Heck, I think detecting sarcasm would be an easier goal, and still tricky.
Well that's actually good news. With a large enough labelled dataset of actually-sound and fallacious text with similar grammatical features you should be able to train a discriminator to distinguish between them using some other metric. Good luck with getting that data set though.
Not when the better metrics are likely alien/incompatible to the discriminator's core algorithm!
Then it's rather inconvenient news, because it means you have to develop something separate and novel.
As the other poster already mentioned, if we can't even get them to reliably count how many objects are being referred to, how do you expect them to also handle logical syllogisms?
Remember NP is equivalent to second order logic with existential quantified. E.g. for any X there exists a Y
And that only gets you to truthy Trues, co-NP is another problem.
ATP is hard, and while we get lucky with some constrained problems like type inference, which is pathological in its runtime, but decidable, Pressburger arithmetic is the highest form we know is decidable.
It is a large reason CS uses science and falsification vs proofs.
Gödel and the difference between Symantec and syntactic completeness is another rat hole.