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A non sequitur (literally, does not follow) is a logical fallacy. Reasoning is said to be non sequitur if the conclusion does not follow from the premises or if a given reason for taking an action is completely irrelevant to taking that action. For example, the police chief's reasoning was a non sequitur when he defended consulting a psychic "to help investigators crack the case" based on the premise that "we tried everything else and haven't solved the case." The fact that the case hadn't been solved using traditional police methods is irrelevant to whether consulting a psychic is a method that should be used. The error in reasoning should become obvious if we substitute "pick a name randomly out of the phone book to identify the main suspect " for "consult a psychic." The fact that you haven't solved the case using traditional methods provides no support for trying a non-traditional method. To justify trying a non-traditional method, one needs direct evidence that the non-traditional method has some merit.
Another example of non sequitur reasoning comes from those who try to justify seeing a homeopath or some other "alternative" therapist because scientific medicine hasn't cured what ails them. Again, to justify using homeopathy or acupuncture, one needs direct evidence that those treatments are effective, not that some other method isn't effective.
One often finds that non sequitur reasoning given by those trying to justify hiring a psychic or going to an "alternative" therapist is accompanied by another fallacy: the questionable assumption. For example, "we had nothing to lose by consulting a psychic" or "I had nothing to lose by going to a homeopath" are questionable claims. You could be losing time and money that could be better spent. (One of the foundations of homeopathy, by the way, is based on non sequitur reasoning. Samuel Christian Friedrich Hahnemann (1755-1843), the founder of homeopathy, reasoned that if a substance produces symptoms similar to those produced by a disease, then that substance, in infinitesimally small amounts, will relieve the symptoms of the disease. Hahnemann's conclusion does not follow from his premise.)
Both police chiefs and sick people justify their fallacious reasoning by claiming that the psychic or the alternative therapist really helped. Non sequitur reasoning can have a true conclusion or lead to a beneficial outcome. Being fallacious means the way of getting to the conclusion is not justified but it does not necessarily mean that the conclusion is false. However, most police chiefs know little or nothing about cold reading or hot reading, and can be easily duped into thinking someone has gained information through psychic means. Most sick people know little or nothing about the placebo effect or natural regression, and, if they live to tell the tale, can be easily duped into thinking an "alternative" therapist cured them of their ailment.
The most egregious example of non sequitur reasoning in recent times comes from those who defend what they call "intelligent design" or ID. They commit what might be called the divine fallacy. Their reasoning looks something like this: we can't figure out how any natural process could account for the complexity of the human cell or the bacterial flagellum or a donkey's eye or some other complex biological entity; therefore, an intelligent designer must have put these parts together in the cell, etc. It doesn't follow from the fact that you can't find a naturalistic explanation that there isn't one. The history of science has many examples of processes we now understand but that at one time we had no explanation for.
For a good general introduction to fallacies I recommend Attacking Faulty Reasoning: A Practical Guide to Fallacy-Free Arguments by T. Edward Damer or Asking the Right Questions: A Guide to Critical Thinking by M. Neil Browne and Stuart M. Keeley.