To deem something “necessary,” it absolutely must occur in order for some order or process to continue functioning. For example, in order for there to be humans, it is necessary that there was a fusion of an egg and a sperm (barring unusual scientific concepts beyond my comprehensive capabilities). There is no scenario where this isn’t true. It was a necessary step.
For there to be a scenario where an argument is necessary, therefore, it must allow for a function that is, itself, necessary.
The trouble with analyzing it from this perspective is that the necessity of the “function” is highly subjective. Assuming that an argument is a confrontation between two ideologies (whose persons are both trying to persuade one another), the contrasting viewpoints can vary from purely ideological principles like discouraging racism, to tangible moments like whether or not to eat a pizza.
Determining whether there are “necessary arguments,” means determining whether or not there is a function that would be inhibited by the absence of an argument.
For some, the inability to choose what they want is an inhibited function. The function of eating a pizza is only possible after arguing with the powers that are restricting this action. In this case, the argument is necessary for the pizza-eater.
For others, the inability to live is an inhibited function. Arguing that capital punishment is wrong is necessary for those on death row, whose function of living depends on the outcome of the argument. In this case, the argument is necessary for the prisoner.
In this sense, the subjective nature of values means there are infinite instances where an argument is necessary for someone.
Yet, I cannot dismiss the thought that, in a grander sense, arguments are never necessary. There are no instances where an argument would allow for a functioning that affected everyone. Until there is, I cannot say for certain that an argument is ever truly necessary.
In order for something to be a “mistake,” a set of internal, intellectual calculations must lead a process to an actual outcome that differs from an expected outcome. For example, making a mistake in math usually involves a false step in calculations that lead to an answer that is different from the actual answer. In another example, making a mistake at a party is leaving your coat on a chair and expecting it to be there at the end of the night, when it isn’t.
For an argument to be a mistake, the argument must take a form that leads to a different outcome than was expected. And using the previously mentioned definition of argument – that it is the clash of two ideologies that both seek an end inhibited by the other – we can consider moments where arguments caused unexpected outcomes. For example, arguing that you are hungry and therefore you should be able to eat the pizza may result in being fed spinach, which, though delicious, is not the expected outcome. This argument was a mistake and actually led the subject away from the expected and desired outcome. In fact, the arguer has given up his strongest position – that eating pizza is a representation of an ability to choose– which he/she may consider a necessary function, and thus it is a necessary argument.
There are instances where an argument may be necessary and a mistake. However, it is not the argument itself (the clashing of inhibited ideologies) that is a mistake, it is the form that the argument takes. If an argument is necessary due to the prior definitions, then for it to be a mistake, it must only lead to an unexpected outcome. Therefore, when he/she wants pizza, and knows that the ability to choose pizza is representative of a function that cannot continue to exist without the argument, then arguing in a poor form can make the argument a mistake. It does not mean that the need to argue it, or the execution of the argument is, itself, a mistake.