How to tell if a Non sequitur does or does not follow?

In argumentation, something is said to "follow" if a) it is the inevitable conclusion of a deductive argument, b) it is causally linked or if the evidence warrants the conclusion. The first option requires common sense. The second requires a good understanding of causality or a good understanding of how evidence works . Consider

If pigs fly, then they must have wings.
Pigs fly.
Therefore, pigs have wings.

Common sense is required here since this is a deductive argument with a necessary conclusion. The non-sequitur would be any other conclusion.

I drank 6 cups of coffee at midnight.
I couldn't get to bed until 4 am.
The coffee kept me up.

Here we need to use reason to conclude that the coffee (or something in it) was responsible (causally linked) to insomnia. If we concluded that coffee was made in China, there is no causal link to establish this given the argument. The question, "How does the fact that you drank 6 cups of coffee and you couldn't go to sleep lead you to the conclusion that the coffee was made in China??" They have the burden of proof to establish this link.

Hi, Jason!

         I will approach your question with deductive arguments in mind. A conclusion that “follows” is just a conclusion of a valid argument. Accordingly, to know that a conclusion “follows” is to know that the argument in which it figures is valid. This implies the answer to your question about methodology: methods for determining that a conclusion “follows” are really methods for determining that the argument for it is valid. 

How can one convince them that it does not follow if they belief it does follow? 

         An argument is valid just in case it is impossible for all the premises to be true and the conclusion false. Some arguments are valid because of their form , while some arguments are valid but not in virtue of their form. 

         Symbolic logic provides powerful tools for discovering whether an argument is invalid in virtue of its form . Symbolic logic involves symbolizing the argument into a formal language and using certain procedures to test whether that argument-form is invalid. For instance, a truth-table can be used to evaluate arguments that are symbolized into the formal language of truth-functional logic. Also, instead of using truth-tables you can can use a tableau test. A tableau test is like a vertical flow chart version of a truth-table. In other formal languages, such as predicate logic, it works to instead construct an interpretation of the argument-form in which the premises come out true and the conclusion false. 

         Outside of using formal languages, an argument valid or invalid in virtue of its form can still be evaluated. You may describe a detailed logically possible story in which the premises of the argument are true but the conclusion false, in order to show that the argument is invalid. This is called giving a counterexample. Not all arguments are valid in virtue of their form, anyways. So, learning to construct counterexamples, rather than purely formal methods, is important in general.

         Let me leave you with some introductory resources to help. In my undergrad, I learned symbolic logic first from “Brain Power” written by Leah Savion (she was my professor, too), which may be for sale on Amazon. Also, “For All x: Calgary Remix: An Introduction to Formal Logic” by Magnus and Button is more advanced, comprehensive and in-depth.

Thank you, Jason. 

From, Kaiden

By adhering to simple if-->then logic, it's easy to avoid an invalid conclusion, or non sequitur. Often a bias attributable a religious belief can can be responsible for the non-sequitur.

Example of an argument which is at best dubious, but the conclusion is still valid  : 

[eg if you have "Conspiracy Theory Syndrome"]

P1: the COVID vaccine is actually a plot to control your brain.
P2: I do not want anyone to control my brain.
C : therefore, I should avoid the COVID vaccine.

Another example of an argument which is at best dubious, and the conclusion is a non-sequitur  : 

P1: I exist
P2: The universe exists
P3: bunnies are lovely
C: therefore, Jesus Christ

No, it’s a process of informed ratiocination. Sadly there are many who cannot fulfill that simple task but that is more of a failure of cognition and disability than fallacy.