Peirce's Problems of Induction

(Traditional Paper submission)

Abstract:  The Humean problem of induction challenges all arguments that use observed cases as premises to draw conclusions about unobserved cases.  These conclusions seem unjustified without a general assumption of uniformity:  that the unobserved will resemble the observed.  However, that assumption itself cannot be justified non-circularly. Peirce's justification of induction avoids this assumption of uniformity.  For Peirce, induction is justified because it can detect and correct its own errors.  The justification of induction therefore does not depend on the likelihood that our inductive conclusions are true now; we can know only that if they are not true now, continued use of induction will uncover their falsity in the long run.  In fact, Peirce argues that induction cannot (in the long run) fail as a source of knowledge in any possible world.  Peirce's alternative to Hume is weakened, however, by his idiosyncratic standards for counting induction as successful.

The Humean problem of induction challenges all arguments that use observed cases as premises to draw conclusions about unobserved cases.  These conclusions seem unjustified unless backed up with a general assumption that the world is uniform:  that the unobserved (say, the future) will be like the observed (say, the past).  However, that assumption itself is unjustified, or justified only circularly by the use of arguments of the very type in question. Peirce's justification of induction avoids making this assumption and asserts that an inductive conclusion is justified not by a general assumption of uniformity but by the inevitable long-run success of the inductive method.  Peirce is thereby licensed to ignore Hume's problem of induction, although it is immediately replaced with two new questions that Peirce must answer.  First, how can a defense of induction be satisfying if it guarantees only long-run, not short-run success?  And second, on what basis is Peirce so confident of induction's success in the long run?  In section one, I'll say a bit about the place of induction in Peirce's epistemology, and sketch out his argument for its justification.  In section two I'll say something about why Peirce held such a surprising view of justification, and in section three I'll examine Peirce's arguments for induction's guaranteed success in the long run. 

1.  Induction is the last step in Peirce's three-step method of science.  In "The Fixation of Belief," Peirce explains four different methods that can be used to move out of the irritating state of doubt and into a comfortable state of firm belief.[1]  Of these four, only "science" insists that beliefs must be fixed in accordance to the way the world really is.  Peirce's method of science applies to a broad range of questions besides those that are narrowly scientific.  It resembles the hypothetico-deductive method.  A likely-looking hypothesis is chosen using the guidelines for abductive reasoning, and then deduction is used to draw out the experiential consequences of that hypothesis.  Induction is the phase in which hypotheses are tested, and therefore the phase in which knowledge is acquired.  Abduction only tells you that a hypothesis is worth testing, and deduction can only tell you that if the hypothesis is true, then something else is true.  Inductive arguments are the only sort that can tell you that a conclusion is true.  According to Peirce, inductive arguments are justified by the fact of their inevitable success in the long run. 

Suppose we define Inductive reasoning as that reasoning whose conclusion is justified not by there being any necessity of its being true or approximately true but by its being the result of a method which if steadily persisted in must bring the reasoner to the truth of the matter or must cause his conclusion in its changes to converge to the truth as its limit. (7.110)

For Peirce, an inductive conclusion is justified because if it is wrong, we will sooner or later learn that it is wrong.  Of course, there is no guarantee as to when we will figure out which conclusions are false.  So to say that induction is inevitably successful in the long run is not to say that it is reliable in the short term.  An inductive conclusion is justified because it is the result of a method that must lead the reasoner to the truth in the long run.  But that justification doesn't provide a reason to think that the inductive conclusion reached now is true.  The point is not just that Peirce's account of justification is externalist, though it is, since one's belief is justified if it is the result of the right method, regardless of what one knows about the method.  The most counterintuitive part of the account is that justification is not tied to any measurement of past experimental verification, and is not an indicator of truth in the short run.

Peirce distinguishes three varieties of inductive arguments, "crude," "quantitative," and "qualitative" induction.[2]  All of these forms consist in comparing a hypothesis to actual experience, and all are justified because of their ability to lead the reasoner to the truth in the long run.  Although the latter two are the strongest, and the most useful in the context of scientific inquiry, I'm going to talk mostly about crude induction here in order to clearly show the distinctive nature of Peirce's justification of induction. 

In a crude induction, the inquirer, having no evidence to the contrary, takes a hypothesis to be true, while watching for indications that it is false.[3]  The inquirer's belief is then (somewhat) justified.  Peirce offers the following entertaining example of crude induction:

I find myself introduced to a man without any previous warning. Now if I knew that he had married his grandmother and had subsequently buried her alive, I might decline his acquaintance; but since I have never heard the slightest suspicion of his doing such a thing, and I have no time to investigate idle surmises, I presume he never did anything of the sort. (7.112)

Peirce's example was clearly chosen in part to make the point that crude induction is commonly relied upon in daily life, rather than in formal science. 

To count as a crude induction, the presumption or belief must be sensitive to new evidence.  Just as we'll see with stronger forms of induction, the method's justification comes from its capacity to detect error.  Peirce does not believe in unknowable truths or undetectable falsehoods.  If one of my beliefs, say, the conclusion of a crude induction, is false, that means that I (or someone) may (sometime) have an experience showing it to be false.  At that point, I will have to either give up my belief or give up my claim to be using science and admit that I'm fixing my beliefs by some other method, not trying to make them correspond to the way the world is.  The proper use of crude induction for a scientist relies on her openness to new evidence; as long as she is open to new evidence, those crude inductions that turn out to be false will always be corrected in the long run, whether or not the correction happens within the individual scientist's lifespan.  "Open to evidence" here means both that the scientist updates her beliefs when confronted with new evidence, and that she does not deliberately try to avoid being so confronted.

This implies that for Peirce, a belief can be justified in the absence of evidence; my inductive conclusions are justified beliefs if I respond correctly to new evidence.  Even a belief taken at random can be (weakly) justified by a crude induction, at least until falsifying evidence becomes available.

Crude induction draws a conclusion based on a lack of evidence to the contrary.  That analysis seems to lump together typical cases of induction by enumeration with cases that resemble arguments from ignorance.  In the first case, there is some evidence regarding whether p, but none of it supports p, and we conclude ~p.  To paraphrase one of Peirce's examples, let p represent "clairvoyance really exists," and suppose that several alleged cases of clairvoyance have been studied carefully, but none were determined to be genuine.  The conclusion that clairvoyance doesn't really exist seems (weakly) inductively justified to us. In the second case, there is no evidence at all regarding whether p, and we conclude ~p.  Here, suppose that no cases of clairvoyance have been studied at all, so that in that sense there is no evidence indicating that clairvoyance is genuine.  In this case the conclusion that clairvoyance doesn't exist doesn't seem even weakly justified according to our usual notion of justification.  There may be no evidence that it doesn't exist, but there is also no evidence that it does, so many of us would conclude that the right response is to withhold judgment.

These two kinds of case seem distinct to us, with our own notion of justification, which is roughly that p is justified if there is evidence that p.  If we searched for evidence that p and found none, that counts as some evidence supporting ~p, so the conclusion ~p seems somewhat justified.  On the other hand, if we did not search, but we conclude that p on the basis of no evidence, it does seem strange to consider ~p justified, even slightly.  If Peirce was working with a notion of justification similar to ours, it would be strange for him to include these two different cases under the same name. 

For Peirce, however, inductive conclusions are justified because they are "the result of a method which if steadily persisted in must bring the reasoner to the truth of the matter"(7.110).  On Peirce's account, a belief's justification doesn't depend on the evidence available when the belief is formed, or on the method's short-term reliability, but on its long-term success. 

2.  This notion of justification may seem unsatisfying; after all, it destroys the connection between justification and evidence, so that it makes no difference whether a belief was arrived at by rational or random means.  In his defense, however, Peirce's notion of justification completely avoids the Humean problem of induction.  On our own notion, an inductive conclusion is justified based on past experiences, thus provoking the question:  why should we expect that future experiences will resemble past ones?  It is notoriously difficult to find evidence for that expectation, and if the future will not in fact resemble the past, then justification based on treating past experiences as evidence is useless.  By contrast, Peirce's justification relies on no expectation about the future; our beliefs are justified as long as we continue to be open to future experience, whatever it might be.

As mentioned earlier, Peirce's escape from Hume's problem of induction lands him with problems of his own.  He holds that inductive arguments of any of his three varieties must, in the long run, bring the reasoner to the truth.  However, bad luck could delay convergence on the truth for a very long time.  One might hold a false belief derived from crude induction for an entire lifetime without happening upon an experience that would shatter it.  Although induction's conclusions are justified in Peirce's sense, even in the short run, that is no indication of their truth in the short run.  Peirce asserts only that in cases of short-term mistake, repeated use of induction will eventually, in the long run, enable one to discover the mistake.  It's no fault of the method if you are unfortunate enough to die of old age before reaching the truth.

Peirce seems to be asking us to accept the conclusions of our inductive arguments (since they are justified now as well as in the long run) while at the same time he admits that we have no reason to think they are true.  Our usual intuition would be that an inquirer presented with no evidence on a particular matter should suspend belief while awaiting evidence; instead, Peirce allows her to choose a belief and hold it justifiedly while awaiting evidence to either support or demolish it.  Notice that this difference is in line with Peirce's definition of "inquiry."  The purpose of inquiry is to get out of the uncomfortable state of doubt and achieve the state of belief.  Ideally, one prefers beliefs that are stable, that won't collapse and dump one right back into doubts, so one prefers beliefs that won't clash with future experience.  But without evidence, there could be no way to get beliefs of that quality.   In that situation, it is preferable to choose a belief, even if at random, than to suspend; at least you'll have a belief, and it might prove to be stable, while a state of suspension might be no better than a state of doubt according to Peirce. 

It's an odd sort of defense of induction that concedes that no inductive argument can now give me a reason to believe its conclusion is true.  Peirce's method is not to respond to Hume's argument, on his own terms, but to change the rules on him.  For Hume, a defense of induction would need to provide a non-question-begging argument that would give us reason to believe the conclusions of our inductive arguments (now) are true.  He doesn't think such a defense is possible, and Peirce agrees.[4]  Peirce's defense aims instead to show that induction is a method that in the long run will lead the community of inquirers to the truth.

3.  Peirce's notion of justification is "pragmatic" in the following sense:  using justified methods is the only way for the human race to get to the truth.  No individual inquirer is assured of getting the truth now, but all can do their parts to help the progress of science along in the long run.  If its long-run success is going to justify use of induction though, what justifies Peirce's confidence in induction's long-run success?

In his 1868 paper "Grounds of Validity of the Laws of Logic" Peirce argues that we cannot conceive of a world in which induction fails.[5]  (I take it that his interest here is in whether such a world is possible, and that he must be assuming that if it is possible, then it is conceivable.)  The world in which induction failed would be one in which inductive arguments can find premises, but their conclusions are true only about half the time.  If our inductive arguments proved correct only 2% of the time, by contrast, we could derive true beliefs by taking all of our inductive conclusions to be false.  If induction is to fail as a source of knowledge, it must work only about half the time; in other words, induction must be no more reliable than guessing.

In the world in which induction is no better than guessing, Peirce argues, there can be no true plural universal propositions.  Here is my reconstruction of his argument:

  1. In universe U, induction works 50% of the time, producing conclusions that are no better than guesses.
  2. Some universal propositions are true of experienced parts of U:  say, proposition (G) "All experienced emeralds are green."
  3. Since G is true, inferences from one experienced particular emerald to another with respect to greenness are invariably reliable[6].  (Simply, what it means for G to be true is that each experienced emerald is green.) 
  4. So, proposition G allows us to make inferences that are better than guesses in universe U.  The inductive conclusion that an emerald is green is a guess with respect to unexperienced parts of U, but it is 100% reliable with respect to experienced parts of U.  On average, then, G is better than a guess.
  5. So, if universal propositions such as G are true in U, induction would reliably result in conclusions that are better than guesses (on the subject matter covered by G).  But by premise 1, induction cannot produce conclusions that are better than guesses in universe U.

The two premises that induction works only half the time and that some universal propositions are true thus produce a contradiction.  If a plural universal were true even within a limited scope it would license inferences that are "somewhat better" than a guess and thus contrary to the premise that in this universe inductive arguments are no better than guesses. 

What can Peirce mean here by "somewhat better" than a guess?  It can't be that the universal's truth within a limited realm (experienced emeralds) gives us any kind of reason to trust inferences outside that realm.  That would just beg the question in favor of induction, and in any case Peirce admits that universals true within the limited realm are still "no better than guesses in reference to other parts of the universe." (5.345)  He must be thinking that truth within a limited realm already counts as a kind of inductive success.  If it's true that all experienced emeralds are green, then if you think of an emerald at random, not knowing if it is experienced or not, your claim that it is green is better than a guess.  If it turns out that it's an experienced emerald, you will be reliably right.  If it turns out that it's not an experienced emerald, then your claim is no better than a guess.  Overall, your claim is "somewhat better" than a guess.[7] 

I don't think this argument works, so I'll briefly explain why before turning to the second one.  The problem arises when we try to interpret the phrase "better than a guess."  Peirce claims that the inductive conclusion "emerald e is green" is "no better than a guess" when e is taken from the unexperienced parts of U.  What does this claim mean?  Consider the following two cases:

  1. A fair coin toss.  My call of "heads" is equivalent to a guess in the sense that the two possible outcomes are objectively equally probable.
  2. The Goldbach conjecture.  My call of "true" is equivalent to a guess in the sense that I myself have no idea whether it's true or false. 

If Peirce intends "no better than a guess" in the first, objective-probability sense, then he is asserting that the randomly chosen unexamined emerald is equally likely to be green or not green.  That would mean that about half the unexamined emeralds in universe U were green.  But Peirce's argument here is supposed to work for any universe U, since the conclusion is that induction won't fail in any universe.  Peirce is not entitled to stipulations about the unexperienced parts of U. 

If Peirce intends the second interpretation, the epistemic or subjective-probability sense, then his claim about the unexamined emerald seems fine:  I can only guess that it's green because I myself have no idea.  In that case the argument goes as follows.  I'm sure the examined emeralds are green, and I'm completely ignorant about the unexamined emeralds.  In Bayesian terms, I assign 1.0 (or nearly) to "examined emerald is green" and 0.5 to "unexamined emerald is green."  But under this interpretation it's hard to see how the conclusion of the argument can be that "e is green" is objectively "better than a guess" overall.  I can average together my subjective probabilities, and claim that I'm more confident that e is green – more than 0.5 overall – but that says nothing about whether I'm right

So if Peirce intends the objective-probability interpretation, his argument doesn't apply to all possible worlds after all; if he intends the subjective-probability one, then the argument doesn't prove that inductive conclusions are likelier to be correct, but only that we are likelier to believe them.  I don't see any way to fix this argument, so I will set it aside for the moment, in the interest of understanding Peirce's whole project.  If the project seems worthwhile, maybe another argument can fill the gap left by this one's collapse. 

Next Peirce argues that it is contradictory to suppose that individuals exist in a universe in which induction fails 50% of the time.  Here's the argument as I reconstruct it:

  1. If there were individuals in this universe, then, there would be at least one class, "that is – there must be some things more or less alike – or probable argument would find no premisses there."(5.345)  As Peirce said before, he is attempting to imagine a world in which inductive arguments fail 50% of the time, so it must be a world in which inductive arguments can be constructed.  If inductive arguments can be constructed in a world containing individuals, then some of those individuals must be "more or less alike," forming some kind of class (C).
  2. If there is one class C there must be at least two classes, the mutually exclusive and jointly exhaustive C and ~C, "since every class has a residue outside it."
  3. Any individual x must belong either to C or ~C, but cannot belong to both.

Therefore, there must be one true plural universal for each individual x, either "None of the members of C = x" or "None of the members of ~C = x."

Therefore, if, as argued above, there are no true plural universals in this universe, then there can't be any individuals to serve as the subjects of singular universals. 

When finished arguing that in the world in which induction fails, there could be no true universal propositions of either type, Peirce describes what the world is like.  Without true universals,

...every combination of characters would occur in such a universe.  But this would not be disorder, but the simplest order; it would not be unintelligible, but, on the contrary, everything conceivable would be found in it with equal frequency.  The notion, therefore, of a universe in which probable arguments should fail as often as hold true, is absurd.  We can suppose it in general terms, but we cannot specify how it should be other than self-contradictory.  (5.345)

In the above argument, Peirce considers a generalization that is true within the limited domain of our experience to count as a success for induction even if we don't have reason to believe it to be true outside that domain.  The conclusion is already "better than a guess overall."  There is something odd in taking an inductive generalization to be successful overall to any degree if it only describes experienced objects, giving us no reason at all to apply it to unexperienced objects.  Usually we think a successful induction just is one that gives us a reason to believe its conclusion to be true outside the domain of our experience.  According to Peirce, though, we cannot reasonably expect the conclusion of any particular inductive argument to be true of the unexperienced world.  Induction makes no short-run guarantees.  It only guarantees that if we persist "long enough" we are bound to reach true conclusions. 

"Success" in our usual terms – a specific individual inductive argument's ability to justify our belief in its conclusion – is not the validity that Peirce is trying to prove induction to have.  In fact, all the "success" that a Peircean inquirer can expect to achieve is a set of generalizations forming a correct description of one's experience so far along with an openness to, and method for incorporating, future experiences – whether they are similar or not.  Rather than avoiding Hume's challenge, then, Peirce's argument concedes the point:  we have no reason to believe that our inductive arguments deliver true conclusions in the short run, and the long-run success of induction that Peirce offers amounts only to a promise that inductive arguments are better than random guessing.

[1] 5.358-387.  All references of this form are to a volume and paragraph number in the seven-volume Collected Papers of Charles Saunders Peirce, edited by Charles Hartshorne and Paul Weiss.  Cambridge:  Harvard University Press.

[2] Quantitative induction is the inference from a sample to the whole and is justified because if the first sample happens to be misleading, repeated sampling will eventually bring the mistake to light.  Qualitative induction is the testing of a "sample" of the predictions made by a given hypothesis.  While quantitative induction is mechanical in that each new sample must affect the inquirer's belief in a determinate way, qualtitative induction has room for individual judgment, since some predictions carry more weight than others.  Both share the advantage that corrections can be made gradually, unlike the corrections of crude inductions which take the form of sudden demolition.

[3] The idea of crude induction may have originated with Bacon, although Peirce renames it.

[4] Indeed, since induction is not guaranteed to be reliable in the short term, Peirce recommends relying on custom for many short-term matters of practical importance, and sounds rather Humean in doing so.

[5] 5.318-357; this argument is at 5.345.

[6] Peirce calls these "inferences by analogy" but he must be counting them as a kind of induction in order for the present argument to work. 

[7] How much better seems to depend on how much of the domain is experienced, the proportion of experienced to unexperienced emeralds.