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One major difference between these two approaches is the agent's responsibility.
Positive consequentialism demands that we bring about good states of affairs, whereas negative consequentialism may only require that we avoid bad ones.
A more strenuous version of negative consequentialism may actually require active intervention, but only to prevent harm from being done.
An alternative theory ( using the example of negative utilitarianism ) is that some consider the reduction of suffering ( for the disadvantaged ) to be more valuable than increased pleasure ( for the affluent or luxurious ).

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