarian claims that happiness is the one and only thing desirable in itself (IV 2).
(b) that the only proof of desirability is desire and proceeds to argue that happiness is the one and only thing desired.
(c) that a person does desire his own happiness for its own sake and that, therefore, happiness as such is desired by and desirable for ...

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