The Kantian ethical formulation, "Always act according to that maxim whose universality as a law you can at the same time will," seems rather vague. Where, exactly, do we decide how to formulate one's actions? Suppose a poor man steals medicine he otherwise can't afford, in order to save his child. Which maxim's universality as a law, exactly, should he be willing? If we examine the maxim as being "It is permissible to steal", then we clearly have a universal law that shouldn't be willed, but if we examine the maxim as being "It is permissible to steal in order to save lives, when there are no other options", then we have a maxim that might hold up under more scrutiny. For any given action, there are multiple possible maxims that are more or less reprehensible, each of which can justify the action in question. So how does Kantian ethics treat situations like this? Do we boil it down to the most semantically/logically simple maxims? Do we go instead for the most nuanced maxims?

That’s an interesting question about how to interpret and apply the universal law test of Kant’s categorical imperative. That test is supposed to apply to our maxims, and our maxims are our subjective ends or goals. So presumably an agent’s maxim is a psychological fact about an agent having to do with her aims. So I don’t think we always choose the most coarse-grained maxim – for instance, stealing – or always the most fine-grained maxim – stealing $10 to feed a blonde six-year old child on a Wednesday because of a lack of other options. It depends on what things the agent -- for instance, the child’s mother – is aiming at and what does guide her actions, and this may depend on the answers to counterfactual questions. Would she steal the money in any case? Would she steal food if this were available? If there were other, morally innocent ways of feeding her child (e.g. going to a soup kitchen or picking berries), would she do so? If the mother would have stolen even if her child was not hungry,...

How does chaos theory effect Mill's philosophical interpretations? It seems that through chaos theory and the butterfly effect, it is impossible for anyone to know what action is the one that will create the most happiness in the world. It doesn't even have to be extreme cases like someone saving a child Hitler from drowning. But even in regular every day cases, every action we take has literally billions of results that are impossible to know about and thus a person would never know which action was the morally right one. Did Mill take this into account in his writings?

Insofar as a system is chaotic, it is impossible to predict all the consequences of actions we perform. Many systems, while not literally chaotic, are still complex, with the result that predictions about the total consequences of one's actions are difficult and not fully reliable. But it's not clear that these facts about complexity and chaos compromise the claims of classical utilitarians, such as Mill. The classical utilitarians (e.g. Bentham, Mill, and Sidgwick) all insisted that utilitarianism -- the demand that we act so as to promote human happiness -- be understood as a standard of duty or right action, rather than as a decision procedure . That is, what determines the moral assessment of one action in relation to others is the comparative value of its actual consequences and not the comparative value of its expected consequences. Whether we should try to apply the utilitarian principle in various contexts is itself a practical question, the answer to which depends upon the comparative...

Why would Plato agree with the claim that there are not any universally valid moral values? Or where can I find information that supports this claim?

I'm not sure why you think that Plato would deny that there are objective and universal moral values. To the contrary, Plato is often taken to be a prototypical advocate of the sort of realism or objectivity about moral value that posits moral truths that obtain independently of the appraiser's beliefs or attitudes about what is right or wrong (cf. John Mackie's Ethics: Inventing Right and Wrong , ch. 1). This would be the common reading of Plato, with which I would agree. In the Euthydemus and Theaetetus Plato offers extended arguments against relativistic views, and the in Euthyphro he extends his realism to virtues, such as piety. You'd certainly be swimming against the scholarly current to read Plato as an advocate of any significant kind of relativism.