I have been spending a lot more time thinking about logic and math than I ever would have anticipated even as recently as five years ago. I’ve read books that were essentially about the square root of two, or how many points are on a line, and my current train reading is a primer on non-Euclidean geometry. Meanwhile, formal logic has become a staple of my teaching, both in the philosophy department and in the Shimer Great Books program. And who do I have to thank for all of this? Hegel.

During the pandemic, I was part of an intense reading group on Hegel’s Science of Logic. I have seldom read a book so hard — reading the section for the next meeting (often as little as ten pages) multiple times over, poring over every page during our long discussions, even poking at the German text. It is well-known to be among the most challenging texts in the Western philosophical canon, and I found it to be among the most rewarding as well. I haven’t spent much time with the text since then, but it often feels like everything I do — or at least everything new in my teaching and reading — is somehow related to it. I initially volunteered to teach logic in the philosophy department both to prepare for the Shimer course on logic and math and to get a handle on the syllogism, which is so central for Hegel. I have increasingly been teaching history of science courses, and I am drawn to non-reductionist accounts of the emergence of life out of physics and chemistry that echo the final passages of the Science of Logic. I don’t know when or even whether I will ever have a level of expertise needed to write credibly on these subjects, but my encounter with Hegel has profoundly shaped the way I look at the world.

It has also shaped the way I teach logic. As I have grown more confident and fluent in my syllogisms, I have tried to give my students a sense of how and why they work — something that neither the Copi/Cohen or Hurley textbook is much concerned with. They just list the valid forms and provide a dense proof (which they all but invite the reader to skip) as to how all of them fulfill the syllogistic rules, which are in turn simply presented as brute facts to be memorized. By contrast, I try to show how all the forms are generated out of the famous Barbara (AAA-1) form, and why the figures other than the first are more limited in their range of possible valid conclusions. (An initial effort at such an explanation can be found in this post.) I don’t want them to simply recognize that the EIO form is valid in all figures, I want them to know why.

At the end of the day, though, this knowledge inevitably collapses into a merely mechanical memorized list. Once I created a kind of algorithm for how to generate the 15 unconditionally valid (or Boolean) forms, the list became a permanent possession for me. To get there, I spent many a walk to the train pondering the mysteries of three overlapping categories — once I “had it,” I lost all patience for further reflection. It became boring, inert.

That tendency is what Hegel is constantly fighting against in the Science of Logic. He wants to show us how logic emerges into life, and he injects genuine drama into the whole affair. Yet as we progressed through the massive tome, the past sections inevitably became subject to rote memorization: first Being, then Nothing, then Becoming, then Existence….. Remember, Quality comes before Quantity! Many people, including no less an authority than Jameson, will tell you that all you “really need” is the Encyclopedia Logic, but in that text, I feel that Hegel has already succumbed to the tendency toward mechanism. Each step becomes something to be rattled off, without bothering so much with the connective tissue. Symptomatic here is the conversion of the “Remarks” — so energetic and tightly argued in the Science of Logic, beginning with scornful and even sarcastic attack on anyone who would deny that Being and Nothingness are the same — into the more disconnected “riffs” (my translation of the German Zusätze) in the Encyclopedia Logic‘s lecture notes.

I keep trying to forestall that moment by giving myself fresh challenges — for instance, trying to teach basic logic directly “out of” Aristotle’s original texts, which is… super hard. I also continue to delve more and more into the project of modern logic and its interpentration with mathematics. In part this is to help my teaching in the Shimer course, which requires me to find a way to help the students through the difficult territory of non-Euclidean geometry or Gödel’s Proof. But in part it’s also to confirm my intuition that the entire project is, in a very real sense, insane. And not only that, but the most fervently held desire — namely, to develop a completely rigorous, self-contained, and autonomous axiomatic system with no necessary grounding in intuition or any external reality — has been rigorously proven to be impossible on its own terms.

Hegel’s approach to logic is often regarded as self-evidently misguided in modern terms, and a whole literature has arisen trying to “save” Hegel through a reductionist reading. The reason is that, from the outset, Hegel is clearly breaking some of the most basic rules of contemporary logic. Above all, he repeatedly violates the existential fallacy (indeed, that is one of his very first moves). Hegel died shortly before the Boolean revolution got underway, but I think it’s obvious that he would have rejected its terms completely. The purpose of logic isn’t to create a purely self-sufficient abstract system, but to found our understanding of the world, which we cannot finally understand without presuming that it is in some sense proto-logical “in itself.”

This is not to say that nature is governed by abstract “laws” or that it has an inherently “mathematical” structure. The concepts we use to grasp nature always have to be won through the struggle of our own understanding — they do not come ready-made. Indeed, I think Hegel would agree with Lee Smolin that the remarkable effectiveness of mathematical tools in grasping the natural world is grounded in the humble fact that at every level we know of, nature presents itself as an ensemble of discreet entities that we can count.

Hegel lived a bit too early to put it this way, but I would say that our logical categories cannot be divorced from our empirical intuitions because our minds evolved out of our natural environment in order to grasp it. Pretending otherwise — no matter how complex and rigorous the results — collapses into contradiction and, worse, triviality.

Even so, I do mostly stick to the Boolean system of syllogisms for the sake of my teaching, because there’s only so much world and time. And in case you’re curious, here is my method for generating the fifteen unconditionally valid forms. First, I presume that you have the forms in the first figure memorized, which is easy enough because they’re all pretty intuitive (see the above-linked post for more of an explanation):

AAA-1
EAE-1
AII-1
EIO-1

Then, starting again from AAA-1, you list out the same forms horizontally, incrementing the figure on each one:

In the second and third column, you then write a form with the premises reversed, and you add those “new” forms to the fourth column, incrementing the figure to 4 for each.

Returning to the second and third columns, continue both down by replacing the non-A lines with Os.

Complete the list by adding EIO to any column that doesn’t already have it:

And there you have it! Once you have this technique down, you will have the 15 valid forms in the bag and never need to think about it again!