Dissolving the problem of induction

Hmmm… okay, I confess I rather liked this excerpt from chapter 6.

This last section is going to be one of the most difficult in the book so far, but it’s going to deal with a famous philosophical idea that often comes up in debates about religion, the so-called “problem of induction.” As it’s often told, the problem of induction is how philosophy shows that science itself is a matter of faith.

To even explain what the phrase “problem of induction” means, I need to start by giving some definitions from Wikipedia:

A necessary condition of a statement must be satisfied for the statement to be true. In formal terms, a statement N is a necessary condition of a statement S if S implies N (S => N).

A sufficient condition is one that, if satisfied, assures the statement’s truth. In formal terms, a statement S is a sufficient condition of a statement N if S implies N (S => N). (Accessed 13 June 2012)

In other words: if X is a necessary condition for Y, you can’t have Y without X. If X is a sufficient condition for Y, then if you have X, you have Y.

Now, validity and soundness. Here, I will be talking in these terms in the special sense used by philosophers, not their ordinary English meanings. Here are the definitions, again from Wikipedia: “An argument is valid if and only if the truth of its premises entails the truth of its conclusion. It would be self-contradictory to affirm the premises and deny the conclusion.” Or, most importantly, if an argument is valid that means that if its premises–the assumptions the argument makes–are true, then the conclusion is true. And “An argument is sound if and only if (1) The argument is valid. (2) All of its premises are true.”

Philosophers define “valid” and “sound” this way because doing so is useful, but is also very confusing because it has no basis in how the words are ordinarily used. Because if this, if you are confused by these terms, I sympathize with you. When this happens, look back to the definitions of the “valid” and “sound” I’ve just given. Don’t try to go on what they seem like they ought to mean.

What makes validity and soundness useful is just that if an argument is sound, then its conclusion must be true. Thus, if you can make a strong case that an argument is sound, you have made a strong case that the conclusion is true. However, it is important to emphasize that neither validity nor soundness, as defined by philosophers, mean an argument is a good argument. In fact, it is pretty uncontroversial soundness is not a sufficient condition for an argument’s being good. In other words, it takes more than being sound to make an argument good.

First, take a closer look at validity. Nothing in the definition of validity prevents the premises of an argument from being completely crazy. “All men are mortal, Socrates is a man, therefore Socrates is mortal” is a valid argument, but so is “All cups are green, Socrates is a cup, therefore Socrates is green.” If the premises of the second argument were true, the conclusion would have to be true, but in fact the premises are completely crazy. The argument is valid but not sound.

It may be tempting to think that if an argument is valid, this must at least count for something, that this must mean the argument is at least not terrible. But this is wrong. The argument that assumes Socrates is a cup is not even halfway good. Also, as one of my professors used to say, validity comes cheap. All it takes to turn an invalid argument into a valid one is to add a premises that says, “if all of the above premises are true…” followed by the argument’s intended conclusion. But obviously it takes more than that to make an argument even halfway good.

Though it is slightly less obvious, an argument can be sound and still not be any good. Imagine arguing with someone who believes that the Sun orbits the Earth rather than the other way around. Now imagine giving them the following argument: “Premise: the Earth orbits the Sun. Conclusion: the Earth orbits the Sun.” If the premise of this argument is true, the conclusion must be true, and the premise is true. Thus the argument is sound.

Yet you couldn’t blame anyone for not being persuaded by that argument. The argument is circular, which is to say it assumes what it is trying to prove. The moral of circularity is that an argument’s being sound is not enough if you, or the person that you’re trying to persuade with the argument, can’t see that the argument is sound.

Thus, the reason it is useful to ask whether an argument whether an argument is sound is not because all sound arguments are good arguments. Rather, the reason is that if an argument can be shown to be sound, then you have shown the conclusion of the argument to be correct.
Everything I’ve said so far is, to the best of my knowledge, uncontroversial, rare as that is in philosophy. But now I’m am going to say something more controversial: soundness is not a necessary condition for being a good argument. That is to say, there are good arguments which are not sound in the special sense of “sound” that philosophers have defined.

Here’s why: Arguments that aim at being sound are known as deductive arguments. However, some arguments do not even try to be sound, for example, the argument, “The sun has risen every day for all of recorded history, therefore the sun will rise tomorrow.” This argument is invalid, because there’s no contradiction in imagining that the sun does not rise tomorrow, even though it has always risen in the past. Arguments like this argument about the sun are known as inductive arguments. (There is some disagreement about how broadly or narrowly to define “inductive argument,” though that won’t matter for my purposes.)

The argument about the sun seems to me to be a good argument, even though it is not valid. Some philosophers disagree. The usual way to frame the issue is in terms of “solving the problem of induction,” but this is a bad approach because it assumes from the start there is a problem with induction. This problem is helped only a little by clarifying what is meant by “the problem of induction.” For example, defining “the problem of induction” as the question of “can induction be justified?” encourages us to skip over questions like “does induction need justification?” and “does it even make sense to talk of justifying induction?”

The real question, in my view, is whether we have any reason to doubt that the argument about the sun, and arguments like it, are good arguments. And philosophers don’t often try to give such a reason. David Hume’s Enquiry Concerning Human Understanding–usually cited as the source for the problem of induction–does try to do something like that, though his actual conclusion is not about which arguments are good, but rather that, “All inferences from experience, therefore, are effects of custom, not of reasoning.”

Hume’s argument for this conclusion, though, is unclear. One thing he says is that “all inferences from experience suppose, as their foundation, that the future will resemble the past,” and from there he argues that there is no way to prove this without circularity. But it’s not clear why he thinks that all inferences from experience suppose this. Maybe what he thinks is that reasoning, to be reasoning at all, must be deductive reasoning, so the only way an inductive argument can count as “reasoning” is if it has a hidden premise that turns it into a deductive argument.

But why think that? It seems to me that some inductive arguments are perfectly good as-is. Because of that, I think soundness is not necessary for being a good argument. That is to say, there are good arguments that are not sound in the special philosopher’s sense of “sound.”

More generally, some people seem to have it in their heads that “prove” can only mean “prove in the way mathematical theorems are proved.” That’s wrong; it’s not how the word “prove” is used in legal cases, for example. But once “prove” is defined in this way, “faith” will be defined as “belief in anything you can’t prove.” Because science doesn’t proceed solely by mathematical theorems, it’s concluded that science is a matter of faith.

If you accept these definitions of “faith” and “prove,” the result would be to define matters of faith as everything outside of mathematics, making the word “faith” useless. The belief that the Earth is round and the belief that aliens have secretly been abducting thousands of Americans for experiments would be equally matters of “faith,” but redefining words wouldn’t make those beliefs equally reasonable. The problem with religious beliefs is not that we lack mathematical proofs for them, it’s that we don’t have any good reason at all to think they’re true.

Plantinga's incredibly weak arguments in Warranted Christian Belief
Bill O'Reilly's argument for the existence of God
There are no good arguments for the existence of God
Peter van Inwagen's argument for Christianity

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