This article expands on an earlier article written by me in 2007. This article explains a two consequentialist theories of morality, that is to say theories of morality that are based on the consequences of actions. This article assumes a background in the axiomatic arithmetic, set theory and probability theory that can be found in certain Stage 2 university mathematics and computer science courses.
Axiom 1: for any event A, P(A) ≥ 0
Axiom 2: P(Ω) = 1
Definition 1: Two events A and B are mutually exclusive if and only if P(A ∩ B) = 0. A collection of events S = {A_{1}, A_{2}, A_{3}, ...} are mutually exclusive if and only if P(A_{i} ∩ A_{j}) = 0 for all i,j ∈ {1,2,3,...,S}.
Axiom 3: for any countable collection A_{1}, A_{2}, A_{3}, ... of events that are mutually exclusive: P(A_{1} ∪ A_{2} ∪ A_{3} ∪ ...) = P(A_{1}) + P(A_{2}) + P(A_{3}) + ...
Lemma 1: Let A,B ⊆ Ω. Then P(A ∪ B) = P(A) + P(B)  P(A ∩ B).
Proof: Firstly note that (A \ B) ∩ (A ∩ B) = ∅ and (A \ B) ∪ (A ∩ B) = A.
Therefore P(A \ B) + P(A ∪ B)  = P((A \ B) ∪ (A ∩ B))  by Axiom 3  
= P(A)  since (A \ B) ∪ (A ∩ B) = A  † 
Secondly note that (B \ A) ∩ (A ∩ B) = ∅ and (B \ A) ∪ (A ∩ B) = B.
Therefore P(B \ A) + P(A ∩ B)  = P((B \ A) ∪ (A ∩ B))  by Axiom 3 
= P(B)  since (B \ A) ∪ (A ∩ B) = B ‡ 
Thirdly let X_{1} = A \ B, X_{2} = A ∩ B and X_{3} = B \ A. Then X_{1},X_{2} and X_{3} are mutually exclusive and A ∪ B = X_{1} ∪ X_{2} ∪ X_{3}.
P(A ∪ B)  = P(X_{1}) + P(X_{2}) + P(X_{3})  by Axiom 3 
= P(A \ B) + P(A ∩ B) + P(B \ A)  by definition of X_{1},X_{2} and X_{3} 
Therefore P(A ∪ B) + P(A ∩ B)  = P(A \ B) + P(A ∩ B) + P(B \ A) + P(A ∩ B)  adding P(A ∩ B) to both sides 
= P(A) + P(B)  by † and ‡ 
Subtracting P(A ∩ B) from both sides yields the result.
Lemma 2: Suppose that A ⊇ B. Then P(A) ≥ P(B).
Proof: Suppose that A ⊇ B. Then A = B ∪ (A \ B) and B ∩ (A \ B) = ∅
Therefore P(A)  = P(B) + P(A \ B)  by Axiom 3 
≥ P(B), since P(A \ B) ≥ 0  by Axiom 1. 
A natural way to define morality is by the consequences of actions, specifically the probability of an individual i ∈ ℘ receiving harm in the form of retribution from another individual j ∈ ℘. Let us define P(H_{i}) as the probability of person i receiving harm. Furthermore let us define P(H_{ij}) as the probability of person i receiving harm from person j. With these two definitions in place then the following result holds:
where ℘ denotes the set of all people and i ∈ ℘. Later on in this article I will make the notion of "harm" less vague and more logically rigorous.
Definition 2: Two events A and B are independent if and only if P(A ∩ B) = P(A) ∗ P(B).
To calculate P(H_{i}) we need an additional assumption that the set ℘ is of the form {1,2,3, ... ,n} so that the set is a finite subset of the natural numbers. This can be done without any loss of generality because the total number of people that have ever lived is a finite set and an arbitrary set ℘ can be bijectively mapped into such a subset of the natural numbers.
Theorem 1: Assuming that the events H_{ij} and H_{ik} are independent for all i,j,k ∈ ℘, then P(H_{i}) = P(∪_{j = 1 ... n} H_{ij}) = z_{n} where
z_{0}  = 0 
z_{n}  = z_{n1} + P(H_{in})  z_{n1} ∗ P(H_{in}) for n ≥ 1 
Proof: The case n = 0 is self evident and the general case where n is arbitrary can be proved using mathematical induction.
LHS_{1}  = z_{1}  
= P(H_{i1})  by definition of z_{1}  
= z_{0} + P(H_{i1})  z_{0} ∗ P(H_{i1})  since z_{0} = 0  
= RHS_{1} 
Inductive step. Assume the equation is true for n.
LHS_{n+1}  = z_{n+1}  
= P(∪_{j=1..n+1} H_{ij})  by definition of z_{n+1}  
= P(∪_{j=1..n} H_{ij} ∪ H_{in+1})  expanding the union  
= P(∪_{j=1..n} H_{ij}) + P(H_{in+1})  P((∪_{j=1..n} H_{ij}) ∩ H_{in+1})  by Lemma 1  
= P(∪_{j=1..n} H_{ij}) + P(H_{in+1})  P(∪_{j=1..n} H_{ij}) ∗ P(H_{in+1})  since H_{ij},H_{ik} are independent for all i,j,k ∈ ℘ *  
= z_{n} + P(H_{in+1})  z_{n} ∗ P(H_{in+1})  by definition of z_{n}  
= RHS_{n+1} 
Theorem 2: P(H_{ij}) > ε implies P(H_{i}) > ε.
Proof: Suppose that P(H_{ij}) > ε.
Then P(H_{i})  = P(∪_{k ∈ ℘} H_{ik})  by definition of P(H_{i}) and P(H_{ik}) 
≥ P(H_{ij})  by Lemma 2, since ∪_{k ∈ ℘} H_{ik} ⊇ H_{ij}  
> ε 
Theorem 4: P(H_{i}) > 0 by itself affords no police protection unless you have a job that comes with police protection such as the leader of a country.
Theorem 5: If P(H_{ij}) > 0 and then person i should terminate their relation to person j at the end of a university course or a school year. In particular, social media such as Facebook or Twitter should not be used to keep a relationship open with individual j. References on person j’s Curriculum Vitae are another way of keeping a relationship open with person i.
Theorem 6: If P(H_{ij}) > 0 and P(H_{ji}) > 0 then both parties should terminate their relationships with each other and only communicate via the medium of lawyers. If P(H_{ji}) > 0 then person i should be careful to make sure their replies to person j should not invoke P(H_{ij}) > 0, in which case a mutual termination would occur.
Theorem 7: If P(H_{ij}) > 0 then person i should not make eye contact with person j. There is one exception to this. When the eye contact is at a distance, person i could look away for a short interval before remaking direct eye contact. This is because there is a nonzero probability that in person i’s mind, they are making eye contact for the first time.
Theorem 8: To ensure that P(H_{i}) = 0, then by Theorem 2 and Theorem 5, for person i to have a lifetime relationship with other persons needs to ensure that P(H_{ij}) = 0 for all j ∈ ℘. As will be explained later this requires honesty. Therefore to live outside of the protection of the law you must be honest. Therefore Bob Dylan's lyric: to live outside the law you must be honest holds true.
Theorem 9: If P(H_{ij}) > 0 but P(H_{ji}) = 0 then person i should tell person j that they have no memories of past engagements with person j, especially if person i hasn't actually spoken with person j. If the person j says I think you are lying about having no memory of seeing me, person i can say: I don't have to stand for this, get out!, thus terminating the relation from person i to person j. Person j can then send person i a single email containing the following text:
To Professor X, [Meaning: do you really deserve to be called a professor?]
I recently visited your office where you claimed to have no memory of seeing me. It strikes me that this is a particularly unpleasant way for us to say goodbye to each other. I respect your intelligence and some of your values. I have no greviance [sic] or grudge against you. [The spelling mistake is unintentional but serves its purpose.]
Thus keeping the relationship from person j to person i open but the relationship from person i to person j terminated.
Theorem 10: If P(H_{ij}) > 0 then person i should not give away their home address to person j.
The problem with Kant's categorical imperative is that different moralities apply to different people. Under the potential harm conception of morality this result obtains.
The value of P(H_{i}) depends partially on the number of underlings a person has. Underlings are people who are underneath a particular person. Examples of underlings are people employed by a given person, or students who are taught by a particular high school or a university lecturer. An underling j can invoke P(H_{ij}) > 0 and therefore by Theorem 2, P(H_{i}) > 0.
The value for P(H_{i}) varies with the number of underlings that a person has because people love to gossip about people who have lots of underlings. Therefore people with a lot of underlings such as leaders of cities, states and countries or university lecturers or school teachers will have a higher value for P(H_{i}). Therefore such people have to be more righteous in their conduct than other people because of karma.
An example of semiunderlings are fans of a particular person, such as the charismatic Jimi Hendrix. Jimi Hendrix could dress the way he did because his charima was high and because semiunderlings are not like true underlings. As Jimi Hendrix said in the song ‘‘If six was nine’’, from the album Axis: Bold as Love:
White collar conservative flashing down the street, pointing their plastic finger at me. They all assume my kind will drop and die, but I'm going to way my freak flag. Hi how are you? Mr. businessman, you can't dress like me ... I'm the one that's going to die when it’s time for me to die, so let me live my life the way that I want to.
Businessmen could not dress like Jimi Hendrix because they often have a lot of genuine underlings, and that implies P(H_{i}) > 0. Businessmen or military men who don't have a lot of underlings yet still dress in a businesslike manner or in military uniform are losers. University lecturers or school teachers who dress like rock stars are likely to be of high charisma because genuine underlings invoke P(H_{i}) > 0 more than semi underlings do.
Not hiring someone for a job can invoke P(H_{ij}) > 0 if C_{i} < C, where C is some critical number and C_{i} is the charisma of individual i. This implies that P(H_{i}) > 0. If C_{i} ≥ C then P(H_{ij}) = 0, in spite of person i declining a job application by person j. The value of C is lower in countries with a welfare system for men, because not hiring someone for a job in countries with a welfare system does not kill that individual. In countries without a welfare system for men does nearly kill that person or at least reduces them to the status of beggars.
If you are a person i of a different race another person j, then in the value of P(H_{ij}) is smaller or zero for than for people j who are the same race as i.
Women are the gatekeepers of sex, meaning that if a women wants sex, she just has to open the legs to her vagina and a penis will soon come along, whereas if a man wants sex he has to have a good job, a good car, a good house and so on to attract a female mate. As women are the gatekeepers of sex, women in positions of power have lower P(H_{i}) than men in equivalent roles, because they can theoretically offer sex to men who have a probability of harming that woman greater than 0. This offer of sex can partially offset the wrongs in terms of P(H_{ij}) invoked by that man to that woman. This explains why the jobs of secretaries in organisations are invariably taken by women. For this reason, the future of women in power is a good one, and it is likely that in the future all positions of power will be taken by women.
Men with a P(H_{i}) > 0 will wear a tie to reduce their P(H_{i}). As women are the gate keepers of sex, they do not need to wear a tie.
In shaking hands men to men's handshakes are strong if both men have had sex with a woman. Women to men's handshakes are less strong, since there is a probability that the woman could have sex with the man.
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