(Edit: I didn't notice this thread said 'Gamma' until way after I had posted it. It was written in the brackets like this [Gamma] and so I ignored it. The thread just showed up in the list of recent posts and I jumped in without paying attention.)
Talking about this issue gets me upset. I'm helpless to change something which I believe is misinformation. The one thing that I believe is wrong is the process where you identify what function you use most strongly, and then decide if you're a J or a P based on 'the highest order extraverted function,' quoting from someone earlier in this thread. That will give you a different result (if you're an introvert) than if you simply take the Myers-Briggs test and ask yourself directly, am I a perceiver or a judger? If you just look at yourself directly, asking whether you feel/behave more like a perceiver/judger, you won't get the same type you would get from that weird messed-up flipping-the-j/p-for-introverts thing. I believe that process is just outright, flat out wrong.
However, there's nothing I can do about it. It's a huge piece of misinformation floating around out there in the world, being repeated and taught to new people over and over again, destroying the system and making it malfunction. I can only just watch helplessly. What exactly would one do? Go on an informational campaign to educate the public about the wrongness of flipping the j/p for introverts if they started out by observing which function they used the most? Unfortunately, I am not immortal or all-powerful or super-wealthy, and I have other problems and things to deal with in my life, so this informational campaign is my lowest priority, and yet, it gets me upset each and every time I have to read about this in a forum, people being confused by this again and again, endlessly.
The two systems match perfectly. Socionics and the Myers-Briggs are talking about exactly the same thing. Your type in one should always correlate to one, and only one, type in the other. The type correlations are: ESFJ/ESE, INTJ/LII, and so on... (I don't need to write them all).
That is all aside from the fact that either one of the tests can mistype you. I always mistyped as INTP in the Myers-Briggs because I strongly disliked the way the 'sensing' questions were worded - they made me sound like a moron. 'Do you blindly, mindlessly, and rigidly adhere to traditions, regardless of whether there might be a better way of doing things? Yes! That's me for sure!' Nope, I always picked the 'intuitive' answers because they didn't make me sound like a retard. So, I had to find socionics and see how they worded the sensing questions differently in such a way that sensors didn't sound like brain-dead morons, and suddenly, being a sensor didn't seem like something to be ashamed of.
So yeah, you can be mistyped by one or the other of the tests. If you can figure out which type you really are for sure, then it will translate directly back and forth - ISTP=SLI, ISTJ=LSI, and so on. That whole thing of flipping the j/p for introverts has got to be destroyed - it's misinformation. I'm not even going to acknowledge any kind of reasoning behind it at all, I'm not going to be persuaded that there is some kind of logic to it, I'm not going to be directed to some kind of reading material that explains the rationale behind it.
Unfortunately, I will have to keep on seeing this misinformation coming up forever and ever. It's sort of like spreading a bad false rumor about someone, and they can't get rid of the rumor, for the rest of their life, and they just have to keep fighting against the consequences of that rumor every time they try to do anything. That's the effect that this piece of misinformation has on the personality typing systems. You might spend weeks and weeks trying to convince one single individual of the reasoning that explains why that method is wrong, and after you've spent all that effort convincing one single person, you see that two thousand more people have all picked up that piece of information and are using it and remembering it and all two thousand of them need to have the information removed from their minds. There is just not enough time, money, and motivation to fight such a huge battle.
And there is no particular person responsible for correcting the misinformation. If it were a bad rumor spread about a celebrity or something, that celebrity might feel motivated to go hunt down all the sources of the rumor and sue them for libel, or whatever. But no particular individual feels harmed by the fact that there is some logical process being used out there by millions of people which causes them to mistype themselves in one system or the other, so that they fail to read the right descriptions of their type, fail to feel understood in the description of their type, fail to know which quadra they're in in socionics, and just generally feel frustrated about personality types.
I know how it feels to be mistyped for a really long time, and to somehow know that personality types really exist, if only I could figure out which one I am, and to be obsessed with figuring it out for years and years, never really feeling comfortable with the type I had. I was trying to understand myself, trying to figure out which kind of job I should do for a living, trying to figure out which kind of man I should marry to give myself the greatest chance of a successful marriage, trying to understand what was 'wrong' with me and why I didn't feel like I fit in in the world, trying to understand why I didn't get along with certain types of people.
I believe it's extremely important to type people correctly the first time, every time, with zero mistakes, but I don't have a method of doing that. If I had my way, the public schools would type children at an early age so that they could be understood better through all their years of school, but then, there are a million things I would want the public schools to do if anybody cared what I wanted. Since the school system doesn't care how I want things done, I say, screw it, just homeschool your kids, because public schools will never do anything you wish they would do.
The point is, there will never be a universal system that types everybody, but if there were, we'd have to get rid of the flipping the j/p for introverts thing, throw that whole method in the garbage, and burn all the books that mention it. It's an information virus. (And speaking of viruses, yes, I myself have a horrible, monstrous flu right now, which is probably why I'm so grouchy.

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Edit: adding more rant, because not enough rant already: I like Bukalov's Model B even though I don't have an English translation of it. Basically it says that you have two information elements in each box, one extraverted and one introverted, with a plus and minus sign. So an ILI would be +Ni/-Ne, -Te/+Ti, +Si/-Se, -Fe/+Fi (in its verbal ring). I like that model because it gives introverts leeway to use some kind of extraverted function sometimes, and extraverts some leeway to use an introverted function sometimes, at a four-dimensional level, as their base function. It also explains why SEEs get mistyped as EIEs - the SEE is using +Fe, and it's *allowed* to use +Fe (+Se/-Si base function, -Fi/+Fe creative function...).
Model B also explains why there could ever be any attraction at all in supervision relations, the type of relation recommended as the most desirable one for romantic relationships in the Myers-Briggs, except it's a one-sided supervision - they say 'Keep sensors together, and intuitives together,' so you have ISTP and ESFJ shown as what kind of relationship I myself am supposed to be in. I do have *some* attraction to Alpha SFs in real life, and Model B explains that, because Alpha SFs have the same kind of -Fe/+Fi that Deltas like. Since I can't get an English translation, I don't know the exact reason why they have it in the order they have it in, and all the details of how the model works. So, Model B explains the intertype relations differently between different quadras. I think it's actually even more complicated than I'm making it out to be, but I can't read about how it all works right now.
Okay, end of rant, nobody is going to read this anyway, I shouldn't even be in the Gamma Quadra stirring up trouble, sorry for jumping into the wrong thread.