You can demonstrate illogical thinking in a persuasive text easily:
The point of a persuasive text is for them to make you believe or do something that they would like you to believe or do. So if they state something illogical than it should stand out as the illogical reference is most likely a main part of the subject or it wouldn't have been brought up they won't tend to think about things logically as in the science behind, or common sense wise they would have their own way of explaining that doesn't really agree with knowledge to say the least they would like think of it this way there are many illogical subjects like
- This is true because it hasn't been proven false - This has to be true because if it was false then that would be bad - This is true because everyone says its true - This has to be a bad idea because bad people believe in this idea - If this isn't completely true than its completely false - This has to be true because smart people say that its true - This happened after that so that MUST HAVE caused this
Look for these examples in your text and you should easily be able to tell when someone is speaking illogically