Download the problem of induction

THE PROBLEM OF INDUCTION
It seems that empirical observations are
crucial to science.
But a key question arises:
 What role can observation play in
coming up with scientific theories?
 Can theories by confirmed by
observation?
Let us see why these issues are pressing.
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Deductive reasoning
The nice thing about deductive logic is that
it licenses conclusions that follow
necessarily from premises.
1.
Socrates is human.
2.
All humans are mortal.
Therefore:
3.
Socrates is mortal.
But deductive logic cannot tell us whether
the premises are true, and so cannot tell us
whether the conclusion is true.
We might summarize this as follows:
deductive logic provides certainty, but at the
price of offering no new empirical
information.
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Kinds of reasoning
Hume suggests that there are two kinds of
reasoning:
Relations of ideas: Deductive subjects such
as logic and math.
 To hold the premises but deny the
conclusion is contradictory.
 E.g., Socrates is a human, all humans
are mortal but Socrates is not mortal.
 Therefore, one can deduce conclusions
from premises by thought alone.
Matters of fact: Empirical sciences.
 Conclusions are not entailed by
premises—can consistently hold the
latter but deny the former.
 E.g., the sun has always risen but will
not rise tomorrow.
 Requires investigation into the world.
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Induction
We can never observe all instances of a
phenomenon across all of space and time.
 So, in investigating the world, we move
from samples to wholes.
 This is absolutely central to empirical
investigation of any kind.
This is known as inductive reasoning (what
Hume called reasoning about matters of
fact).
Hume: Induction can never lead to justified
beliefs.
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Temporal induction
Of particular importance to us is drawing
conclusions about the future on the basis of
past or present experience. For example:
 I’ve always liked Mars bars, so I’m
sure I’ll like this one.
 Air Canada has had very few
accidents, so it’s safe to fly with them.
 I’ve given blood for years without
problem, so I’m sure it’s safe to do so.
 Smokers have been shown to be at
high risk for lung cancer, so if you
don’t smoke you’re less likely to get
lung cancer.
We reason in similar ways all the time.
 Indeed, it is hard to see how we could
get on in life without induction.
So why doubt that induction is justified?
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Hume’s doubt
Hume: all such reasoning works only if
things go on as they have before.
 I.e., only if past experience is a reliable
guide to future experience.
 Without this assumption, the reasoning
is unjustified.
But how can you be justified in believing the
future will be like the past?
This can’t be proven deductively because it
is not a contradiction to assume that the
future will change.
 E.g., we can consistently hold that the
sun has always risen but won’t rise
tomorrow.
 So we need some other way to justify
inductive reasoning.
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First answer
In past experience, the future has
resembled the past.
 When I have reasoned inductively (as if
the future would resemble the past), I
wasn’t disappointed.
 So I won’t be disappointed in the future.
Hume: What could this prove?
 Just because the future has resembled
the past in the past, it doesn’t follow that it
will resemble the past from now on.
 If you assume this, you are arguing in a
circle!
 I want to know what the justification is for
assuming the future will be like the past.
 If you simply assume the future is like the
past, you haven’t given me a reason to
believe you.
This answer isn’t even an argument. It is
just a repetition of the conclusion we need
to prove.
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Second Answer
The past/present causes the future. If we
know the (past/present) cause we know the
(future) effect. So, induction is justified.
Hume:
 You only have the idea of causation
because, in the past, you have seen
things go together all the time.
Hence, the very concept of causation is
based on past experience.
 To assume that causation will continue as
it has is to assume the future will be like
the past.
 This is just circular reasoning again.
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Summary of the problem
Hume: the attempts to answer the problem
of induction cannot succeed.
1. Deduction won’t work.
2. Induction just begs the question:
 This argument is: the future will
resemble the past because in the past
the future has resembled the past.
 But this is just an instance of induction,
so we need to justify it.
This leads to an infinite regress.
 If we stop the regress anywhere, then
we are simply stating: “Induction works
because induction works”.
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Sceptical “solution”
Hume: we can’t stop ourselves from
reasoning inductively.
 It is human nature.
 Experience forms in us the habit of
assuming the future will resemble the
past.
However, there is no reasoning that can
justify this habit.
 We just do it, and that’s all there is to
say.
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What about laws of nature?
Third answer: but contemporary science is
very successful.
 Its success is based on discovering
exceptionless laws of nature.
 So, we are justified in concluding that
nature follows exceptionless laws.
Reply: This is still circular reasoning.
 Laws have been exceptionless up until
now—how do we know that will
continue?
Secondly, even if there are exceptionless
laws, how could we know we’ve discovered
them unless we’ve experienced phenomena
throughout all time?
What’s at issue is not the nature of laws, but
the nature of our knowledge of laws.
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Probability and induction
Answer 4: the more we notice a uniformity,
the higher the probability it will continue.
 The more often we see two things
together, the more probable that they
will appear together next time.
 We can’t prove that they always appear
together, but as we gather enough
evidence the probability will approach 1.
Problem: The number of observed cases is
always finite.
 It is possible that the number of
unobserved cases is effectively (or
actually) infinite.
 Hence, the probability may never reach
anything close to 1.
 At any rate, there is no way of proving
that it will.
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A disastrous conclusion
If Hume is right, we are in trouble.
All empirical science seems to depend at
least in part on induction to draw
conclusions about the world.
 If induction is irrational, so is all of
science.
Everyday reasoning depends on induction.
If induction is irrational, it is no more
rational:
 To take an elevator than jump off a roof.
 To eat bread than ingest arsenic
 To take an aspirin for a headache than
commit suicide
 Etc.
So something has gone wrong here, but
what?
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Popper’s “solution”
Popper’s argument:
 Science is the best means of gaining
knowledge.
 Science does not proceed by induction.
 Science proceeds by “conjectures and
refutations”.
 This only requires deductive logic.
Therefore, the problem of induction needn’t
be solved.
To begin, recall Popper’s view of science:
 Scientific theories must take risks—they
must make predictions that can be
falsified by experiment.
But how do we test a scientific hypothesis?
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Testing theories
For Popper, testing a theory, T, means
using deductive logic to determine what the
theory entails. I.e.:
If T, then Q.
Then, one tests to see whether Q is true.
 We know that if not-Q, then not-T.
This is because the following is a valid
argument form:
If T, then Q.
Not-Q
Therefore, not-T
If Yogi is a bear, then Yogi is a mammal.
Yogi is not a mammal.
Therefore, Yogi is not a bear.
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Example: Testing Einstein’s Theory
The star is in fact located here.
The star looks like it coming from here.
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Popper’s reconstruction of the test
Here:
 T = General Relativity.
 Q = light will be deflected by angle 
Had not-Q been observed, we’d have
concluded not-T.
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Passing the test
What can we conclude if Q is observed (as
it was)?
Well:
If T, then Q.
Q
Therefore T
is an invalid argument. Compare:
If John is in Guelph, then John is in
Ontario.
John is in Ontario.
Therefore, John is in Guelph.
So, observing Q does not confirm T.
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Corroboration
Popper: If we observe Q, then T is
corroborated (we’ve seen this before):
 Still in the running.
 Not falsified.
 Worth investigating further.
Key point: none of this requires induction!
Empirical investigation proceeds by
deductive conjecturing (and refutations).
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Summary
The problem of induction casts a great deal
of knowledge in doubt.
Hume: we must remain skeptical—much
knowledge is impossible.
Popper: we can give up induction: proceed
by deductive conjecture and refutation.
Question: is Popper’s stance a solution or
an evasion of the problem?
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Some further issues
Is Popper’s view oversimplified?
Suppose we had observed that the angle
that light deflects ≠ ?
 Would we have really given up on
General Relativity?
Probably not (recall Kuhn, Lakatos).
Even Popper admits this but then the
question becomes:
 What part of the theory do we give up?
Can we ever perform a test to determine
which hypothesis should be surrendered?
We’ll examine this issue soon.
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