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Bentham's Utilitarianism is defined as pursuing "the greatest happiness of the greatest number", but it contains two optimization goals, "greatest happiness" and "greatest number" and in common cases, you cannot guarantee both are satisfied. The decision that causes the greatest happiness for some people maybe is not the one causing happiness for the greatest number of people, and the decision causing happiness for the greatest number of people maybe is not the one causing the greatest happiness among some people. Is there a math expression showing what exactly Bentham refers to?

It seems better defined as "the greatest summation of happiness". For the set of decisions $d_1, ..., d_n$, and for m people whose utility functions are U_1(), ..., U_m() respectively, choose the decision d that:

d = argmax_{d \in {d_1, ..., d_n}} \sum_{i=1}^m U_i(d)

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  • "The greatest happiness for the greatest number" was just a shortened motto, and it is a bad idea to take mottos as literally expressing what theories say. It is clear from Bentham's "felicific calculus" that what he meant is just the aggregated utility and there were no "two optimization goals". Moreover, he later refined even the aggregated utility version, see SEP, Bentham: Later Improvements.
    – Conifold
    Commented Feb 1 at 0:12

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Many people have pointed out that "the greatest happiness of the greatest number" doesn't have any definite meaning.

Attempting to state it as a sum doesn't really work either. Who is to say whether the happiness of different people is comparable, let alone additive? Is my happiness on a similar scale to yours? How would you know? If they are not identical, how should they be weighted?

How does pain, or negative happiness, subtract from happiness?

How do possible future people who haven't been born yet figure in a calculation? It would seem unfair to exclude them from consideration, but do they count the same?

Does the happiness of animals count, and if so are they on the same scale as us?

Is happiness even the right measure of things?

The problems are so many and so serious it is difficult to take utilitarianism seriously as a moral theory.

Arguably, thinking of utilitarianism as a theory of metaethics is missing the point. For Bentham, "the greatest happiness of the greatest number" is more in the nature of a political slogan. He was opposed to the idea of a ruling elite running the country for their own benefit and ignoring the good of the masses. His concern was to stress the importance of making moral and political decisions that benefit everyone on a more equitable basis.

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  • Those are all things one can or should consider, yes. "The problems are so many and so serious" - you didn't present any problems, you just asked some questions. Questions have answers. If you want to argue against utilitarianism, you'd need to make the case that none of the questions you asked have reasonable answers (because if there are reasonable answers, then just go with that - how is that a problem?). Also, I'm still waiting for literally anyone who says utilitarianism is a bad moral theory to present anything better.
    – NotThatGuy
    Commented Feb 1 at 0:03

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