THE PROBLEM OF INDUCTION
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
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 do you know 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 depends on induction.
 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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Justifying induction pragmatically
Reichenbach: Hume is right:
 We can’t prove that the conclusion of an
inductive inference must true.
But the real question is: is induction useful
when trying to decide how to act?
It does seem possible to justify induction
according to this standard. For example,
perhaps we can show that:
Induction is the best guide to action that we
have (even if not that good).
If this is the case, then the principle is
justified.
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An analogy
Suppose S is dying.
 The doctor says: There are no known
drugs or therapies that will save S.
 However, it is possible that this
operation will remove the disease, but I
don’t know that with any certainty.
Reichenbach: It is justified to perform the
operation.
 It is the best (and only) hope.
Similarly: if induction turns out to be the best
way to guide our lives, then it is justified to
follow it even if we can’t prove it will work.
 Okay, but can we show that induction is
the best principle here?
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Reichenbach: Two possibilities:
1. The world is ordered.
2. The world isn’t ordered.
If #2, then no method of prediction will work.
 So, using induction is no better but also
no worse than anything else.
So, using induction is as justified as
anything can be.
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Induction in an orderly world
Assume, now, that the world is ordered.
Then: any method of determining patterns
of event occurrence will have to:
 Examine past frequencies.
 Project them into the future.
In other words, all methods of empirical
science depend on induction.
 So if any works, induction will work.
 If induction doesn’t work, nothing else
will.
Either way, induction is justified as the best
(only) approach we have.
So: whether the world is ordered or not, the
best approach you can take to guiding your
actions is the inductive approach. It is,
therefore, justified.
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Problem
This doesn’t show very much.
 Even if induction is necessarily the best
principle we have, it doesn’t follow that it
is a very good one.
Reichenbach: that’s just our predicament!
 The best we can do is to make our bets
to the best of our abilities.
 There’s nothing we can do about the
fact that future is at best uncertain.
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Popper’s evasion
Popper’s argument:
 Science is the best means of gaining
knowledge.
 Contrary to Hume and Reichenbach,
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.
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Popper on the scientific method
Popper: Some argue that the hallmarks of
science are observation and verification.
But:
1. Many non-sciences are based on
observation:
 E.g.: Astrology, alchemy, palm reading
2. Excessive verification is a weakness of a
theory.
 If anything can be interpreted to fit a
theory, this is reason to doubt it.
 We should be suspicious of anything
that is impossible to refute.
Popper’s conclusion:
Scientific predictions should be risky: they
should be refutable.
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Example: Testing Einstein’s Theory
The star is in fact located here.
The star looks like it coming from here.
General Relativity predicted the angle, ,
with precision. This was risky: it could have
been wrong.
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Conjectures and deductive logic
For Popper, testing a theory, T, means
using deductive logic to determine what the
theory entails:
 If T, then Q
then trying to falsify that:
 Test for Q
 If not-Q, then not-T (as a matter of
deductive logic)
Problem: what parts of T do we reject?
Remember underdetermination?
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Corroboration
Suppose we find that Q. What have we
learned?
 If T, then Q; Q; therefore T
is invalid, so what does Q tell us about T?
Popper: T is “corroborated”.
 Still in the running.
 Not falsified.
 Worth investigating further.
Key point: this does not require 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.
Few philosophers have found satisfactory
answers to the problem.
Reichenbach: we can give induction a
pragmatic justification: it will work if anything
does, so stick to it.
Popper: we can give up induction: proceed
by deductive conjecture and refutation.
These seem more like “evasions” of the
problem than solutions to it.
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