``How does one develop a sense of `normal'?'' I asked myself this morning as I stare out of my window into the coterie of trees immediately in front of it as the quiet chirpings of birds can be barely heard in the background.
Context: I was listening to the cover of Goodbye 宣言 by Takanashi Kiara, and thinking about the definition of ``normality'', and linking it back to how we have prescriptive definitions of various forms of ``abnormal psychology'' but no definition of what ``normal psychology'' is. These thoughts came from the recent news that Simone Biles (US Gymnastics representative in this year's rendition of the Japan 2020 Olympics) has withdrawn from her events citing her mental health concerns, and more importantly, various people who criticised her decision.
No, I'm not going to talk about mental health today. I've talked as much as I cared to with however little I know, and I think that's good enough [for now]. Instead, I would like to talk about ``normality'', how we define it, and the associated ramifications.
Apart from various mathematical meanings of the word ``normal'', I posit that the layperson's definition of ``normal'' is more of a definition by exclusion than a definition by inclusion.
A definition by exclusion is characterised by providing a set of properties that, if present, excludes the object from being part of the set. So statements like ``If you have X, then you are not normal'' fall into this category. Definitions by exclusion are often used to reduce the the rate of false membership---if a set L of properties of exclusion has even one that is applicable for the object being considered, then it is immediately excluded. So the final set as defined by exclusion with L is the really small number of objects that fail to trigger any property in L.
Conversely, a definition by inclusion is characterised by providing a set of properties that, if present, includes the object into being part of the set. Suppose then we have two statements, statement A: ``If you have X, then you are normal'' and statement B: ``If you have Y, then you are normal''. A person &alpha can claim membership by fulfilling statement A and not fulfilling statement B, person β can also claim membership by not fulfilling statement A but fulfilling statement B, and person γ can also claim membership by fulfilling both statements A&B. The only person who cannot claim membership is when he/she cannot fulfill both statements A&B at the same time. Definitions by inclusion end up reducing the rate of false exclusion---if a set L of properties of inclusion has even one that is applicable for the object being considered, then it is immediately included. So the final set as defined by inclusion with L is the really large number of objects that can trigger any property in L.
The more astute reader will realise that there is really a bit of trickery involved in these definitions. In both definitions, the set of properties L are applied in a disjunctive manner. If f(x) is a boolean function that returns true if some property p∈L is applicable to x, then the only time that f(x) is false is when there is no such p∈L where p is applicable to x. Thus on its own, we expect f(x) to return true for ``many members of x'', since of all the possible boolean outcomes of L with n=|L|, we expect only 1 out of 2n (i.e. the one case with all n elements being false) to not be a member.
The tricky part comes in the definition of the set whose membership we are determining. In the case of exclusion, it is the complement of the set. So what began as ``many members of x fulfilling L'' ends up becoming ``many members of x are part of the complementary of normal'', which ends up becoming ``few members of x are normal'' after pushing some of the negations in. Conversely, this does not happen in the case of the inclusion.
So clearly, the definition by exclusion is too strict, and cannot be used to define normality, because that would mean that most people are not normal, which contradicts the layperson's definition of normality (that it satisfies some very vague notion of being an ``expected'' behaviour, or ``standard'', or ``generally acceptable''). Conversely, the definition by inclusion is too lenient since almost everyone is normal, which sounds great in theory (it fits the layperson's definition of normality), but it does paradoxically provide an excuse to not look out for those who have issues (and are thus ``not normal'').
There are at least a couple of ways to rigorise this, but I am uncertain of their practical applicability---that last bit can be considered a good thing, because it can form an argument against automated rule enforcement over human society in general. One way might be to start with L the set of properties (we'll use the definition exclusion for simplicity's sake), and on it, define a new set S whose members σ∈S are defined as σ⊆L, which is fancy notation for saying that each member of S is some sub-set of properties of L. Then we define f(x) returning true when all properties of a set σ∈S are met. This relaxes the strictness of the exclusion definition, since it only excludes when all of a particular subset of properties from L is met, as opposed to any one of the properties in L. If L has a cardinality of n, then S potentially has 2n members, and we choose how many of those 2n members are to be excluded/included.
The catch of course is that that 2n is one very large number that shouldn't be laughed at---it doesn't matter if one wants to use the properties for a less rigid exclusion or a more discriminative inclusion, the same quagmire of defining these subsets cannot be run from. One can apply some simplifying assumptions by saying that up to r<n properties need to all apply before it happens, but this number is still asymptotically exponential the larger r is.
But then again, most people just ignore this rigorisation and stick to the shortcut of defining by [simple] exclusion or inclusion. The predominance of which definition choice is largely determined by a society's value system. Those that value more individual choice will tend to use definitions of inclusion, and those that prefer a more homogenous society will use definitions of exclusion.
Are either forms of definitions for ``normal'' considered good? No, neither are, and that is, like most cases, due to the extremism innately encoded. But a fairer means of defining normal is prohibitively complex, with the effort taken incommensurate with the outcome (which is to operate within society).
Why do something theoretically correct when the vast majority will do fine with the laziest application?
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