Facebook statuses aren’t really known for lingering on in the memory, yet a few days ago I saw one that really stuck with me and has been bouncing around my head ever since. The status was about a friend of mine who was sick of religious people who constantly updated the definition of their own community through a perpetual rejection and exclusion of members who could sully the good name of their group. Or, in simpler terms:

I was unfortunate enough to be a part of a conversation today about sexual assault committed by followers of a religion (not just Catholicism). Somebody threw out the familiar line “Oh, they can’t have been real [insert religion here]”. Regardless of your faith/lack thereof, I ask you all to stop saying this. Don’t push all your shitty people on me and my non-religious friends. Instead, imagine for a second that they ARE [insert religion here], they’re just bad at it and, moreover, are just shitty people. Shitty people exist in every community, yours included.

Now, I have never heard of this kind of argument from a religious point of view, yet I am very familiar with the thought process behind it, as it is pretty famous in philosophical circles. What my friend had come up against was a classic logical fallacy known as the “No True Scotsman”. The fallacy’s name comes from the prime example used to explain the concept.

Person A: “No Scotsman puts sugar on his porridge.”

Person B: “But my uncle Angus likes sugar with his porridge.”

Person A: “Ah yes, but no true Scotsman puts sugar on his porridge.”

– I stole this from Wikipedia

So you can see, this discussion could go on forever, and Person A will always be right, because their argument is based on some subjective opinion of what a Scotsman is, that they can update at will, and exclude from the reference group any counterexamples. Person A is not just moving the goalposts, he won’t even tell Person B where the goalposts are.

If we think about it hard enough you could probably all think of a time when you were frustrated with this type of argument (or more than likely we have all done it ourselves), and there is nothing new about formally explaining something that you probably will now say you have known all your life. Yet my friends experience with religious people using this argument above stuck with me, as it introduced the concept of morality into the equation, and the desire that certain groups define, and indeed pride themselves on comprising solely of universally moral and good people. What I am going to try and (quite literally) prove to you here is that this idea makes things very interesting indeed.

A Simple Model of Morality and Exclusion

Let’s say a group Y exists. The group contains n members, where n is some number above zero. It could be 10, it could be 20, it could be 3 million: it doesn’t matter. Each of the n individuals is represented by ${{x}_{i}}$, where $i=1,2,....,n-1,n$. A group is not much more than the sum of the attributes of its members, so we can express the value of this group in the expression below:

$Y=\sum\limits_{i=1}^{n}{{{x}_{i}}}$

That’s the formal mathematical notification, but we can expand this to get rid of the Greek letters and have this, which means exactly the same thing:

$Y=\sum\limits_{i=1}^{n}{{{x}_{i}}}=({{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....+{{x}_{n-1}}+{{x}_{n}})$

So we have an expression for the total value of group Y. Now let’s assume that this group values one thing above all else: moral goodness. They pride themselves on how good they are, and believe that every other person in the group has a similar level of goodness to them, making the group whole. We will represent the goodness of an individual ${{x}_{i}}$ with ${{g}_{i}}$, where ${{g}_{i}}$ is a value between -1 and 1 ($-1\le g\le 1$), where the closer the individual ${{x}_{i}}$ is to 1, the better the person that individual is. Conversely, the closer that persons is to -1, the worse a person is.

${{x}_{i}}={{g}_{i}}$

This represents the goodness of individual i. So we can now rewrite the value of Y based on the moral goodness of its members

$Y={{g}_{1}}+{{g}_{2}}+....+{{g}_{n-1}}+{{g}_{n}}$

For the sake of simplicity let’s say that all members of this group are assumed to have the same level of g, ${{g}^{*}}>0$ which might be the goodness level achieved just by living life according to their prescribed rules etc. If we assume this, then we can simplify further.

${{Y}^{*}}={{g}^{*}}+{{g}^{*}}+....{{g}^{*}}+{{g}^{*}}$

${{Y}^{*}}=n{{g}^{*}}$

The value of ${{Y}^{*}}$ simplifies to two expressions: the number of its members multiplied by their theoretical constant level of moral goodness.

The funny thing about moral goodness is that it cannot actually be observed directly. We cannot for certain say that the individual ${{x}_{i}}$ is a good person, all we can say is that all the information we have about him up to that point indicates that this is the case. Therefore we can assume that ${{x}_{i}}$ is a good member of the group and should continue as a member. There can only be a theoretical level of moral goodness ${{Y}^{*}}$, and faith must be maintained within the group about the true nature of each other member, that ${{g}_{i}}={{g}^{*}}$ .

Bad behaviour, on the other hand, is completely observable, particularly when an individual, which we will designate as ${{x}_{b}}$ performs some despicable act such that he can no longer be referred to as a morally good person. Once this act is committed, the individual has revealed himself to be a false ${{g}^{*}}$, as his actual value of ${g}$ is not more than zero, and has in fact a negative value of ${g}$, where $g<0$. We will designate this negative value of g as ${{g}_{\psi }}$.

Because of this revelation, our group goodness value has changed:

$Y=(n-1){{g}^{*}}+{{g}_{\psi }}$

Everyone else still has their constant level of goodness ${{g}^{*}}>0$, while our bad person ${{x}_{b}}$ has been separated out because he is of a different moral integrity  ${{g}_{\psi }}$. I’ll remind you here that  ${{g}_{\psi }}$ is negative and will therefore drag the groups goodness down. Obviously in a group that values moral goodness above all else, the individual ${{x}_{b}}$ must be removed and excluded from the group, as he is No True Godsman. Therefore our updated value for the groups morality is

$\tilde{Y}=(n-1){{g}^{*}}$

Discussion

Let’s now compare the different values for Y we have had so far.

${{Y}^{*}}=n{{g}^{*}}$

$Y=(n-1){{g}^{*}}+{{g}_{\psi }}$

$\tilde{Y}=(n-1){{g}^{*}}$

It’s clear that ${{Y}^{*}}$ is the highest value of all our Y’s, and that after all the revelations of wickedness, we are left with a group goodness level that is below our originally perceived theoretical level

${{Y}^{*}}>\tilde{Y}$

We must not forget however that the individual who was removed from the group was always a bad person, we just did not realise it at the time, and Y* never actually existed, and all we had before the unpleasantness was Y, the second of our three Y values given above. Think about this and compare it to our new value ${\tilde{Y}}$. Since ${{g}_{\psi }}$ is negative, this means that $\tilde{Y}$  will always be greater than Y.

$Y<\tilde{Y}$

The groups goodness has actually increased as a result of the expulsion of a member who revealed himself to be bad. Further, this will always be the case, as any further hidden non-${{g}^{*}}$’s who are revealed as such will be removed and therefore increase the moral goodness of the group as a whole. This might sound like quite an obvious and innocent statement, but think about how someone can actually reveal themselves to be a bad person. They will never reveal it voluntarily, but only through bad behavior and acts. If a group values the unobservable moral integrity of it’s members above all else, it will always be good for the group if it’s secretly bad members perform an act so depraved that it reveals them as what they are: a non-${{g}^{*}}$. It is only through horrible acts that the group can actually edge closer to what they truly want: maximising their Y, and therefore their moral goodness.

Conclusion

So, to use these stylised facts on the situation that offended my friend and prompted his Facebook status, he could have argued that this ‘[insert religion here]’ who performed a sexual assault actually did [insert religion here] a favour by telling all the ‘true’ [insert religion here] that the assaulter was no [insert religion here] at all, and that this was good for the [insert religion here] community as a whole. That sexual deviant was living amongst the [insert religion here], passing as a [insert religion here], and now because he revealed his true nature, [insert religion here] is all the better for it. Therefore anyone in that religion should be happy that it occurred. The application of logic to a logical fallacy will always reveal its true nature, and what I hope anyone who read this far gets from this is that it can lead to interesting results, and will more than likely end up backing the offender into a tight, (hopefully) logically sound corner.