In order to believe X falsely, one has to construct a plausible-ish world where X is the case. This distorted-world construction can happen piecemeal, in a sort of auto-sum-threshold attack.
In other words, suppose you want to rationalize your belief in X. You could simply abdicate all logic, and assert X while also asserting everything else that comes from your ordinary truth-seeking beliefs. (Well, that's not simple, because what do we mean by believing in X, then? Something about actions? I'll leave this as an open question here.)
However, that method is kinda hard sometimes. For one thing, if you are trying to truth-seek somewhat, you might follow the rule of believing logical implications of your beliefs. That could be out of pure ethics, or it could be for instrumental reasons. So you would very quickly run into contradictions. For another thing, you'd look silly, and have a hard time mind-melding with people on X-related things.
Better might be to construct a "logical counterfactual" where X is the case. You construct a plausible world nearest to our own, but where X is the case. That world may be logically / conceptually / strategically incoherent, but it could be less incoherent.
X is false, and X is related to many things that you know about. So there are many logical rends to mend.
This can happen piecemeal. You don't have to construct the whole X world in one go. Indeed, doing so would draw a lot of attention, and in particular self-attention. If you try to mend a bunch of rends all in one go, where they all relate to your believe in X and would all be solved by just believing not-X, you might strongly tend to just make that update.
Instead you can do it piecemeal. You mend one rend today, one tomorrow. Any given contradiction can be smoothed over
* by coming up with just slightly implausible explanations,
* by just slightly questioning some logical links,
* by just slightly revising your concepts about X to take a little dent from the contradi