The Limits of Logic
Cheng argues that applying logic requires accepted starting points and human judgment about how arguments are communicated and checked. She uses mathematical proof to explore the difference between logical justification and the social process of convincing other people.
THE IDEA
HOW TRUTH IS ACCESSED, CONVEYED AND RECEIVED
WE HAVE SEEN THE power of logic in producing rigorous unambiguous justifications. Now we are going to address the limits of that power. It is important to acknowledge those limits and not pretend that logic is the ultimate answer to everything: it certainly isn’t.
When you discover your bicycle can’t fly, should you throw it away? No. A bicycle is a wonderful device, as long as you don’t try to use it beyond its limits, or beyond your limits with it. Cycling on the motorway or up Mount Everest might not go very well. Cycling in traffic would be a terrible idea for me but is a great way for better cyclists than me to get around. Sometimes in the rush hour you can even go faster than the cars. It does take more effort, but you might regard that as a good thing if you like being fit, or like burning up your own body fat as fuel rather than petrol.
Logic also has limits, especially in this messy, human, beautiful real world of ours. That doesn’t mean it has failed, and it doesn’t mean we should give up on it in some situations. But it does mean we shouldn’t push it further than its limits. Rather, we should understand those limits and understand what we can do when we are beyond the reaches of pure logic. Understanding how and why logic has limits is the subject of this second part of the book.
I’m going to start by discussing something a little uncomfortable: the extent to which even mathematical proof is, in fact, a social construct, and so logical justifications in life are doomed to be as well. This might seem to fly in the face of everything I’ve said about mathematics being completely rooted in logic, but the situation is more subtle than that.
In Chapter 2 we started discussing the fact that logic has no beginning – it has to start somewhere, and the starting point has to be some truths we assume, without justification. These are called axioms, and they are one aspect of the limits of logic. In Chapter 11 we will discuss how we come up with those axioms, as it must be by some means other than logic.
But another limit to logic is the end – when do we stop justifying things? Mathematical proofs are entirely rooted in logic, and they certainly don’t go against logic. It’s just that strictly logical proofs are impossible to write down for two reasons. One is that the logic they use depends on the rules of logic, and where did those come from? We have to assume some rules of logic even to use logic in the first place. We will come back to this paradox in the next chapter.
The other problem with writing strictly logical proofs, even once we’ve agreed to accept the basic rules of logic, is that they are utterly impractical to write down in full rigour, beyond a certain (rather low) level of complexity. And they would also not be illuminating even if we could do it. So what do we do instead? I think we can look at what mathematicians do to convince each other of their arguments, and extend those ideas to learn about how we can and should justify ourselves to other human beings in the wider world. Logic by itself is not enough. We do something a bit like trial by jury.
When to use it
When debating a controversial issue (e.g., ethics, politics), acknowledge that pure logic cannot settle everything; appeal to shared values or experiential evidence.
When presenting a mathematical argument, make the reasoning valid and the assumptions explicit, then provide enough detail for others to check it.
When teaching reasoning skills, emphasize both the power and limits of logic to avoid dogmatism.
In interdisciplinary research, combine logical analysis with qualitative methods to address complex real-world problems.
FROM THEORY TO EVERYDAY LIFE
In practice
The Bicycle Analogy
Cheng compares logic with a bicycle: useful within its limits, though unable to do everything. Her point is to understand what logical analysis can establish and what assumptions or other information a situation also requires.
Mathematical Proof as Social Process
A mathematician presents a proof at a level of detail that others can inspect and challenge. Peer scrutiny helps establish confidence in the argument. The social process of checking it does not make agreement a substitute for valid reasoning.
A MOMENT TO REFLECT
Put it to work.
What axioms or unproven assumptions do you rely on in your own reasoning? How might they be contested?
Think of a time when a purely logical argument failed to convince someone. What social or emotional factors played a role?
How does the social nature of proof affect our trust in scientific or mathematical claims?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)