Here is a description of complex argument diagramming from Matthew Knachel.
From: Knachel, Matthew, "Fundamental Methods of Logic" (2017).
Philosophy Faculty Books. 1. http://dc.uwm.edu/phil_facbooks/1
Creative Commons Attribution 4.0 International License
The arguments we’ve looked at thus far have been quite short—only two or three premises. But of course some arguments are longer than that. Some are much longer. It may prove instructive, at this point, to tackle one of these longer bits of reasoning. It comes from the (fictional) master of analytical deductive reasoning, Sherlock Holmes. The following passage is from the first Holmes story—A Study in Scarlet, one of the few novels Arthur Conan Doyle wrote about his most famous character—and it’s a bit of early dialogue that takes place shortly after Holmes and his longtime associate Dr. Watson meet for the first time. At that first meeting, Holmes did his typical Holmes-y thing, where he takes a quick glance at a person and then immediately makes some startling inference about them, stating some fact about them that it seems impossible he could have known. Here they are—Holmes and Watson—talking about it a day or two later. Holmes is the first to speak:
“Observation with me is second nature. You appeared to be surprised when I told you, on our first meeting, that you had come from Afghanistan.”
“You were told, no doubt.”
“Nothing of the sort. I knew you came from Afghanistan. From long habit the train of thoughts ran so swiftly through my mind, that I arrived at the conclusion without being conscious of intermediate steps. There were such steps, however. The train of reasoning ran, ‘Here is a gentleman of a medical type, but with the air of a military man. Clearly an army doctor, then. He has just come from the tropics, for his face is dark, and that is not the natural tint of his skin, for his wrists are fair. He has undergone hardship and sickness, as his haggard face says clearly. His left arm has been injured. He holds it in a stiff and unnatural manner. Where in the tropics could an English army doctor have seen much hardship and got his arm wounded? Clearly in Afghanistan.’ The whole train of thought did not occupy a second. I then remarked that you came from Afghanistan, and you were astonished.” (Also excerpted in Copi and Cohen, 2009, Introduction to Logic 13e, pp. 58 - 59.)
This is an extended inference, with lots of propositions leading to the conclusion that Watson had been in Afghanistan. Before we draw the diagram, let’s number the propositions involved in the argument:
1. Watson was in Afghanistan.
2. Watson is a medical man.
3. Watson is a military man.
4. Watson is an army doctor.
5. Watson has just come from the tropics.
6. Watson’s face is dark.
7. Watson’s skin is not naturally dark.
8. Watson’s wrists are fair.
9. Watson has undergone hardship and sickness.
10. Watson’s face is haggard.
11. Watson’s arm has been injured.
12. Watson holds his arm stiffly and unnaturally.
13. Only in Afghanistan could an English army doctor have been in the tropics, seen much hardship and got his arm wounded.
Lots of propositions, but they’re mostly straightforward, right from the text. We just had to do a bit of paraphrasing on the last one—Holmes asks a rhetorical question and answers it, the upshot of which is the general proposition in 13. We know that proposition 1 is our conclusion, so that goes at the bottom of the diagram. The best thing to do is to start there and work our way up. Our next question is: Which premise or premises support that conclusion most directly? What goes on the next level up on our diagram?
It seems fairly clear that proposition 13 belongs on that level. The question is whether it is alone there, with an arrow from 13 to 1, or whether it needs some help. The answer is that it needs help. This is the general/particular pattern we identified above. The conclusion is about a particular individual—Watson. Proposition 13 is entirely general (presumably Holmes knows this because he reads the paper and knows the disposition of Her Majesty’s troops throughout the Empire); it does not mention Watson. So proposition 13 needs help from other propositions that give us the relevant particulars about the individual, Watson. A number of conditions are laid out that a person must meet in order for us to conclude that they’ve been in Afghanistan: army doctor, being in the tropics, undergoing hardship, getting wounded. That Watson satisfies these conditions is asserted by, respectively, propositions 4, 5, 9, and 11. Those are the propositions that must work jointly with the general proposition 13 to give us our particular conclusion about Watson:
Next, we must figure out how what happens at the next level up. How are propositions 4, 5, 13, 9, and 11 justified? As we noted, the justification for 13 happens off-screen, as it were. Holmes is able to make that generalization because he follows the news and knows, presumably, that the only place in the British Empire where army troops are actively fighting in tropics is Afghanistan. The justification for the other propositions, however, is right there in the text.
Let’s take them one at a time. First, proposition 4: Watson is an army doctor. How does Holmes support this claim? With propositions 2 and 3, which tell us that Watson is a medical and a military man, respectively. This is another pattern we’ve identified: these two proposition jointly support 4, because they each provide half of what we need to get there. There are two parts to the claim in 4: army and doctor. 2 gives us the doctor part; 3 gives us the army part. 2 and 3 jointly support 4.
Skipping 5 (it’s a bit more involved), let’s turn to 9 and 11, which are easily dispatched. What’s the reason for believing 9, that Watson has suffered hardship? Go back to the passage. It’s his haggard face that testifies to his suffering. Proposition 10 supports 9. Now 11: what evidence do we have that Watson’s arm has been injured? Proposition 12: he holds it stiffly and unnaturally. 12 supports 11.
Finally, proposition 5: Watson was in the tropics. There are three propositions involved in supporting this one: 6, 7, and 8. Proposition 6 tells us Watson’s face is dark; 7 tells us that his skin isn’t naturally dark; 8 tells us his wrists are fair (light-colored skin). It’s tempting to think that 6 on its own—dark skin—supports the claim that he was in the tropics. But it does not. One can have dark skin and not visited the tropics, provided one’s skin is naturally dark. What tells us Watson has been in the tropics is that he has a tan—that his skin is dark and that’s not its natural tone. 6 and 7 jointly support 5. And how do we know Watson’s skin isn’t naturally dark? By checking his wrists, which are fair: proposition 8 supports 7.
So this is our final diagram:
And there we go. An apparently unwieldy passage—thirteen propositions!—turns out not to be so bad. The lesson is that we must go step by step: start by identifying the conclusion, then ask which proposition(s) most directly support it; from there, work back until all the propositions have been diagrammed. Every long argument is just composed out of smaller, easily analyzed inferences.