(ducks before accusations of misusing "isomorphic")

Old joke:

Q: Are these two objects isomorphic?

A: The first one is, but the second one isn't.

Sort of! But not exactly. This is a topic I've been meaning to write a long post on for ages, and have given a few short impromptu presentations about.

Consequentialism, deontology, and virtue ethics are classifiers over world-histories, actions, and agents, respectively. They're mutually reducible, in that you can take a value system or a value-system fragment in any one of the three forms, and use it to generate a value system or value-system fragment in either of the other two forms. But value-system fragments are not equally naturally expressed in different forms; if you take a value from one and try to reframe it in the others, you sometimes get an explosion of complexity, particularly if you want to reduce value-system fragments which have weights and scaling properties, and have those weights and scaling properties carry through.

Henry Sidgwick in "The Methods of Ethics" actually makes the argument that Utilitarianism can be thought of as having a single predominant rule, namely the Greatest Happiness Principle, and that all other correct moral rules could follow from it, if you just looked closely enough at what a given rule was really saying. He noted that when properly expanded, a moral rule is essentially an injunction to act a certain way in a particular circumstance, that is universal to any person in identical circumstances. He also had some interesting things to say about the relationships between virtues and Utilitarianism, and more or less tried to show that the various commonly valued virtues could be inferred from a Utilitarian perspective.

Of course Sidgwick was arguing in a time before the clear-cut delineation of moral systems into "Deontological", "Consequentialist", and "Virtue Ethics". But I thought it would be useful to point out that early classical Utilitarian thinkers did not see these clear-cut delineations and instead, often made use of the language of rules and virtues to further their case for Utilitarianism as a comprehensive and inclusive moral theory.

I agree. In principle, you could construct a total order over all possible states of the world. All else is merely a pretty compression scheme. That being said, the scheme is quite necessary.

I've been wondering if it makes sense to think of ethical philosophies as different classes of modeling assumptions:

Any moral statement can be expressed using the language of consequentialism, deontology or virtue ethics. The statements can therefore be translated from one framework to another. In that sense, the frameworks are equivalent. However, some statements are much easier to express in a given language.

Sometimes, we make models of ethics to explore the underlying rules that make an ethical statement "true". We try to predict whether an ethical statement is true using information from other, closely related ethical statements. However, ethical statements are multidimensional and therefore vary across many different axes. Two ethical statements can be closely related on one axis, and completely different from each other on another axis. In order to learn about the underlying rules, we have to specify which axis we are going to make modeling assumptions on. The choice will determine whether you call yourself a "consequentialist" or a "deontologist".

Problems such as "the repugnant conclusion" and "being so honest that you tell a murderer where your children are" occur when we extrapolate too far along this axis, and end up way beyond the range of problems that the model is fit to.

Isn't that a necessary step for the claims Derek Parfit makes about convergence in his "On What Matters"?