= is transitive. ≈ is not.
≈ is also not well-defined for general usage, but notice that no matter what error ε>0 one chooses for which one designates two quantities to be approximately equal (≈) if and only if their positive difference is less than ε, one can find quantities a, b, and c such that a≈b and b≈c, but a≉c. (For instance, use b=a+0.8ε and c=a+1.6ε. Note that we are assuming that the domain of quantities has a largest and smallest element more than ε apart, but this quite goes without saying considering otherwise the ≈ operator is useless.)
The important point here is that the sum of insignificant changes can be significant.
In particular, when one is trying to categorize a set of items into categories, it is possible that any sort of meaningful classification upon a particular quality can be defied.
Consider the typical definition of a species, that is, a set of individuals that can interbreed with each other. Interbreeding is something that can occur between organisms of slightly different genomes (thankfully, for the sake of life existing), but the genomes still need to be similar enough for two organisms to be able to produce fertile offspring. As such one could imagine that individual A can successfully breed with individual B, who can successfully breed with individual C, who can successfully breed with individual D, and so on, but, say, individuals A and H cannot successfully interbreed. This phenomenon can of course be generalized to not just individuals but groups of individuals, where no members of one group can successfully interbreed with members of another group. Thus, by the interbreeding definition of a species, these are clearly not in the same species. And yet, every step along the way we find individuals that can interbreed with each other, all the way from A to H, so each should be in the same species as the previous one. What gives?
It turns out this is not just a hypothetical. There are actual cases of this occurrence found on Earth, in what are known as ring species. It is an occurrence that fundamentally challenges the concept of a species itself, an instance where there is very clearly no acceptable line to draw to divide the individuals into members of different species. It appears necessary to accept that sometimes there is neither a line nor an equivalence, but rather a gradual continuum in species membership.
It also turns out this is not just a problem with species. Let’s turn to linguistics. What is a dialect? Let’s use a definition of a dialect as a version of a language with possible slight differences in phonological, morphological, and grammatical specification, such that these differences are small enough that what is spoken by people of different dialects of the same language is mutually intelligible, that is, people can readily understand what the other is conveying despite the differences in their speech. Well, then what constitutes a language? It turns out there’s several cases of what we consider different languages that are mutually intelligible, like with Norwegian and Swedish, or with Czech and Slovak. But also, there exist languages like Chinese for which there are so-considered dialects that are mutually unintelligible but for which there exist a set of intermediate dialects that are mutually intelligible with the next dialect in the chain, stretching the entire span of linguistic change between these dialects.
(Actually, mutual intelligibility gets even weirder. Mutual intelligibility is not only intransitive but also asymmetric. All sorts of weird relationships between languages that are different flavors of mutual intelligibility arise.)
Here’s the most important point that I want to get to in this post: behavioral and social acceptability.
Suppose you want to support a community, and thus you lay out a set of behavioral guidelines, maybe a terms of service or a code of conduct, to ensure that the members of your community feel welcome and comfortable being in, participating in, and contributing to this community.
But suddenly you realize something. It is definitely possible for someone to act in a way completely deemed acceptable by the word of your behavioral code, but still seemingly clearly unacceptable. (Worse off, someone actually does this.)
They appear to be exploiting a loophole in your code, and the problem is that closing this loophole means they could just back off a bit more and act within these new bounds of your code, and if you kept pursuing them and making your behavioral code tighter and tighter, before they actually act acceptably you will have stymied the normal activity of most members of your community, because so many things are deemed unacceptable by your code now that the most innocent things can be considered to trigger some clause of it.
(And maybe for a moment, you feel you understand why dictators exist now.)
But what’s the fundamental issue? It’s that behavioral acceptability is a spectrum*. A roughly ordered set of behaviors does not have a point in at at which such actions suddenly make people of the community feel uncomfortable, or ostracized, or attacked, for which all behaviors before this point do not cause this problem at all and all behaviors after this point cause the problem in fullest effect. There are steps, for which behavior at one level of acceptability makes people feel slightly more uncomfortable than behavior at the previous level. Having a behavioral policy that involves rigidly enforcing a code of conduct or a terms of service treats the problem of bad behavior as a black-and-white issue when in realistic manifestation it is a matter of a spectrum. Thus, what it manifests in the behavior of those who find objection with the generalized behavioral standards of a community is that it strongly incentivizes acting just inside the delineated bounds of acceptability and strongly disincentivizes acting just outside the bounds.
(*Actually, I’m very much oversimplifying things by calling it just one spectrum. It is quite arguable that behavioral acceptability is very much a multispectrum, with several orthogonal and inconvertible dimensions.)
Does this mean all attempts to be transparent and consistent, and to be clear about what one or a community finds unacceptable, are futile, and in the end due to the exploitability of transparency every community must end up being a dictatorship in the administration of behavioral enforcement?
No. Codes of conduct and terms of service can have grey areas and acknowledged spectrum areas built in. They could state example bounds of where behavior is most likely okay but may start to be problematic and where behavior is most likely unacceptable but for which there may be extenuating circumstances, for each behavioral issue desired to be addressed.
Is it a problem that this allows a community to render different treatments to different groups due to there being grey areas in which it is prescribed that different reactions to behaviors of comparable nature can occur? No, because as always, the outside world could still see what the actions taken are and whether a particular sub-community is given disproportionately different treatment. Sure, there will be times when it seems someone was punished more for doing something less bad, but one could acknowledge that this is quite a low cost to pay to keep a community healthy and functioning while still maintaining a broad sense of transparency.
Having a clear policy and being resistant to loophole exploitation are not completely incompatible. It is possible to lose a small amount of one to gain a large amount of the other. And what it comes down to is acknowledging that some things just can’t be categorized into bins, and trying to work with the intransitivity of ≈.