Read this tutorial about how to construct an argument map. Argument maps are a way of visually representing the logical structure of an argument.
An (simple) argument is a set of one or more premise with a conclusion. A complex argument is a set of arguments with either overlapping premises or conclusions (or both). Complex arguments are very common because many issues and debates are complicated and involve extended reasoning. To understand complex arguments, we need to analyze the logical structure of the reasoning involved. Drawing a diagram can be very helpful.
§1. Argument Maps
An argument map is a diagram that captures the logical structure of a simple or complex argument. In the simplest possible case, we have a single premise supporting a single conclusion.
Consider this argument:
Death is inevitable. So life is meaningless.
This can be represented in an argument map as follows:
We can also use numbers to label the premises and conclusion. (1) = Life is meaningless, (2) = Death is inevitable:
Let us now look at another example:
(1) Paris is in France, and (2) France is in Europe. So obviously (3) Paris is in Europe.
Here is the corresponding argument map:
Note that the two premises are connected together before linking to the conclusion. This merging of the links indicate that the two premises are co-premises which work together in a single argument to support the conclusion. In other words, they do not provide independent reasons for accepting the conclusion. Without one of the premises, the other premise would fail to support the conclusion.
This should be contrasted with the following example where the premises are not co-premises. They provide independent reasons for supporting the conclusion:
 Smoking is unhealthy, since  it can cause cancer. Furthermore,  it also increases the chance of heart attacks and strokes.
Instead of writing the premises and the conclusion in full in the argument map, we can label them and write down their numbers instead:
This diagram tells us that  and  are independent reasons supporting . In other words, without ,  would still support , and without ,  would still support . (Although the argument is stronger with both premises.)
Finally, it is also possible to have a single reason giving rise to multiple conclusions:
 Gold is a metal.  So it conducts electricity.  It also conducts heat.
§2. More Complicated Examples
Now that we know the basics of argument maps, we can combine the templates we learn above to represent more complicated arguments, by following this procedure:
- Identify the most important or main conclusion(s) of the argument.
- Identify the premises used to support the conclusion(s). These are the premises of the main argument.
- If additional arguments have been given to support any of these premises, identify the premises of these additional arguments as well, and repeat this procedure.
- Label the premises and conclusions using numerals or letters.
- Write down the labels in a tree structure and draw arrows leading from sets of premises to the conclusions they support.
Let us try this out on this argument:
Po cannot come to the party because her scooter is broken. Dipsy also cannot come because he has to pick up his new hat. I did not invite the other Teletubbies, so no Teletubby will come up to the party.
We now label and reformulate the premises and the conclusions:
- Po cannot come to the party.
- Po's scooter is broken.
- Dipsy cannot come to the party.
- Dipsy has to pick up his new hat.
- I did not invite the other teletubbies.
- [Conclusion] No teletubby will come up to the party.
We can then draw the argument map like this:
This is an example of what we might call a multi-layered complex argument, where an intermediate conclusion is used as a premise in another argument. So  and  are the intermediate conclusions, which together with  lead to the main conclusion . This complex argument is therefore made up of three overlapping simple arguments in total. Of course, in this particular case you can understand the argument perfectly well without using this diagram. But with more complicated arguments, a picture can be an indispensable aid.
Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/arg/complex.php
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