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When the reasoner begins with what they are trying to end with. Example: 'We know that God exists because the Bible says so, and we know that the Bible is accurate because it was inspired by God.'

Introducing an irrelevant topic to divert attention from the issue at hand. Example: 'We can’t be worrying about the validity of this mathematical proof when we don’t even know if there are extraterrestrial beings.'

Arguing that a claim is true just because a valid authority or expert on the issue said it was true, without other supporting evidence. Example: 'Well-known mathematician John Doe says that this is the best approach to solve the problem, so it must be.'

Assuming something is true or correct because everyone believes it. Example: 'People long believed the earth was flat, so it must have been a reasonable assumption.'

Presenting two opposing options as the only possibilities, when in fact more possibilities exist. Example: 'You need to decide whether you worship Pythagoras as the greatest mathematician, or you reject math entirely.'

Drawing a general conclusion from a small or unrepresentative sample. Example: 'All odd numbers are prime. Just look at 3, 5, and 7.'

Presuming that a real or perceived relationship between things means that one is the cause of the other. Example: 'Every time I wear my lucky shoes, I solve mathematical problems more efficiently. Thus, the shoes must enhance my mathematical abilities.'

Using double meanings or ambiguities of language to mislead or misrepresent the truth. Example: 'The sign in the park says 'Fine for littering.' So, it must be fine to litter here, as in, no problem.'

Arguing that a lack of evidence proves something. Example: 'No one can prove that the Hilbert's problems are all solvable, therefore they must be unsolvable.'

Assuming a relatively small first step will lead to a chain of related events culminating in some significant effect. Example: 'If we allow students to use calculators, next they'll want to use computers, and before you know it, they'll forget how to think for themselves.'

Misrepresenting someone’s argument to make it easier to attack. Example: 'Professor Smith says that we need to read more Plato. What a ridiculous idea! We shouldn’t be giving students homework at all.'