by nonsense-argument I mean a line of argument that is based on either wrong assumptions or wrong inferences (or both), and this wrongness can be easily demonstrated, based on common-knowledge. See linguistics blatant nonsense for examples.
It may seem that nonsense-arguments are not a problem, because they can be brushed aside easily. However, there is very large number of possible nonsense-arguments. Thus when a person uses a nonsense-argument to support their position, once you prove this is wrong, they can always find another argument to support their position. Since the number of arguments that can be discussed in any kind of setting is finite and actually quite small, a person that uses nonsense-arguments will never run out of arguments. Thus when a person holds a wrong position and supports it with nonsense-arguments, they can never be convinced that their position is wrong (see Trying to publish the Brain-symbols paper for an extended example).
This problem is much more widespread than most people (or maybe even all people) are aware of. In particular, in the field of human thinking it is almost universal. In general, you cannot assume that anybody is free from using nonsense-arguments, including yourself.
Additional problem is that while people don't like finding that the position they hold is wrong, they hate much more finding that their arguments are nonsense. Thus, when a person supports a position by a nonsense-argument, they will be even more reluctant than usual to admit that their position wrong. They will also find any person that points to their nonsense-arguments irritating. This causes most of people to avoid pointing to other people when they use nonsense-arguments.
When a person holds a sound position, they can avoid the unpleasant situation of having to admit a nonsense-argument by using sound arguments to support their position. Thus, normally we should not expect a person to defend a sound position by nonsense-arguments. Hence, when a person uses nonsense-arguments it is not only not supporting their position, it is actually suggesting that their position is wrong.
If you are not actually interested in understanding the subject of discussion, then there is no problem. However, if you are interested, you should monitor yourself to see if you make nonsense-arguments. When you find that you have used a nonsense-argument (i.e. somebody refuted your argument with a short counter-argument), you should avoid doing the common thing, which is either changing the subject or coming with a new argument. Instead, you should re-think what you know about the subject, assuming that your position (the one you were trying to defend) is wrong. The point is that once you have defended your position (which is probably wrong) by a nonsense-argument, you will find it even more difficult to find that it is wrong in the future, and will hinder your understanding of the subject.
As explained above, once a person uses a nonsense-argument, they are unlikely to be ever convinced that their position is wrong. In principle, you may try to point to the nonsenseness of their argument, and hope that they will do the right thing as in the previous section. In practice, however, it is more likely that they will either pick another nonsense-argument or change the subject, and the only effect you will have is irritating them. As mentioned above, most of people already worked out the latter point, and avoid pointing other people's nonsense-arguments.
It should be noted that if you think that you have spotted a nonsense-argument, it is not necessarily because the other person used a nonsense-argument. It may also be because:
If you avoid pointing the nonsense-argument, or you do and the person you discuss with picks up a new nonsense-argument or changes the subject, then the discussion becomes useless. You are not going to learn anything useful from the other person, because their position is probably wrong, and they are not going to learn anything from you, because they are going to defend their position. Of course, there may be reasons why you will want to continue the discussion (politeness, you want to get the other person to do something, you are arguing for an audience), but it is better described as a 'psychological wrestling' rather than a discussion.
While the fact that a person uses a nonsense-argument strongly suggests that their position is wrong, it gives only very very weak support to your position. That is because yours and the other person's position are not the only possible positions, so even after the other person's position is eliminated, there are competing positions.
As discussed above, that suggests that the position this person holds is unsound. This suggestion becomes stronger when the person knows a lot about the subject, because in this case they would have found it easier to find the sound arguments, if there were any. The latter point may seem paradoxical, but that is only if you assume that knowledgeable people don't hold wrong positions.
It should be noted that because in this case the person does not have a chance to correct or defend themselves, you need to be more careful in checking that it is really a nonsense-argument. On the other hand, it is important that you take into account the possibility that the person uses a nonsense-argument, no matter what their authority is, because otherwise you will be misled by the Blatant nonsense effect.
As discussed above, it is not actually possible to show that somebody that uses nonsense-arguments is holding a wrong position, because they can always come with new nonsense-arguments. What is possible (in a discussion that is for the sake of other people) is to show that they use nonsense-arguments. Thus when arguing with somebody that uses nonsense-arguments, it is important not only to show that their current argument is wrong, but also to show that it is nonsense.
That is a problem because people regard it as impolite, and most of them don't understand the distinction between saying that somebody's position is wrong and between saying that their arguments are nonsense. As a result, most of people cannot be convinced that somebody that talks nonsense is holding a wrong position, unless they already figured it out themselves. This, with the blatant nonsense effect, explains how so many nonsensical theories have such a strong following.