There is a famous argument by Blaise Pascal, a historical scientist and theologist, intended to convert atheists and agnostics into belief by ways of a wager.
He asserts that you have two choices: belief in God or no belief in God. If you believe in God you will be gifted with endless pleasures in the afterlife in heaven. If you don't believe in God then you have only the pleasures in the finite world around you. Believing in God, he claims is the most rational choice because the payoff is so great and the loss is only finite (loss due to being pious and maintaining that lifestyle). Therefore the best choice for a rational individual would be to believe in God.
However, the flaws in this argument are great, if not immediately apparent. First of all:
It is not an argument for the existence of God.
The whole argument is about the potential payoff for either choice and not about the likelihood of either outcome. In this case, I think some of his basic assumptions are true for only some people out there.
First of all: He assumes that pleasure is the only goal or value of a rational individual by including no other values in his wager. However, I personally value the freedom of thought and the pursuit of truth over a potential paradise. If I were to believe in God simply to get the end 'payoff', then I would be blindly believing and not seeking truth or allowing myself freedom of thought. If I got the 'payoff' in the end, I would still know that I had blindly followed an idea for selfish reasons and that would be a sort of self-betrayal.
Second of all: He assumes that it is possible for all people to simply choose their beliefs. My mind doesn't work that way. When I was a child I went through an existential crisis that I couldn't handle yet, so I desperately tried to believe in God. It didn't work; I couldn't just choose to believe, all I could do was pretend to others that I did. My beliefs are based on my experiences and interactions with the world around me and though I allow myself a modicum of doubt on everything, doubt does not equal belief.
Third of all: One still has to make a choice between all the possible Gods out there in the world of religion. All the Gods who would really care if I believed in them or not have personalities I cannot abide (since that very trait is attention-seeking) and their techniques for dealing with atheists tend to be brutal (another trait I cannot stand) and, even if I would regret it later, I could not stand to bow to them. The 'nice-guy' gods, if they turned out to be real, wouldn't care that I hadn't believed in them and some of them would congratulate me for having made the rational conclusion with the information I had available.
As logical fallacies go "Pascal's Wager" includes an "Appeal to Consequences" (the consequence of not being admitted to heaven if it turned out to be real) and a "False Dilemma" (there are many choices because many of the other potential God's would be angry if you believed in the 'wrong one', thereby making the liklihood that one's belief in God would payoff infinitesimally smaller).
Resources:
Stephen's Guide to Logical Fallacies
What is Pascal's Wager?
The Argument from Pascal's Wager
Classic argument. You pegged the fallacies.
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