Reasoning allows to produce statements that are more likely to be true based on statements that are known to be true. You'd need to structure your "falsehood training data" in a specific way to allow an LLM to generalize as well as with the regular data (instead of memorizing noise). And then you'll get a reasoning model which remembers false premises.

You generate your text based on a "stochastic parrot" hypothesis with no post-validation it seems.