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I’d be interested to see what an example of constructing a logical argument that proves something that isn’t true. In my experience the logic is sound, but the assumptions going into it (often unstated assumptions) are garbage. The other thing I see ALL the time is people using inductive as opposed to deductive reasoning to prove that something is “true”.
When I mentioned the model not matching reality, that's basically the same thing as saying your assumptions are incorrect. That's the fatal flaw of logic: you can't really use it to model the real world without making tons of assumptions about how the world works. For instance, technically speaking, science as a whole is illogical. If 9 people who jump off a roof fall to their death, there is no purely logical reason that the 10th person to jump won't fly instead.
As to "an example of constructing a logical argument that proves something that isn’t true" I'm going to be a bit pedantic here and point out that I said
And draw attention to those quotes. You can construct a logical proof, but that doesn't mean that actually says anything about the world. What I'm meaning is when someone uses a logical proof to claim something untrue is true.
Part of developing a logic an argument is stating your assumptions.
Also, the example you pointed out is actually something I see all the time. You’ve tried to draw a deductive conclusion by using inductive logic. This is why we need ACTUAL logic training in schools. Most schools don’t even teach the difference.