Use Bad Logic as the Villain's Character Flaw
Logic is an abstract concept. Bad logic is a character flaw.
In most characters, character flaws lead to conflict and growth. They influence the outcomes of that character's actions, and a character might reflect on them or change because of them. They help readers empathise as characters struggle.
This matters because a reader won't put up with an egregious character flaw no matter what kind of flaw it is. You don't read many books with characters as racist as H.P. Lovecraft's any more. Flawed logic that's so bad a reader can't put up with it isn't just bad logic: it's bad characterisation.
The Villain is Committed to Bad Logic
You've mentioned this is specifically about a villain, which introduces one complexity. To quote an article by Stephen Pressfield, "the villain never changes". If they changed, they'd be the hero. That means we spend a lot of time with the villain's logic.
Normally, a character with a flaw like bad logic would cause some problems for themselves. Their assumptions would cause conflict, such as having a plan go awry because what actually happened didn't match their assumptions. One might even be able to say that heroes survive mostly because the villain failed to think of everything. :)
But, since this character is a villain, no matter what the results of their flawed logic are, they should not see the problem as being with their logic.
Basically, villains don't change, they double down. Jafar's plan in Aladdin to make Jasmine fall in love with him failed not because it was impossible in the first place, but because he didn't have ultimate cosmic power. We as an audience know that his logic is wrong, but there's nothing trite about the way his logic appears.
So how do you present bad logic as something a character can be committed to?
How is this Flaw Reasonable?
Bad logic is just like any other flaw: a character is the way they are because being that way makes sense to them. The character should see their flaw as reasonable or even necessary. This is what internal consistency means, and it's absolutely required for us as readers to empathise with the villain.
In particular with logic, even if we as readers can see that the villain isn't thinking correctly, we should be able to say "I wouldn't do that" and not "Nobody would do that." For logic, that means agreeing with his premises and disagreeing with his conclusion (see examples below), as well as appreciating the thought that went into arriving at his conclusion.
In fact, you rarely want a character to come off only as illogical. Illogical characters are ones that readers don't understand and don't empathise with, and are more commonly comedic. Find something that's close but more emotionally charged: When Billy in Shazam goes to find his mother and expects to be a family again, we don't think he's illogical, we think he's naive (and it warms our hearts).
Flaws should help us as readers empathise with the character, not isolate us from them.
Tying it Back to Motives
Logic and motives are completely separate. Jafar is illogical because he believes he can accomplish something that's impossible; he's evil because he's motivated by conquest of Jasmine instead of love. The best villains want something universal but reason about it in abhorrent ways, such as how Thanos from the Avengers movies wants sustainability at all costs.
Evil isn't dumb, it's corrupt.
Bad logic that the reader can see is bad logic can be used for comedy or villainy, but only if the character with bad logic is completely unaware of how bad his logic really is, he refuses to change even when his bad logic is pointed out, and we as readers can believe it even if we don't agree with it.
The Tragic Approach
A villain railing at the limits of their understanding is a powerfully moving thing. Yet, he cannot succeed.
See the first two and a half minutes of this video from Fullmetal Alchemist (Brotherhood): "What more could I have done?"
Note that it isn't even necessary for us to agree with the author's idea of logic - it's possible to view this exchange as cruel, for example, even if we don't agree with the villain either. The important part is how the world itself works, and the statement the author wants to make with their view of what's logical.
The Comedic Approach
With comedy, the result of bad logic has to be relatively benign. Otherwise, a character who is unaware of their affects on others is calloused, not funny.
See this scene from the Big Bang Theory: "I accept your premise, I reject your conclusion."
Bad logic is an excellent flaw that can be used to emphasise willing blindness of consequences. Bad logic contrasts the evil things a villain values with the good things that the protagonist values, rather than separating the villain from reality (and the reader). Bad logic is logic we disagree with rather than logic that misses a few steps.
Show us what your character values by what they're willing to ignore.