Calling something a “bullshit argument” is just a vulgar way of rejecting an argument. What is an argument? And what are the reasons for rejecting arguments?
An “argument” is a group of statements – two or more – of which one statement is allegedly supported by the other statement or statements. The allegedly supported statement is called the “conclusion” of the argument, and the allegedly supporting statement or statements are the “premise” or “premises” of the argument. I say “allegedly” because the conclusion may not, in fact, be supported by the premises. However, whether the premises do or do not support the conclusion, an argument exists because the premises are offered in support of the conclusion.
If the premises do not, in fact, support the conclusion, we can say that the argument is a bad argument, i.e., a bullshit argument.
Logic is the discipline which studies the nature and the goodness of arguments. There are very many good textbooks on logic, and I have nothing to add to what they cover.
However, in logic books written by philosophers, there are sections which are called “fallacies” which present a problem. The problem is this. On the one hand, a fallacy is simply a bad argument. In that case saying that the argument is unsound or uncogent should end the matter. [A deductive argument which is invalid or has a false premise is unsound. An inductive argument which is weak or has a false premise is uncogent.] But, on the other hand, various utterances are called fallacies which aren’t even arguments. So, it seems that under the category of “fallacies” is subsumed something other than arguments, and this needs explanation.
An excellent explanation was given by C. I. Hamblin in his book Fallacies. The gist of his explanation is that we have to understand fallacies from the standpoint of formal debates as practiced in antiquity and in the middle ages. Fallacies, from this perspective, are the fouls which can occur in formal debates.
An interesting collection of fallacies is found in W. Ward Fearnside and William B. Holther, Fallacy: A Counterfeit of Argument, 1959.