The Problem of Induction

There are two main types of reasoning in logic and philosophy: deductive and inductive. In a deductive argument, one deduces a conclusion from a set of premises. The argument is a deductive one if the truth of the premises makes certain the truth of the conclusion. A typical example:

Premises: All men are mortal, Socrates is a man

Conclusion: Socrates is mortal.

It is often said that a deductive argument tells you nothing more than the information that is contained in the premises. All mathematical arguments are deductive ones.

In an inductive argument however, the conclusion contains more information than the premises, here’s an example:

Premises: Electron 1 has a charge of e, electron 2 has a charge of e, electron 3 has a charge of e, … and so on

Conclusion: All electrons have the same charge e.

The sentences that make up the premises are called “particulars”, or particular instances. The sentence that is the conclusion is called a generalisation. Inductive arguments argue from the truth of particulars to the truth of general statements.

The example above I’ve given of inductive reasoning was chosen on purpose to make it obvious how the scientific method relies on inductive reasoning.

Science, especially physics, is built upon a bedrock of completely general, immutable laws like the First Law of Thermodynamics, or Maxwell’s Equations of Electromagnetism. Physics looks at the results of experiments on particular entities like electrons and forms a general law about the way that all of those entities behave.

It’s obvious that we haven’t encountered every electron in the universe, but seeing as every electron we’ve encountered so far has the same charge and mass, we reason inductively that all electrons have the same charge and mass.

But is inductive reasoning rational? In deductive reasoning, the truth of the premises guarantees the truth of the conclusion, but this is not so for an inductive argument. In fact, induction has been on philosophically unstable ground ever since Hume’s work in the 18th century.

Principle of Uniformity

If we see green apples fall from a tree every day, why should we reason inductively and conclude that every apple that falls from the tree will be green? Why not conclude instead that the next apple to fall from the tree will be red?

The conclusion that we pick for our inductive argument is the one that preserves a regularity. If we’ve seen the sun rise every morning, we’re going to conclude that the sun will rise again tomorrow morning, we’re not going to conclude based on this that the sun will not rise tomorrow.

Okay then, but we’re making an assumption here. What we’re assuming is that nature is uniform, that regularities exist in nature (more commonly known as laws of nature). We think that the Sun will rise every morning because we believe that nature is governed by the laws of physics, and the laws of physics dictate that the sun will rise every morning, as it has done so for billions of years.

The assumption we’re making to justify our inductive reasoning is that nature obeys certain immutable laws.

To contrast, imagine this inductive argument instead:

Premises: Every person in this room has two children

Conclusion: Every person has two children

We don’t think this is a valid inductive argument because there’s no “law of physics” that entails that every person must have two children. In the case of the Earth and the Sun however, the laws of mechanics and gravity mean that our inductive argument about the Sun rising every morning is legitimate.

So it seems induction is OK so long as we can use this principle of uniformity.

But how do we know the principle of uniformity is true? How do we know that the laws of physics are actually laws that don’t change?

The Problem of Induction

How would we justify the principle of uniformity?

Could we propose a deductive argument for it? Well… no we can’t. If there was some deductive argument that guaranteed the truth of the principle of uniformity, then induction would just be a glorified form of deduction – which it’s not.

Could we propose an inductive argument for the principle of uniformity? We can’t do this either. If we try an argument like this:

Premise: Induction and the Principle of Uniformity has always worked in the past

Conclusion: The Principle of Uniformity will always work

Then our argument is circular – we’re using an inductive argument, and hence the principle of uniformity, to argue that the principle of uniformity is true. This argument is circular – we can’t use it to justify the principle of uniformity.

So now we’re stuck, we can’t use induction or deduction to justify our use of the principle of uniformity, and hence all of our inductive arguments are unjustified.

And yet, science is entirely based on an assumption that uniformities exist in nature. Does this mean that the entirety of science is philosophically unjustified?

In a way, it is unjustified. Since Hume’s discussion of the problem, no satisfactory solution has ever been found to the problem of induction. But compare this problem to the problem of skepticism in epistemology:

We can never know if we’re brains in vats or not, but even if we can’t know that the world around us is real, we’re not going to stop acting as if it is.

In the same way, it seems we can’t ever know if inductive reasoning is justified, but despite this, we’re not going to stop using it and science to find out about the world around us.