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.

  • Caravelle

    But hasn’t Bayesian reasoning dealt with the problem of induction, and pretty rigorously at that ? Is there a fundamental philosophical flaw in it that I don’t know of, or some other reason why you don’t bring it up at all ?

    • http://www.facebook.com/chris.hallquist Chris Hallquist

      There’s the issue of where the hell you get your priors from.

      • Caravelle

        But doesn’t the successive addition of further evidence swamp out the original priors’ influence eventually ? Theoretically speaking at least, I suppose in practice “eventually” can be a long time.

        (I don’t actually know this, I’m rather new to those ideas, that’s why I’m asking)

  • Kevin

    Anything that we haven’t defined to be true would ‘suffer’ from the problem of induction. This doesn’t mean that we can’t have reasonable expectations for certain things. I feel like the proponents of this ‘problem’ need to learn some probability theory.

  • Dunc

    The way I like to deal with this is to make the distinction between “truth” (in the ordinary, practical sense) and “Truth” (in the absolute, philosophical sense). Science does not deal in absolute Truth. Yes, conclusions from induction are not provably, rigorously, absolutely True, but they’re a lot better than nothing, and we can gain a lot by acting as if they’re true, at least until some contrary evidence turns up.

    It’s at times like this that I recall a joke about a version of Zeno’s Paradox of Achilles and the Tortoise, involving a mathematician, an engineer, and a beautiful naked woman… I won’t bore you with the details, but the punchline is “But I can get close enough!”. ;)

  • mikespeir

    I’ve always assumed that the problem of induction stems from our inability to know for sure why a thing is happening. If the Sun has risen in the sky every morning throughout recorded history, that points to the likelihood that it will tomorrow, too. But until we know why the Sun rises we can never be absolutely sure. We can point to gravity and laws of motion and whatnot, but until we know why those are like they are we’ve just delayed the problem. We still won’t be able to claim with better than, say, 99.999% certainty that the Sun will in fact rise in the morning.

    Am I right or all wet?

    • mnb0

      You are quite close. The reason we may assume the sun will rise tomorrow again is not only based on induction and extrapolation (it did the last couple of times) but also on the theory we have formulated to explain this. I think CH doesn’t stress this point enough (yet?).

  • –bill

    You provide definitions for the terms “valid argument” and “sound argument”, but you never provide a definition for the term “good argument”. What makes an argument good? All we get from this post is that sound argument aren’t necessarily good, and that good arguments aren’t necessarily sound.

    • http://www.facebook.com/chris.hallquist Chris Hallquist

      I’m doing my best to use it in a non-technical sense. But will try to include a note about that in the final version.

  • http://aigbusted.blogspot.com Ryan

    I happen to think that induction can be justified logically. Induction can be likened to the sample-taking done by scientists. Scientists will often take a very large sample of something, and then reason that what is true of the sample is probably true of the whole (of whatever they are sampling). For example, if I interview 10,000 random people, and 90% of them inform me that they will re-elect Barack Obama, I can be reasonably sure that this is true of the entire population of voters. It is logically possible that somehow my sample wasn’t representative of the entire population. Maybe, out of the entire population, only 10% want Obama re-elected. But it is extremely improbable that my sample would be that far off the mark.

    Likewise, when we reason inductively, we observe something so many times (equivalent to taking a sample) and we assume that that sample is probably representative of all cases of that kind of event.

    This type of reasoning can and has been given a rigorous mathematical justification (i.e. see Timothy McGrew’s paper on the problem of induction).

    • rayndeonx

      McGrew’s paper on that is great. Link here: http://homepages.wmich.edu/~mcgrew/kyburg7d.htm. Another good, but rather technical paper relavent to the issue is a paper on data compression in Kolmogorov complexity: http://homepages.cwi.nl/~paulv/papers/mdlindbayeskolmcompl.pdf I unfortunately do not have the requisite mathematical background to digest the latter paper, but am trying to work through it.

      On a note I feel almost compelled to point out, it’s somewhat depressing that for as intelligent as Tim McGrew and Lydia McGrew appear to be, the latter seems to harbor a number of more or less batshit insane opinions. I’d like to think the former doesn’t think the same.

      • http://aigbusted.blogspot.com Ryan

        Thanks Rayne! I was feeling too pressed on time when I wrote that to dig up the link. And yes, the two of them are Christian apologists, and rather unimpressive ones at that. THey do such good work in epistemology that it is sad that they can be so blind about Christianity (read the paper they authored about the resurrection of Jesus to see what I mean, it’s available on lydiamcgrew.com).

      • rayndeonx

        Hi Ryan,

        I’m not talking about their essay in the Blackwell Companion, which I found unconvincing.* Obviously, I find all Christian apologetics and theistic philosophy at large to be patently unconvincing, but that doesn’t mean that I don’t think a number theists do work in both good work philosophy at large and philosophy of religion. One of my favorite current philosophers of religion is Peter van Inwagen, who seems to recognize that a lot of philosophy is in fact absurd but seems committed to that fact, as opposed to someone like Rorty who moved past analytic philosophy. The fact that his opinions in, for instance, the problem of evil are pretty unconvincing is fine. It’s wholly unlikely to agree with anyone and I think there are a bunch of great theistic philosophers: Peter van Inwagen, Keith Derose, Robert Adams, John Hick, Michael Bergmann, etc. And there are of course a number of great philosophers who also happen to be theists i.e. Saul Kripke, Hilary Putnam, Bas van Frassen, etc.

        What I’m really talking about Lydia McGrew’s various highly distasteful opinions on her blog, and the sheer disconnect from the formidable intelligence and insight that goes into her work with her husband in epistemology – from the, to put it plainly, fucking insane opinions she entertains on her blog. I’m just hoping her husband isn’t the same way.

        *I once thought about constructing a similar type of argument for the truth of Islam, since the narrative chain and testimony of the early days of Islam and the like are orders of magnitude better well attested to than anything I’ve seen in Christianity. Whatever; I think both are patent nonsense anyway.

        • rayndeonx

          Er, that should be “wholly unlikely completely agree with anyone.”

    • mnb0

      I refer you to superconduction. Until say 1985 physicists based on many samples assumed that this only could happen at temperatures near 0 K. Then Bednorz and Müller proved them wrong (and received the Nobel Prize within a year).
      That’s the problem of induction and why it cannot be justified logically.

  • David Evans

    Of course the Sun has not, in fact, risen every day for all of recorded history. There is nothing that the Sun does exactly once a day. What has happened is that every location on Earth (outside the Arctic and Antarctic circles) has been carried every day by the Earth’s rotation into a position where the Sun is above its horizon.

    Ignoring that, there are ways in which it could happen that no-one would see a sunrise tomorrow:

    Worldwide clouds from an asteroid impact or volcanoes

    An interstellar cloud coming between the Sun and the Earth

    The Earth being destroyed by a cosmic catastrophe

    so it would be wrong to regard the inductive argument that it will rise tomorrow as 100% certain.

    In fact the inductive argument that the Sun will rise every day for the next 10 billion years is certainly wrong.

  • Alex SL

    Maybe what he thinks is that reasoning, to be reasoning at all, must be deductive reasoning

    Well, this is kind of what I always assumed to be the stance of Mr/Mrs generic philosopher.

    Now I am a scientist with no training in philosophy, but I reason thusly: Induction will pretty much by definition work in a universe that shows some rule-like behaviour. A universe that does not show some rule-like behaviour would at best not be conductive to complex life that can wonder about induction and is at worst entirely unimaginable. (I mean seriously, what would it even mean to have a universe without physical laws and constants?) Ergo, induction will work in every universe containing beings that ask themselves how induction can be justified.

    • mnb0

      I refer you to superconduction as well.

  • Alex SL

    By the way, the sun is perhaps a bad example although the most common one because it is so close to our everyday experience. And David Evans has pointed out that it does not behave as regularly as would be necessary to arrive at valid conclusions through induction just by watching it rise.

    Still, the knowledge that the sun will not rise as it does today in ten billion years is also derived from induction.

    It could also be argued that the problem of induction fails in a similar way that positivism and postmodernism fail[1], in that the philosopher cannot even ask the question for the justification of induction without using language acquired by inductive processes.

    1) You know the drill: The idea that all ideas must be falsifiable is not itself falsifiable, and if all knowledge is merely ideology and discourse then the knowledge that all knowledge is merely ideology and discourse is also suspect.

    • mnb0

      “Still, the knowledge that the sun will not rise as it does today in ten billion years is also derived from induction.”
      No, it’s also derived from the theory we have developed to explain it. That’s called deduction.
      You are right about falsifiability. The materialist and sceptic should strive for as few metaphysical assumptions as possible. God is not a necessary one.


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