| The Problem of Induction | |
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| Primary Sources | |
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Hume, D. (1739) A Treatise of Human
Nature.
Popper, K. (1953) The Logic of
Scientific Discovery
Goodman, N. (1950)
Fact, Fiction, and Forecast.
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Introduction: Aristotle's Inductions |
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Putting aside the
extraordinary (and ultimately unconvincing) doubts of the Academic
Skeptics and Descartes’s Method of Doubt, let us go back to the view of
scientific knowledge that Aristotle had proposed and consider what other
problems we might have in coming to know the truth about the external
world.
You’ll recall
that according to him scientific
knowledge is a structure of demonstrations: deductions leading upwards
from more general truths to more particular truths, giving the causes of
things. The sorts of demonstrations that constitute scientific knowledge
are things like
The statue is bronze
Bronze is brown
--------------------------
The statue is brown And we know that at
the bottom of the structure are conceptual truths that are undemonstrated
(and thus ‘unscientific’) which we come to know through induction (epagoge/epagwgh) on data given by our
senses. There’s some debate about precisely what Aristotle meant by
induction, but it is usually taken to be some kind of inference from
particular facts to general ones. He does say at one point that “induction
proceeds through an enumeration of particular cases”
[1] which
suggests that what he has in mind is the sort of inference one makes when
one observes some number, n, of
bronze things and, noting that they are all brown, concludes that all
bronze things are brown (or that bronze is brown.) He says that such
inferences are possible for us because
“the soul is so
constituted to be capable of this process.”[2]
The possibility of
an ‘inductive’ method of discovering new facts from observational data is
reflected in the possibility of an
inductive argument in which the observational data supports the new
fact as a conclusion. For example, in the case of the bronze things we can
form the ‘enumerative’ inductive argument
Bronze thing
number 1 is brown
Bronze thing
number 2 is brown
…
Bronze thing
number n is brown
----------------------------------------
Bronze is
brown which will be taken
to be a good argument if we think that the inference is a good one. But
remember from our previous brief introduction of inductive arguments that
we distinguished inductive from deductive arguments on the ground that
inductive arguments are fallible whereas deductive arguments are not. It
takes only one observation of a
green bronze thing to falsify the conclusion to this argument. Even in
a good inductive argument the
best that we can hope for is that if the premises are true then they make the conclusion probably true.
Most modern views of
science also include some element of deriving regularities from
observations. The regularities we seek may be described as laws of nature; as when repeated observations of gases under variation in
pressure and volume are used to justify Boyle’s Law:
p1V1 = p2V2.
We too have a role for
deductions from such inductively derived general statements; but we use
them as predictions or explanations of events in the world – not to
provide the static structure that Aristotle sought. So, given that we have
a gas at 10kPa in a 1 litre bottle, we can predict what happens to the
pressure if we move all the gas to a smaller ½ litre bottle. Or, we can
explain why the pressure increased when we moved all the gas to the ½
litre bottle. In either case, we would make a deduction that looks like
this:
p1=
10kPa
V1
= 1l
V2
= ½l
p1V1
= p2V2
----------------
P2
= 20kPa So for Aristotle and for ourselves, it seems that Science really can’t get started unless we accept induction. But can induction actually be justified? On what grounds can we believe that induction is a good method of discovering facts about the world, and that inductive arguments are rationally acceptable? This is the problem of induction.
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The Humean Problem of Induction |
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The exact nature of the problem was first made
clear by David Hume[1]. (Though
he never actually mentioned induction – being more concerned with
understanding how we can acquire the concept of causation – we can
paraphrase his argument in our terms quite easily.) Consider first how we
might justify induction through experience. We could argue that induction
has been a very successful way of getting the facts about the world that
we now believe that we have. Through it we have learnt such things as that
bronze is brown, and that there is a simple relationship between pressure
and volume in a gas, and many other good things. Why should we doubt that
that it will be a reliable source of true general statements or physical
laws in the future? Couldn’t we just take induction to be a generally
successful method on the basis of experience?
No. We couldn’t. Consider the actual form of
this justification. It appears to be this:
Induction was successful in case 1
Induction was successful in case 2
…
Induction was successful in case
n
------------------------------------------
Induction is a generally successful method
And this is just another induction from
particular cases to a general claim. But this would only be an acceptable
way to get to that conclusion if the method of induction was already
justified. Unjustified induction can’t be used to prove that induction is
justified.
But if that’s not an acceptable way to justify
induction, then what is? We only know of one other kind of argument:
deductive arguments. Could there be some sort of deductive argument to
justify the use of induction? It would seem not, because the conclusion of
any valid deductive argument is going to be a necessary consequence of the
premises, whereas we have defined inductive arguments specifically as
those (‘good’) arguments that do not establish that there is a necessary
connection between reasons and conclusion. If there was a valid deductive
argument to say that we are right to use induction, then the conclusions
of inductive arguments would have to be necessarily true – and they’re
just not.
This might be clearer with some expansion.
Suppose that there is a deductive argument to justify induction; then
there would be a valid argument that looks like this.
Premiss 1
Premiss 2
…
Premiss m
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Induction works
So if the premisses 1 to
m are true the conclusion must be true. And we can now include that
as a premiss in further arguments, like our erstwhile inductive arguments
above. Thus:
Bronze thing number 1 is brown
Bronze thing number 2 is brown
…
Bronze thing number n is brown
Induction works
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Bronze is brown
And since ‘bronze is brown’ is got by induction
from the premises of the argument, and that induction is reliable (because
all inductions are good,) it follows that ‘bronze is brown’ follows
necessarily from the premises. At which point we notice that it really
isn’t logically impossible for the premises to be true and the
conclusion to be false (because it only takes one unobserved green piece
of copper to make it false,) and so the conclusion about induction being
reliable can’t possibly be true. So there can’t be any sound deductive
argument to show that it is true.
Retreat to Probability
One very obvious response is to restate the
principle of induction as a probabilistic rule. Thus inductive arguments
would look like
Bronze thing number 1 is brown
Bronze thing number 2 is brown
…
Bronze thing number n is brown
----------------------------------------
Bronze is probably brown
In this case, there would seem to be no difficulty about seeing one or two green bronze things, because the conclusion allows for that. Unfortunately, this really doesn’t help very much, because there’s no logical reason why we couldn’t see n brown bronze things followed by n+1 green bronze things. Europeans saw any number of white swans before Australia was discovered, but the claim that swans are probably white would have been wrong if there were more than that number of swans in Australia that were black. And of course we get all the same problems with trying to justify the principle of probabilistic induction as we just saw with justifying the original version.
[1]
Treatise of Human Nature 1.3.6, Enquiries Concerning Human
Nature 4
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Popper’s Solution |
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On the face of it then, it seems that induction
is unjustifiable and scientific knowledge impossible. That’s unfortunate,
to say the least; and naturally, given the success of science, and the
desire of philosophers not to have only nonsense to say about history’s
most successful epistemological enterprise, there have been a number of
attempts to get around this problem. One that has been particularly
fruitful is due to the Austrian philosopher Karl Popper.[1]
Popper’s solution is more or less to sideline
the ‘problem’ of induction. We only think there is a problem because we
take induction to be necessary to the success of science. Induction, so
the story goes, is required to derive ‘good’ or ‘true’ generalisations
from observational data – and if there can be no assurance that any such
process can be justified, then there can be no assurance that the results
of such a process can be relied on, and no assurance that science has got
at the truth of the world. Popper argues that in thinking this we are
locating the source of the power of science in entirely the wrong place.
To see what he has in mind, let’s look again at
the model of scientific knowledge proposed by Aristotle. We noted that the
procedures of science were left somewhat vague by him, but that they
seemed to include a vital role for dialectic in distinguishing amongst the
proposed solutions to scientific problems. And we had also earlier seen
that dialectic, as practised by Socrates, could provide a model for
investigations into matters of concern to science. In the Socratic method
of elenchus, a victim was asked to provide a definition for some
concept in dispute, and Socrates would then go to work drawing the
consequences of that definition, and finally determining that if it led to
falsehoods, that the definition could not be correct. The victim would
then offer further definitions which might suffer the same fate. Just so,
we pointed out that when trying to discover an explanation for some
observed fact about the world, it would be a reasonable procedure to first
propose some answer and then to test that answer by comparing what would
be true if that hypothesis were true again what is actually seen to be the
case. If the hypothesis is contradicted by the observations then it is
declared to be false and another explanation is proposed. We used the
example of continental drift to illustrate this. The observed fact to be
explained was the distribution of similar fossils in areas separated by
large seas. The first hypothesis to be proposed was that the plants and
animals migrated across those seas by means of now-vanished ‘land
bridges’. It was noted that another consequence of that hypothesis was
that there would be traces of some of those bridges. When it was observed
that no such traces remained, the hypothesis was said to be falsified and
another had to be proposed. The process went something like this:
Hypothesis 1:
There were land bridges
Consequence:
If there were land bridges there would be traces of them
Observation:
There are no traces of them
Conclusion:
There were no land bridges
Hypothesis 2:
…
The significant point for us to note is that
Socrates didn’t much care about how those who claimed to know what courage
is or piety or whatever came to their opinions. The important thing was
that they could be shown to be false, and that that false opinion could
then be dropped. In this way we could make progress towards discovering
true opinions. Similarly, we can imagine there being better and worse
guesses at solutions to scientific problems, produced by whatever
psychological process – be it induction or anything else – but what
matters is that bad solutions can be progressively discarded and so
science can approach the truth about the world ever more closely.
This is Popper’s solution to the problem of
induction. Science approaches truth not because there is a mysteriously
reliable inferential process of induction, but because perfectly
straightforward deductive arguments are able to eliminate falsehoods.
Induction is therefore irrelevant to the success of Science and no
‘problem’ then arises from recognising the effectiveness of science.
The Falsificationist View of Science
You’ll note that we said nothing about what
happens if the observations do not
falsify the hypothesis. Does that prove the theory correct? No, no more
than if Socrates had been unable to derive a contradiction from
Euthyphro’s definition of piety that would have proven Euthyphro correct.
All that anyone could say would be that Euthyphro’s definition had not
been falsified. It would remain a merely
provisional definition. Just so,
when observations are consistent with a theory they are said to
confirm it, but they can never
be said to prove it. To see why
this is, consider the form of the argument that you would be making if you
wanted to say that an observation
proved a theory. It would have to be something like this:
If the theory T is true then we should observe X
We do observe X
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Theory T is true
But this is a fallacy (called the ‘fallacy of
affirming the consequent.’) To see that it is a fallacy and not a
trustworthy form of argument, put it into more familiar terms.
If my battery is flat then the car won’t start
My car won’t start
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My battery is flat
And I know to my cost that the premises of this
argument may be true and the conclusion false. There was a loose
connection in the ignition circuit, but the battery was just fine.
So how
does a scientist prove his theory? Well, according to Popper, he
doesn’t. Scientific theories are never more than hypotheses. They can be
better or worse confirmed hypotheses, but they are always open to being
disconfirmed or falsified by the next experimental observation. No matter
how many confirmations we might have of continental drift or of the Earth
orbiting the Sun (rather than the other way around) they remain hypotheses
that may be falsified by the next observation. We shouldn’t think then of
scientific knowledge as a structure of necessary truths as Aristotle did,
but as a structure of hypotheses that are, as far as we can tell, the ones
that are best supported by the
evidence. It might be that many of those hypotheses are in fact
not true – but the process of
science, by continually testing more and more complex consequences of
these hypotheses, will eventually find them out. Falsehood can not survive
contact with evidence forever.
Objections
This is rather a neat solution, and has been
very influential in modern thought, especially because it seems to
describe the way that science progresses from ‘bad’ theories to ‘better’
ones. We may begin with, for example, the Ptolemaic Earth-centred
universe, replace that with the Copernican Sun-centred system with
circular orbits, replace that with the Keplerian system with elliptical
orbits, and so on until we arrive at our current view of the universe –
which we are sure is much closer to the truth than any earlier version
and which we fully expect to be
replaced by a better one at any time. Nevertheless, there are those who
find Popper’s elimination of the problem of induction less than
satisfactory.
The problem as they see it is that science really does give us positive knowledge about the world, and yet Popper’s solution allows us only to have negative knowledge. We think we can know things like ‘the stars are large gaseous bodies far off in space burning by nuclear fusion,’ but Popper will only allow us to say that we don’t know that that is false (or we know that it hasn’t yet been shown to be false, or etc.) Similarly, when we make predictions on the basis of experience we accept that there are more and less reasonable predictions. It is more reasonable, for example, to believe that if you touch a hot plate you will burn yourself than to believe that if you touch a hot plate you will be turned into a duck. The preference is clearly based on knowledge of hot plates and their effects that has been derived by induction from experience, and is clearly not justifiable unless somehow induction itself is justified – but Popper says there is no justification for induction. This sort of restriction on our knowledge claims is ok if you’re going to be a super-sceptic like Descartes, but we’ve already accepted that that sort of scepticism is pretty pointless and makes the word ‘knowledge’ useless. No, we want to be able to say that we have knowledge even about some things that are – just conceivably, in some possible world – not true. And if that’s the case, then induction seems to be how we go about getting it. We do naturally treat induction as a reliable way of getting new facts from old. So how do we justify that use of induction. Popper has nothing at all to say about this: he can still see no reason to support the claim that induction is reliable.
[1] Popper, K.
(1953) The Logic of
Scientific Discovery
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Goodman’s New Problem |
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But there are other arguments against the
reliability of induction that do not require Popper’s almost Cartesian
levels of scepticism. One of these was proposed by Nelson Goodman in 1950
and goes by the name of the ‘New Problem of Induction.’ It will be easiest
to see what this involves by means of a now-famous example. First let’s
remind ourselves of how standard enumerative induction is supposed to
work. We identify a property that exists in all observed instances of some
thing, and claim that that property will exist also in all
unobserved instances. The
example we’ve been using is the process by which we conclude that bronze
is brown; thus
Bronze thing number 1 is brown
Bronze thing number 2 is brown
…
Bronze thing number n is brown
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Bronze is brown It turns out that a
good deal of the intuitive appeal of this sort of induction depends on our
having a pre-existing notion of ‘brown’ that somehow ‘works’ in
inductions. And the same is true of other properties on which we use
induction. But suppose we defined a property ‘brue’ in the following way: Something is ‘brue’ if it is first
observed before [tomorrow’s
date] and is brown, or is not first
examined before [tomorrow’s date]
and is blue.
Then it seems we can equally well apply
induction on all the bronze things that we’ve ever seen to discover that
they are all brue and not brown.
Thus
Bronze thing number 1 is brue (it was seen before tomorrow
and was brown)
Bronze thing number 2 is brue (ditto)
…
Bronze thing number n is brue (ditto)
----------------------------------------
Bronze is brue
It’s clear that something has gone wrong here.
We now have two equally good inductions on the same set of observations
which have contrary conclusions[1].
And, of course, you can think up as many of these sorts of predicates as
you like, each of which would give another contrary conclusion.
It’s pretty clear, too, that the problem is in
the predicate, and that if we are going to have any faith in an induction
we will need to be sure that the predicate is not a problematic sort.
Goodman called the non-problematic predicates ‘projectible’ because they
named properties whose presence in further instances could reasonably be
projected from previous examples. But what is it that makes a predicate a
projectible one? You can’t just dismiss the problem by saying that
non-projectible predicates are badly-formed property words, or that you
can’t have time-dependencies in properties. To make that dismissal work
you have to be able to say why
it’s badly formed, or why a
property can’t be time-dependent like that.
Unfortunately, Goodman does not come up with a really convincing reply. He just says that the sorts of predicates that are projectible are just the ones that we’ve come to accept in our inductive practices in the past – i.e. we can conclude by induction that they are non-problematic. You might suspect this definition is susceptible to all the same objections as were made to inductions of other sorts, but I think that the point about the problematic nature of induction has been sufficiently made and we can move on.
[1] They can’t both
be true
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