Basically, there's a realm of "Logic", which contains all the thoughts and ideas of humans within it. Even the most minute of thoughts, no matter how insignificant, actively influence this realm in countless incomprehensible ways. Each human has their own "Logic" (for example, Sally has "Sally Logic" and Billy has "Billy Logic") which defines who they are. Their abilities are influences by their personal "Logic", and the way their thoughts fluctuate influences how their abilities function based on the situation (For example: Jimmy believes that he can produce fire when he's in a state of calm. As such, he can. Whenever he's not calm, as in, angry or sad, his personal "Logic" that he applied to himself fluctuates, and he can no longer produce fire until he returns to a state where he believes he can do it, since he's made himself believe that he can only produce fire when he's calm).
Hmm, reminds me a lot of the Orks from the verse. Assuming you've seen the memes and the few ork pages of 40K in VSBW, they are capable of causing reality to shift based on what seems plausible. If I recall the best example of this are people pointing guns at an ork at close range and making gunshot noises and the orks thinking they get shot, end up suffering gunshot wounds somehow. It does have limits though, they can't just turn a log into an orbital lazer with belief for example.
The summary of
this page should give a decent explanation
I can see a lot of last-minute power-ups arising from this logic
I imagine there's gonna be this character that's super OP but has been gaslit into only being able to activate a portion of their power in extremely specific situation with the climax of the story involving said character realizing this ruse and instantly eliminating all the enemies or something.
Or a government that uses propaganda to convince their civilians to be powerless while the anarchists attempt to stop this and bring power back to the people.
Anyway this seems solid, are there any dangers to using this magic? Because something being really OP can be mitigated by certain crippling weaknesses. Like horrifying entities ruling the realm that may leak into the world when this happens, or this...
Essentially in a place where thoughts and belief are the source of power, there are certain people who would be extremely powerful. Like megalomaniacs and narcissists believing themselves to be better than everyone else would probably be capable of crazy feats. Someone who is insane may be able to cast a variety of very powerful spells due to a variety of reasons such as a different perception of reality, hallucinations, paranoia, etc.
A perception manipulator or illusionist may be able to change how reality looks and their victim's belief makes it possible like make them see scrap as powerful weapons or make a person think they have been depowered by ensuring they don't see their casted ability and its effects. A mind-controller can probably make their victims perform feats they normally can't assuming the controller isn't aware of the usual limitations of their victims.
And lastly, Social Influencing and Memory Manip are probably some of the strongest abilities in the verse for obvious reason