1) A fallacy is a form of argument but importantly a bad argument. It is an argument where the premises do not support the conclusion to the needed degree. Thus, a fallacy not simply a matter of being wrong about certain facts - say for example I believe premise 1: the car is on fire to be true, whereas it is in fact false. Rather, a fallacy is in general a form of error in reasoning. Hence, what determines whether an argument is fallacious or not is whether the premises support the conclusion to the needed degree. If it is a valid inductive argument the premises must make the conclusion make likely. In a valid deductive argument on the other hand it follows that if the premises are true the conclusion must also be true.
2) There are numerous forms of fallacies, from the blatantly obvious to the very sophisticated. The most common form of fallacy is probably the so-called ad hominem argument. An example of the ad hominem argument could be the following: A right-wing politician says that he doesn't believe in 'global warming' and that we should rather rely on oil as our energy resources. Now we might think that the only reason for this right-wing politician to have these opinions is because he has financial interests in the oil industry and thus we might argue: 'You are only saying this because you want protect your own financial interests rather actually evaluating the validity of the scientific theories concerning global warming'. However, even though we might be in fact right about this, it would be a fallacious argument. The fact that he has financial interests in the oil industry is a right-wing politician and so forth does not support the conclusion that his opinions about global warming will necessary follow from these facts about him.
3) Now why would anybody deliberately use fallacious arguments, that is, why would anybody use what we know as a bad argument. Well, as the above example shows, even though it is easy to understand why the argument is fallacious, the argument might still very likely have the effect of arising suspicion concerning the right-wing politician's motives. Political debates are therefore often endless row of fallacious arguments purely questioning opponents' motives, characters, specific circumstances and so forth. And the reason why they use these bad arguments is simply that it works; it shifts the public's attention from the real and often complicated political issues to the much easier but often quite irrelevant questions of who is good and who is bad.
4) The slippery slope arguments are very influential in modern controversial ethical and political debates. An example could be debates concerning freedom of speech. Some liberals might for example fear that any limitation of free speech is the beginning of a slippery slope to tyranny and censorship, while others might say that if we do not heavily censure speech, no matter how reasonable the speech, it could incite to public disturbances, and that with uninhibited free speech we shall soon end in a Hobbesian state of nature, where life as we know it from Leviathan is 'solitary, poore, nasty, brutish, and short'. Another usage of the argument that is very common is in bio-ethical debates, where one side wishes to engage in some form of say gene-experiment and the other side objects by saying that this is the beginning of a slippery slope to say human clones - just put in your favourite horror scenario. However, though this fallacy is very common, it is in fact a fallacy and that before painting too horrific consequences of our opponent's views, we should remind ourselves that there is always more than one slippery slope. Or to put the point in another way, you cannot get off the slippery slope altogether, the task is rather to decide where on slope to stay. This reminder seems especially pertinent in the debate over the free speech debate, where the one side claims that free speech is not a matter of degree and the other side holds that any kind of insult to say aholy religious figure should be prohibited. Thus, a slippery argument will paint a too black or white portrayal of the consequences of the opponents position, though these horrible consequences do not in fact follow from he or her holding this specific position.
5) A hypothetical syllogism is a valid argument of the form: If P then Q, If Q then R, Thus, If P then R. For example, if I do not wake up, I cannot go to work, I cannot go to work, I will not get paid, Thus, if I do not wake up, I will not get paid.

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