## Thursday, August 10, 2023

### Fallacies of Diagrammatic Reasoning

In recent decades there has been an increasing recognition that we can reason directly with diagrams -- that the role of diagrams in reasoning need not merely be illustrative of some kind of non-diagrammatic inference but may also carry some genuine weight in the reasoning itself. (This is an idea that has gone in and out of style; the last time it had widespread acceptance was in the nineteenth century.) If we assume this, however, and recognize that we can go wrong in diagrammatic reasoning, it follows that there will be fallacies of diagrammatic reasoning. What are the fallacies that can arise directly in diagrammatic reasoning? I can think of at least four.

(1) Improper Form: A diagram may be irrelevant. Your diagram, as used in reasoning, may mischaracterize how different kinds of information used in the reasoning are related to each other, so that it mischaracterizes the situation; in this sense, the diagram is just the wrong diagram, however much it might look like it is right. Every kind of reasoning allows for some kind of fallacy of irrelevance, so it's unsurprising to find such a fallacy of irrelevance.

(2) Smuggled Assumption: When people attempt to come up with systems of diagrammatic reasoning, like Venn diagrams, or Lewis Carrol's literal diagrams, or Peirce's existential graphs, a great deal of  effort is put into constraining how a diagram is used so that the diagram does not add assumptions extraneous to reasoning. Diagrams are potentially very information-rich; this means it is easy for diagrammatic reasoning to add in an assumption that should not have been made. In geometrical diagrams, it is very easy to assume that you have shown that lines or arcs intersect when all you've actually done is show that your diagram doesn't rule out that they do. This is a smuggled assumption -- it looks like they intersect in your diagram, so you assume that they do, even though it may be the case that they just pass arbitrarily close to each other without intersecting. In this fallacy, an accidental feature of the diagram is treated as if it were non-accidental.

(3) Inconsistent Interpretation: When used in reasoning, one may end up giving more than one distinct interpretation to some diagrammatic feature. This means that it is possible for diagrammatic reasoning to have a kind of fallacy of equivocation.

(4) Incomplete Diagram: A diagram may not represent all the information that its use in reasoning requires it to represent. This is the opposite of the Smuggle Assumption fallacy; in that fallacy, the diagram is introducing assumptions that shouldn't be there, in this fallacy, the diagram is failing to introduce assumptions that should be there.

Beyond thinking of obvious cases of diagrammatic reasoning, it's worth noting that it's unsurprising that the kinds of fallacies you find in diagrammatic reasoning include versions of fallacies that we find in syllogistic reasoning, and it highlights something that is easily missed if you just assume that diagrams are mere illustrations: syllogistic systems are simultaneously verbal and diagrammatic. Indeed, this accounts for several features of Aristotle's syllogistic system, most notably those having to do with syllogistic figure.

If we look at a simple Barbara syllogism, which is First Figure, we explicitly locate subjects and predicates spatially:

All M is P.
All S is M.
-------------
All S is P.

If we change the locations around, we get a different (and invalid) argument; we have improper form, and are using (say) a Third Figure syllogism to do First Figure work.

Aristotle's actual syllogistic puts a huge amount of emphasis on converting Second Figure and Third Figure to First Figure. Later logicians added Fourth Figure; the primary reason for adding it is to account for all diagrammatic representations of subject and predicate, but medieval logicians also put considerable emphasis on transforming other figures to First Figure. This makes complete sense in diagrammatic terms; First Figure shows minor term and major term in the premises in the same spatial order as they are found in the conclusion, and this is a clear diagrammatic superiority of First Figure over the other figures.

In the nineteenth century, C. H. Hinton wrote a book, The Fourth Dimension, in which he claimed that other logicians had missed a valid syllogism in the Fourth Figure, IEO-4. He gave an example:

Some Americans are of African stock.
No one of African stock is Aryan.
Therefore Aryans do not include all Americans.

Hinton is right that this is valid, but wrong that it is Fourth Figure; he has misdrawn the diagram for a counterexample. It's actually Ferio (EIO-1). Hinton has confused the grammatical order of the conclusion with the logical order it should have in the syllogism; the conclusion is an O proposition, but it means "Some Americans are not Aryans", not "Some Aryans are not Americans", which would be required for it to be Fourth Figure. Thus he has treated an accidental feature of the words as a substantial element of the reasoning, and committed the fallacy of Smuggled Assumption.