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Logic and rhetoric
Morton's fork is the term given to a tricky bit of sophistry in which a conclusion is drawn in several different ways that contradict each other. It is named after Cardinal John Morton (Henry VII of England's sidekick). Morton went around England "persuading" people to give money to the government. If they were lavish in their spending it was proof they were rich and could afford to pay what was asked, if they were less forthcoming, it was evidence that they must have a great deal of savings from living such an austere lifestyle, and hence could likewise afford to pay, although the irony at the time was that more likely they actually did not even have the money to save, and hence could likewise not actually afford to pay.
Variations of this are often used by creationists, who will use any observation, as well as its inverse, as "proof" of Biblical literalism. This is also a favorite of conspiracy theorists, who believe evidence against the conspiracy is actually evidence for the conspiracy because those perpetrating the conspiracy are obviously covering up the necessary evidence.
This fallacy can be neatly summed up by the phrase "Heads, I win; Tails, you lose."
Fine Tuning Argument
The Fine Tuning Argument attempts to prove the existence of a god by showing that the likelihood of a universe arising in which our life is supported is so small that our existence in a universe that supports us shows something must have created the universe for us. Essentially it states that because we are in a universe which can support our life, but it seems likely that most "possible" universes could not support our life, a god must have ensured this universe supports our life.
However, if we existed in any of the multitude of "possible" universes that clearly shouldn't support our life, this would then become evidence that some god had somehow protected us so we formed despite all possibilities, rather than the universe forming to support us despite all the possibilities. So either a god exists because we are in one of the few and rare "possible" universes that can support our life without a god, or a god exists because we are in one of the many universes that can't support our life without a god.
During many periods of history, including the Salem Witch Trials, the European witch-hunts, and the McCarthy Era, the question has been posed to anyone presumed guilty of a crime: if they did confess guilty, then they would be relieved of their belongings (which would, more often than not, end up with their accuser) and set free to die on their own, whereas if they declared themselves innocent, they would be tortured to death, then relieved of their belongings, as a truly guilty person would never confess. This was later applied in Arthur Miller's play, The Crucible, which helped to end the activity in those times.