The Art of Logic

I thought this book was dreadful, and it is one of the few books I’ve tossed in the recycling bin after reading.

In the 90’s, I went through notepads and erasers grinding through the work required to earn my philosophy degree’s required symbolic logic credit. I took simple proofs to dead-end after dead-end. I remember silently scratching away at sheet after sheet of green engineering paper that had been gifted to me by my roommate Justin on many late nights trying to get those damnable proofs to work out. Every failure or wadded-up paper was an opportunity; it was a chance to start over and see the problem differently. It was laborious, but I felt better after every surmounted obstacle.

Every textbook section or class would introduce a new spell: DeMorgan’s Theorems, modus ponens, modus tollens, “existential instantiation,” etc. and then throw terse, dense premises at us that we must transform to a conclusion. It’s in this class, under the guidance of my then-TA, Tracy Lupher, that I learned debugging in its most essential form. I eventually earned an A in the class, acing the final. My 5^th edition of Irving M. Copi’s “Symbolic Logic” will travel with me as long as I have a pulse as a testament to that challenge.

As such, I take logic pretty seriously. Decades on now, I still love the topic and, sadly, have gotten a bit rusty. While visiting Foyle’s in Charing Cross last year, I saw The Art of Logic and thought a refresher from a “lay” book would be nice. As a writer of curriculum, I’m always keen to see how others take the difficult palatable as well, so I thought the book might serve in both of those capacities.

Sadly, my high expectations for this book were not realized. Instead, it:

• Failed to provide substantial footing as an introduction to logic
• Lost its way both by structure and poor editing
• And had an odd fixation with using distracting and contentious social-justice scenarios for demonstrating the poorly-introduced logical concepts — and I’m a liberal!

In total, my dislike for this book was so great I had to write a lengthy post about it. I’ve included notes about the few bright spots as well.

The Essence of the Good Part (Part I)

The book is structured in three parts. The introductory third (“The Power of Logic”), is the strongest. Once reduced to its essence it hardly merits a book though, although it would have been a very fine Medium long-form blog post. The summary is below:

Summary of Part I of Eugenia Cheng's _The Art of Logic_: The Good Parts

• Logic is a system of organized thinking [it tells us nothing as to the truth of the premises, but asserts that, if granted] they necessitate certain conclusions

• To engage in logic we omit incidental features of things to the essence of what we’re considering in a process called “abstraction”

• Implication is the basis of logic

• By chaining implications, we move from premises to logical conclusions

• Chains often break down due to two categories of problem:

  • Problem of knowledge
    • Unstated assumptions
    • Incorrect definitions
  • Problem of logic
    • Leaps: breakdown of syllogism
    • Incorrect inferences: misapplication of a logical rule
      • Necessary versus sufficient condition for truth
    • Handwaving (not really a logical flaw)
    • Incorrect logic (seems to be the same as the Incorrect references?)

• Many problems of logic result from assuming that converse implications are necessarily true e.g “Love is blind. Ray Charles is blind. Ray Charles is love.”

• Arguing against is done best by asserting not(given-statement). Anything else runs risk of a logical error.

• Implication sets can be expressed by truth tables

• Implication can be delivered as a contrapositive: A => B === ~B => ~A

• Avoid sweeping statements like “all” or “every”

• We should pursue truth (“illumination”) over scoring points a la the sophists

Other Good Parts: Defining “Logic,” Defining “Abstraction”

Two particularly strong points in Part I is her coverage of what Logic and Abstraction are.

What Logic Is

• Maths isn’t just about numbers and equations: it’s a theory for justification. It provides a framework for having arguments and is so successful that in maths people actually agree regularly about upon conclusions. (x)
• Mathematics in school may well be mostly about numbers and equations, but higher-level mathematics is about how to think. (xi)

Both of these are wonderful and helpful. It’s too bad the rest of the book didn’t maintain this clarity.

What Abstraction Is

• …in mathematics we forget some details about the situation in order to get into a place where logic does work perfectly [across specifics]

These ideas should have been expanded and other less-relevant ideas (see Part II/III) should have been cut.

The Persistent Absentee Editor Fly in the Ointment

As nice as those phrases were, we already see small flaws ruining the foundation that the rest of the book should ride comfortably atop. I blame the editor here. Who was asleep at the wheel who allowed this howler to get through?

“Most real objects do not behave according to logic [OK, I’m with you]. I don’t [Sure. Me neither]. You don’t. [Also true.] My computer certainly doesn’t [record scratch].”

What? Computers are entirely logical. They are, quite possibly, the best example of pure logic we have access to. Cheng’s text justfies this statement that they are not logical because “I do the same thing every day and then periodically my computer refuses to connect to the wifi.”

This is stupid. Assuredly Cheng knows that this is a horrible example. If not, someone should have told her. In trying to get a jokey, friendly voice, she writes something that demonstrates confused reasoning that might well encourage the ignorant to emulate this garbage line of reasoning.

To clarify, the computer is perfectly logical. Interactions of systems sometimes give rise to errors. But to say that “despite regular success doing so, sometimes my wifi won’t reconnect” justifies concluding that “my computer isn’t logical” is itself (as the book teaches, poorly) neither necessary nor sufficient causation. It’s modeling bad reasoning in a book about…better reasoning! While the computer remains logical its runtime state changes every millisecond: you’re never using the same computer twice (apologies to Heraclitus tech support) that’s why consistent actions occasionally lead to surprising (irritating) failures — but never illogical failures.

Poor Choice of Examples

In between the bones described in the summary, Cheng makes the bizaare pedagogical choice of using hot-button social justice issues to explain the concepts.

This is a horrible idea.

First, when someone’s mind is trying to integrate new knowledge, we should choose examples that are as trivial as possible. In my team’s programming curricula at Flatiron School we reach for food, movies, pets, world-historical individuals. It allows learners to focus on the abstraction, not the content. Learning is hard enough without engaging some sort of amygdala response.

Based on my work teaching to under-represented groups, choosing issues like this doesn’t not help. Instead it makes them aware that their non-minority friends have just been re-primed to think about it. This increases stereotype threat and team cohesion. Many minority-identity individuals like programming and maths because it’s abstraction’s a realm free of these social dynamics that they have to fight every day. And here comes a trainload of that stuff right back in. This was horribly unwise.

Secondly, by choosing social justice / political issues as examples, Cheng’s inviting the reader to engage with that instead of engaging with her lesson. I’d love for a lot more red-hat-wearing nationalists to learn some logic, but this book has torpedoed its own market reach.

Per my outline above, if-then reasoning or “implication,” is one of the most essential tools of logical behavior. To introduce the venerable tautology (curiously she never introduces this vocabulary word of the craft when it is one of the most important ideas in logic: A = A), Cheng provides the following example:

So much for things that are not logical implications. What counts as logical? “If you have white privilege then you have privilege. (25)”

Wait, huh? As a liberal I recognize that I have privilege and a lot of it comes from conforming to a non-geographically rooted, manufactured, incidental-to-my-skin-color standard called “white.” But why bring that example in here, now? What’s wrong with?

The red car is red.

or

The red car is a car.

These both scan more quickly, are vastly more obvious, and don’t instantly shut down the minds of some of the audience who most needs this message. It’s a poor authorial/editorial choice.¹

It’s not unusual to find Mein Kampf cited as a source in logic classes so that students learn to apply their freshly-honed logical skills against fascist propaganda. Cheng certainly has a pedigree to follow here, but the reader’s interest and the interest of progressive social policy might have been better served by creating an appendix of exercises that challenged the reader to apply their learning into a “ripped from the headlines” context. Although for that to be possible, Part I would have had to have carried a whole lot more heft. It would have been a vastly better book had it done so.

Similarly, to teach the venerable “argument by syllogism,” Cheng argues whether calling a man “feminine” is an expression of of misogyny (32). I was taught that by “All men are mortal / Socrates is a man / Socrates is mortal.” While moving from “men” to “human” is reasonable enough, Cheng’s chosen example makes the simple needlessly complex and controversial.

Failure of Title

Furthermore given the weak skeleton of argument I outlined in the first third of the book, Cheng has not provided “the art” of logic. Her prose provides no breathless poetry of the way that logic inexorably, uniformly moves from a loose array of data to a simple, refined conclusion. That joyous bounce of Beethoven as you finally get propositions to work. It’s dull.

Can logic be described beautifully? In the perfectly-average pulp sci-fi Macroscope by Piers Anthony, intelligent beings, when seeing a certain transmission signal, are described as entering a flow state of perfect logical harmony. Each conclusion flows into another: truths about geometry, mathematics, and color all lead towards a final (and in Macroscope, fatal) end.

Cheng infuses neither the Romantic, passionate love of the art of logic (Beethoven) nor the Baroque watchmaker rigor of a textbook (Bach). It’s miserable.

The Logical Part is Shorter than the non-Logical Parts

The last two thirds (“The Limits of Logic” and “Beyond Logic”) read like a self-help book. We’re given bromides about logic’s limits and are treated to musings that some types of meaning are not to be found in logic but in a well-lived life filled with meaningful emotional connections. While I’d not disagree, it’s damned confusing to be reading a book called “The Art of Logic” where the majority of the content is about logic’s shortcomings. While appropriate to call out the limits of logic in itself e.g. mentioning Godel’s Incompleteness Theorem and Elean paradoxes, the author gives up an inexplicably large portion of the book to the limits of logic.

In retrospect, I should have tossed the book when I got to this line in its fifth paragraph. There was going to be no “Art of Logic,” clearly.

But like any lifebelt [flotation device, UK-style], it will only help us if we use it well. This means not only understanding logic better, but also understanding emotions better and, most importantly, the interaction between them.

Conclusion

I wrote “Do not read this book. It is bad.” on the front page. I tossed the book in the recycling bucket.

Footnotes

1. To be clear, I thoroughly believe that “white privilege” exists and I have it. I can walk into dollar stores on Flatbush and no one follows me. When I accidentally set off an inventory control at the TJ Maxx, I’m greeted with “Oops, sorry!” instead of a wary security guard. It exists, it’s real, and it’s willful ignorance to dismiss concerns about it as “SJW” (social justice warrior, a pejorative term used by right-wingers to deny the existence of concepts like “privilege”) as claptrap. BUT it’s horribly unwise to use it as the first introduction to logical implication.