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I am aware of Bentham's hedonic calculus, for which I feel is insufficient for describing ethical propositions outside of its weak implicature. There likely have been many critiques of it, so I shall not linger on this subject.

There have also been attempts to establish Kantian equilibria in the context of superrational symmetric games, which provide a novel attempt but ultimately fail to describe the capacity for an agent to 'model' their ethical reasoning; that is, this line of reasoning is only useful insofar as we assume by necessity that agents will use Kantian utility in order to maximize their collective payoffs. This is the only attempt at applying formal techniques to ethics that I know of, which of course only assumes cooperative behaviour as a consequence of the categorical imperative.

My question is this: what are the requirements for constructing an ethical calculus? Is it possible or futile, and if so why?

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Ethics can be framed as a decision problem. You have various options A, B, C ... and need to determine which one of these is "best". This is the framework of Mathematical Optimization or Mathematical Programming.

Given this, to construct an ethical calculus you need three components:

  1. Dimensions to consider - what aspects of reality are ethically relevant.
  2. Tradeoffs to be considered.
  3. Quantity to be optimized.

There are an infinite number of solutions to this but I think at a minimum you need to address these in an ethical calculus.

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One problem, perhaps not rendering this a rather futile task, but perhaps so rendering it, would be that a sufficiently general such calculus would have to apply to the ethics of belief and justification in the foundations (such as they are) of mathematics. There seems to be a difficult or even inescapable circularity involved in this project, then: we need an ethical calculus to evaluate the principles of the very same calculus! One wonders so if the desire for formal rigor must be satisfied in every domain of discourse (there is aesthetic rigor, granted, but must this always be formal? and is even that rigor, here, always "required"?).

Or: how compelling will increasingly complicated ethical reasoning prove? If we want an ethical calculus to try to settle debates over various moral questions, yet if this is too complicated a matter (eventually), how many people will be culpably responsive to such complications? When will we need to call in local intuitions to solve local moral problems?

Further reading (selected examples):

  1. Moral Particularism and Moral Generalism
  2. Moral Particularism
  3. Formal ethics (Wikipedia synopsis)
  4. Deontic Logic
  5. The Logic of Action
  6. Practical Reason and the Structure of Actions

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