In my previous posts, The Foundation of Morality and A Pattern of Moral Refinement, I presented a new theory of morality based on the natural selection of sustainable behaviors over long periods of time. In its current state, the morality I have presented is nebulous and difficult to put into practice. It needs a more rigorous foundation, and I will attempt to do that here. Starting with self-evident axioms and clear definitions, we will proceed with logic to demonstrate that objective morality as defined in this theory exists and is well-defined.
As I explained before, morality cannot be defined without free agents, and so we must start with the following definition and axiom:
Definition: An individual capable of freely making decisions and acting is called an agent.
Axiom: A population of agents exists.
Next, these agents must have opposing choices to choose from, otherwise there is no morality to judge their choices. So let’s write some definitions to build up situations to which rules can be applied. We will accept as an axiom that the universe exists. We will then define circumstances, situations, and rules, to start building up the logic of moral choices.
Axiom: The universe exists.
Definition: A circumstance is any statement describing all or any part of the universe.
Definition: A situation is a set of circumstances and a set of possible actions which an agent can choose from, denoted by 
Definition: A rule is a situation with a selected action, denoted by 
We now have the foundation we need to start building a good definition of morality. Morality will be a set of rules, but not just any set, so we’ll define any arbitrary set of rules as a “code of ethics,” rather than a “morality,” since we’re reserving that term for the true morality.
Definition: A code of ethics is a set of rules.
Now we need to incorporate the underlying refinement tautology in some way. Certain rules will preserve themselves indefinitely, while others will end up eliminating themselves. This means that eventually the entire population of agents in a society will be obeying the rules that preserve themselves. We call these special rules “convergent rules,” and the rules that are eliminated are called “divergent rules.”
Definition: A convergent rule is a rule which is followed by the entire population of agents in the limit as time approaches infinity.
Definition: A divergent rule is a rule which is not convergent.
The convergent rules exactly satisfy the requirements for morality that we discussed previously: they preserve themselves indefinitely. These convergent rules must be at least a subset of morality. The divergent rules are not all necessarily immoral, but may simply cease to exist because certain situations cease to exist as a society refines itself. We can call these divergent situations, as opposed to convergent situations. Think of medical care. As society perfects itself, perhaps people won’t get sick or hurt anymore, so the daily actions of doctors may eventually cease to exist. But medical care tends to support the progress of morality by keeping the population of agents (on whom morality depends for its existence) alive, so it ought to be considered a good thing. Given this insight, we can’t define morality solely as the set of convergent rules, so we will refer to that set simply as the “Convergent Code of Ethics” or the CCE.
Definition: A convergent situation is a situation which continues to exist regularly as time approaches infinity.
Definition: A divergent situation is a situation which is not convergent.
Definition: The Convergent Code of Ethics (CCE) is the set of all convergent rules.
The divergent rules must be split into at least three categories: those that support the progress of the CCE, those that neither support nor impede the progress of the CCE, and those that impede the progress of the CCE. We will call these supportive rules, neutral rules, and destructive rules, respectively.
Definition: A supportive rule is a rule which supports the CCE.
Definition: A neutral rule is a divergent rule which neither supports nor impedes the CCE.
Definition: A destructive rule is a divergent rule which impedes the CCE.
In any given situation, many rules are possible. More than one of the rules could be considered supportive, but whatever best supports the progress of the CCE ought to be considered the moral choice. We will call this best choice the “optimal rule.”
Definition: The optimal rule is the rule which best supports the progress of the CCE in a given situation.
What happens in the case where two or more rules equally support the CCE? In this case the agent can freely choose from among the best options. Suppose there is a situation with options A, B, and C, where A and B equally support the CCE and C impedes the CCE. Clearly C is immoral, and either A or B would be a moral choice. In this case, the optimal rule is “A or B.”
We are now at the point where every situation, convergent and divergent, has a morally correct choice. In the case of convergent situations, it is the convergent rule. In the case of divergent situations, it is the optimal rule. There is a possibility for inconsistency here: what should one do in a convergent situation where the optimal rule and the convergent rule are different? Which type of rule is the most moral? As it turns out, this question is irrelevant because it can be shown that in all convergent situations, the convergent rule is the same as the optimal rule.
Theorem: For a given convergent situation S, the optimal rule is the convergent rule.
Proof: As time approaches infinity, the population wants to support the CCE, so whenever situation S arises, they will choose to follow the optimal rule. Therefore, the optimal rule is convergent.
In order to be sure that what we’re doing actually makes sense, we need to prove that at least one convergent rule exists. Consider Hamlet’s famous dilemma, “To be, or not to be.” Should a person commit suicide or not? Those who commit suicide can only do so once, and by doing so, they make it impossible to do it again in the future. They limit their ability to convince other people to follow their example, because they are gone. This is clearly a self-destructive behavior. On the other hand, the act of not committing suicide is one that can be accomplished every second of every day as long as a person lives. Those who choose to live this way continue to live and reproduce and influence society, while those who choose to die will eliminate themselves. Eventually, our society will be refined to the point where people don’t kill themselves anymore. Therefore, not committing suicide is a convergent rule and forms part of the true morality. This proves the following theorem:
Theorem: At least one convergent rule exists.
With the example of suicide, we have shown that convergent rules and optimal rules exist and are well-defined. The option that was not convergent is divergent, so divergent rules have also been shown to exist. With this knowledge, we can finally give a definition of morality that covers all situations and is always consistent with itself.
Definition: Morality is the set of all optimal rules.
Notice that since all convergent rules are optimal, the CCE is a subset of morality, as we said it should be. Morality, as defined here, is consistent because it never disagrees with itself. It is adaptable because every detail of a given situation must be considered. It is also objective because any outside observer analyzing the situation and correctly applying the definition would come to the same conclusion about what is right. The fact that morality is always consistent with itself, combined with the fact that at least one optimal rule has been shown to exist, means that, as an abstract concept, morality exists. It is just as real as the laws of physics.
Truth Reconciled 