Deduction, Induction, and Axioms

You cannot think critically without understanding the difference between inductive and deductive reasoning. It’s not difficult to understand, but it’s crucial.

Inductive Reasoning

A basic argument using inductive reasoning looks like this:

Premise: Every swan I have ever seen is white.

Conclusion: Therefore, all swans are white.

Here’s another way to phrase it: “I have observed the world, recorded my experiences, and the data shows that all swans are white.”

This argument isn’t unreasonable, even though the conclusion is false. Black swans exist; they are rare exceptions to the rule.

Inductive reasoning is the primary methodology of science. Scientists gather data; they formulate theories based on their data, and they come up with plausible conclusions. Whenever new data arises (whenever a black swan is discovered) their theories have to change accordingly.

Induction starts with concrete data, then draws generalized conclusions from that data. Say a new pharmaceutical drug has been through a series of trials. In 90% of cases, participants experienced nausea as a side effect. It might be reasonable to say, “90% of people will experience nausea as a side effect of this drug”, but that might not be true. After all, only the people in the scientific trials were tested. Perhaps they were anomalies, and nobody in the general public will experience nausea at all. It’s an inductive leap to say, “What happened in situation X is what will happen in situation Y.”

Induction has a key feature: you can have true premises and false conclusions. Even if all of your evidence suggests something is true, the conclusion does not necessarily follow.

As an extreme example, consider an idea we take for granted: when we drop things, they fall to the ground. Why do we believe this is true?

Well, all of our experiences suggest it’s true; take a bunch of things, drop them, and observe what happens. But even though we have a strong belief that “things fall to the ground when dropped”, it’s certainly possible that some day, we might drop something and it floats up into space. It doesn’t seem plausible, but it’s logically possible.

Deductive Reasoning

Contrast this with an example of deductive reasoning:

Premise A: All men are mortal.

Premise B: Socrates is a man.

Conclusion: Therefore, Socrates is mortal.

If our premises are true, is there any possible way the conclusion is false?

No. This is the nature of deductive reasoning. With true premises and sound deductions, you can know with certainty that your conclusions are true. This is because deductive reasoning appeals to logical necessity.

Deduction is also the methodology of mathematics. Propositions like, “One apple plus one apple equals two apples” is certainly true; it’s a logical necessity. You don’t need to “go out and test” whether or not addition is true in the world.

A Powerful Pair

As somebody interested in accurate reasoning, I prefer certain conclusions over plausible ones. In an ideal world, we could arrive at all of our conclusions via deductive reasoning. Unfortunately, this isn’t possible. In fact, most of our knowledge is gained through inductive reasoning; deduction is beautiful and powerful, but it isn’t always possible, given the uncertainty of the premises we begin with.

However, there are a few, profound exceptions.

Some areas of thought begin with a self-evident premise, called an “axiom”. Then, they use deduction to build out a larger theory from the initial axiom – leading to logically-necessary conclusions.

For example, in economics, consider this seemingly-benign axiom: “humans act”. Such a statement is hard to deny – denying it would be an action itself.

But Austrian economist Ludwig von Mises used this axiom as his starting point and deduced an incredibly profound economic theory from it.

He built a logically-certain framework for economic thinking. Once the framework was in place, he then filled it in with all kinds of empirical and non-logically-necessary assumptions. The result is an incredibly powerful explanation for how the world works.

This technique for thinking has a name: axiomatic-deductive reasoning. You find an axiom(s), and you use logical deduction to discover what necessarily follows from your axioms. This method is less popular than the standard “scientific method” – though it is enjoying a bit of a renaissance – and it leads to far more precise conclusions.

To see how this way of thinking can be applied to other areas of thought, consider a couple examples. First in the philosophy of mind: “Perception happens”.

Seems simple, but what follows? What concepts are necessarily presupposed and implied by the axiom of perception?

Or, consider an axiom in the philosophy of language: “Humans attempt communication.”

Again, what can we know for certain based on this truth? It’s not easy, but use your mind to try and discover some of the implications.

In the future, I imagine a world where most intellectual disciplines can state their foundational axioms along with the conceptual framework that logically follows. But re-organizing our thoughts around axioms and tautologies takes a significant effort, and lots of work needs to be done.

If you’re interested in any particular field of thought, try to discover sturdy axioms and see what can be deduced from there. You might be surprised how far you can get.