If I’m not careful, I am prone to approaching any kind of argument as a zero-sum game, transforming the meeting of minds into a competition I want to win. When discussing controversial topics, the competitive mindset is not only unhelpful but dangerous to the relationships I hope to maintain. In The Art of Logic in an Illogical World, Eugenia Cheng provides some mathematical tools to help reasonable conversations be meaningful and prevent squabbles. Of all the lessons learned from her book, there is an essential goal: the point of an argument is not to win but to reach a mutual understanding of the other’s position.
The reason Cheng draws from her background in mathematics is reasonable: the field has been around for over 2,000 years and consensus on basic mathematical principles are not controversial. In fact, there is a very large body of work that isn’t so much debated as it is practiced, so if you’re thinking (like I did) that Cheng’s approach would turn out to be equations on crafting a logical argument then you haven’t met a career mathematician. Instead, Cheng sets out to demonstrate the principle that math — and arguments — isn’t about determining what is right or wrong, but rather the sense in which something is right or wrong. When I disagree with someone, the disagreement is a result of a differing set of fundamental beliefs and points of view. Because arguments aren’t about determining a winner or declaring someone as wholly right, I have had to relearn how to approach disagreements so that my end goal is to create a bridge for mutual understanding.
These lessons I’ve learned were the ones that left the biggest impression on me.
Lesson 1: Precision vs. pedantry
I enjoy language. In the immature days of my late teens and early twenties I used to correct people’s grammar habitually and I was commonly known as a grammar nazi. I didn’t mind being associated with knowing how to properly use words like ‘whence’ or ‘whom’ but I eventually realized that my unsolicited English advice was a form of pedantry. Most of the time, the people I was correcting were using language well enough that their points were being made clearly, so my insistence on the words being perfect didn’t help anyone communicate better.
Logic requires more structure than language and so communicating with someone else logically asks a lot from a language as fluid and flexible as English. As a software developer, I’ve practiced thinking logically as a career and so it feels easy for me to move from abstract thought to precise detail and back again. But just because it feels easy doesn’t mean it always is, and using the word ‘feels’ is more precise than saying that abstract thought is easy for me. In fact, it is difficult for me to think abstractly when I have strong emotions like fear or anger, which was why this lesson was important to me. I am more capable of being precise with my arguments when I am operating with my rational mind and not my emotional mind.
There is a question I ask myself when I’m concerned about being precise or wondering if I’m being pedantic: is the difference so meaningful that it illuminates the argument? Pedantry offers alternative words or phrases but doesn’t add any understanding, whereas precision ensures that my argument is being heard for what it truly is.
Lesson 2: How to be right
There are words I have tried to remove from my vocabulary, especially when making any arguments, e.g.: always, everyone, no one, never, forever, etc. If anything I say is to be taken seriously, it should first be qualified believably.
Compare these two statements:
Statement 1: Everyone in the United States hates Canada and Mexico. Statement 2: Some people in the United States hate Canada and Mexico.
It is easy to negate Statement 1 because as a citizen of the United States I can attest that I do not hate Canada and Mexico. In contrast, Statement 2 could be supported by polling data demonstrating that there is a population of United States residents who have disdain for their North American neighbors. Using exaggerated language weakens arguments and does not strengthen them.
This type of statement uses qualified populations to simply assert the existence of something in a set. Using quantitative language in this way helps prevent arguments from being negated and dismissed because it is a completely different problem to assert that something does not exist in a set. To negate Statement 2 one would have to poll every single person in the United States to determine that none of the population hates Canada or Mexico. To reiterate Cheng’s main principle in arguments: it is not whether something is right or wrong, but rather the sense in which something is right or wrong. Qualified language that specifies exactly what you assert to be true is a way to guarantee that you are right and is a natural extension of importance of precision.
Lesson 3: Understanding opposites
Using qualified language protects an argument from simple negation, but it’s also important to understand the difference between the negation of an argument and an opposing argument. Negation is a broader statement than an opposite argument, as demonstrated in this example:
Original statement: Sugar is good for you.
Opposite: Sugar is bad for you.
Negation: Sugar is not good for you. But it’s also not directly bad for you as a small amount every day probably won’t do you any harm, it’s just that in large quantities it’s probably bad for you.
An opposite argument can be thought of as the polar opposite to the original statement, whereas the negation usually represents something between the two. Arguments can tend to lean toward black-and-white opposing statements but logical arguments should properly account for gray areas. Cheng provides more nuanced examples of how to deal with gray areas in arguments than I will deal with here, but the greater understanding of the difference between making opposing arguments and negative arguments helps me frame my thoughts more clearly. The point is to create an unambiguous and logically rigorous argument.
Lesson 4: Logic has limits
“Logic turns out to be a good way to verify truth, but this is not the same as convincing others of truth” —Chapter 8, “Truth and Humans”
The mathematical analogue for an argument is a proof. Proofs are written to demonstrate what is thought to be true. Any gaps in an a proof or an argument can be explained or justified, but Cheng says that we should only keep going until the other person is convinced your argument is correct or until you realize that the other person’s fundamental beliefs are so different that it is not possible to convince them until their beliefs change. While it might seem impossible to change someone else’s beliefs, it is still helpful to frame their skepticism within reason. In general, reasonable skepticism comes about in one of two ways:
- Someone might think there’s a gap or error in your logic.
- Your conclusion might contradict someone’s intuition.
The first objection is simple because it is logical. The second objection is more difficult because it deals with a person’s internalized intuition, which can include emotions or differing points of view. This is where an argument requires two people to understand each other because it is impossible to confront an intuitive objection if you cannot empathize with the other’s perspective.
There are also moments in life where logic is not helpful. In cases of emergency, it isn’t that logic should be contradicted but it cannot be relied on. Logic requires information to be productive and there are often situations where information is inaccessible or there isn’t enough time to fetch the information needed. For example, a doctor treating someone whose heart has stopped cannot take a full and logical assessment of the patient’s body in the short 5 minutes it takes for permanent damage to occur: she must act immediately or the patient will die.
Issues of trust cannot be solved logically, either, since you cannot know for certain how someone else will behave. It is beyond the scope of this article, but the prisoner’s dilemma is an excellent thought experiment demonstrating the limit of logic to make the best decision.
Lesson 5: Good humans move beyond logic
“A logical human is one who uses logic.” Chapter 16, “Intelligence and Rationality”
To build logical arguments, you have to move from the real world to the abstract world where logic works. Since humans don’t live in the abstract world, a good human will use logic as a tool to understand the world around him. At the core of a person there should be axioms (or principles): things someone believes and does not need to justify in every argument. Logic helps a person understand and define the fine lines and gray areas of differing topics in order to generate insights and foster understanding with others.
Sometimes it is necessary to understand someone else’s point of view in order to help them understand your own. Humans learn by analogy very effectively, so a good human will provide helpful metaphors or stories from someone else’s world to build a bridge from the other’s thoughts to his own. This requires empathy, which will also be helpful when a person is dealing with someone who is captive to an overriding emotion like fear. A good human will understand that often people respond to something with an emotional reason or an intuition, and no amount of perfectly logical arguments can deal with the emotions of a moment.
Most importantly, a good human will understand that even two logical people can disagree. For instance, one person might believe in helping others where another believes others should help themselves. Differing fundamental beliefs create differing logical arguments, but there is always a limit to those arguments and eventually you have to deal with the person standing in front of you as a person.
My hope is that more people read The Art of Logic in an Illogical World and come away with lessons learned. These five lessons encapsulate the big ideas I took away from the book.