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Traditional Problem of Induction

Induction (by enumeration): Infer from past cases where a law

or generalization held to future cases.

or

Induction: (by enumeration): Infer from all (or most) A's have

been B's in the past to all (or most) A's will be B's in the future.

(A law may be seen to assert a generalization: e.g., all A's are

B's — in the past, present and future.)

(e.g., Tomatoes have been nourishing in the past, law of gravity

has held in the past)

How to justify (scientific) induction?

A justification takes the form of an argument:

________________________________________ _____________________________________

Conclusion: The inductive method is reliable (it will work

in the future): inferring from past to future success is a

reliable method.

What will the premises be?

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a. Use an inductive argument to justify induction:

Premise: The inductive method has worked, has been

reliable, in the past. ______________________________________

Conclusion: The inductive method is reliable (it will work

in the future), i.e., inferring from past cases to future cases

is a reliable method.

Problem: circular. (It uses the method in need of justification to

justify that method.)

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b. Use a deductive argument to justify induction:

Premise: If a method has worked (been reliable) in the past,

then it will work in the future.

Premise: The inductive method has worked, has been

reliable, in the past. ______________________________________

Conclusion: the inductive method is reliable (it will work in

the future), i.e., inferring from past cases to future cases is a

reliable method.

Problem: not a sound argument.

In order to infer the truth of the conclusion of a deductively

valid argument, the premises must be true, i.e., the argument

must be sound. But one cannot know the first premise of this

argument is true, unless one already knew the very thing which

the argument was supposed to justify!

Alternatively put in terms of assuming the uniformity of nature

Attempts to dissolve the problem

It’s asking for justification beyond where it’s appropriate,

(i) It’s converting induction to deduction (i.e., it’s asking

for a certainly true conclusion from true premises)

(ii) That’s just what we mean by rational

For (ii), consider my friend the crystalgazer…

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Counterinductive method:

Infer from All A's have been B's in the past to the next A will

not be a B.

In terms of a method:

Infer from the fact that a method M has worked well (been

reliable) in the past that M will work poorly in the future.

or

Infer from the fact that a method M has worked poorly (been

unreliable) in the past that M will work well in the future.

(The crystal gazer)

If a method M has worked poorly (been unreliable) in

the past then M will work well in the future.

M has worked poorly (been unreliable) in the past

Therefore, M will work well in the future.

So, unless we allow this justification of counterinduction, we

should not allow the following justification of (scientific)

induction:

If a method M (induction) has worked well in the past

then M (induction) will work well in the future.

M (induction) has worked well in the past.

Therefore, M will work well in the future.

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You might say, you just can’t argue with someone who accepts

counterinduction as rational.

It’s like someone who would not accept any argument, thereby

rejecting machinery necessary for fruitful discussion.

You can’t convince them that induction is superior to

counterinduction, but all we care about is showing it’s rational for

us to accept scientific induction

Leads to the program of building a rational inductive logic:

(a) A system of inductive logic is rationally justified if it

captures inductive rules of science and common sense that

we take as a standard of rationality.

(b) A stronger possibility under this umbrella would be to

show scientific inductive logic is better suited for

accomplishing its goals than any other.

This was the hope at least into the 80s and even now in some

circles.

This was called the New Problem of Induction (confirmation

theory, formal epistemology—a bit more general).

Every fairly obvious rule winds up with everything confirming

everything.

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Instance confirmation

Probabilistic confirmation

An instance of All A’s ‘are B’s is an example of an A that is a B.

All ravens are black.

For all x, if Rx then Bx.

Also plausible, if S = S’, then anything that confirms S confirms

S’.

All ravens are black = All non-black things are non-ravens.

So a white shoe confirms all ravens are black.

(Armchair ornithology)

This was the kind of problem that leads to requiring severity, or

something like it.

It’s too easy to get confirmations even if the generalization is

false.

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Mayo: Brief Popper Notes

Demarcation Criterion: Popper’s problem is how does one determine (demarcate) a

scientific theory from a non- or pseudo-scientific theory? This

is the problem of Demarcation.

The common answer was based on science having a unique

“empirical method”. This was essentially inductive proceeding

from observation or experiment to theory.

But Popper wants to distinguish between empirical methods that

are scientific from those that are not, and methods may appeal to

observation and experiment (e.g., astrology) but fail to come up

to scientific standards.

The theories that got him involved: Marx’s theory of history,

Freud and Adler’s theories of psychology, and Einstein’s theory

of General Relativity.

His friends were impressed with these theories’

“explanatory power”: once your eyes were opened, they

seemed to explain everything, studying them caused one to

undergo a psychological conversion.

He presented Adler with a case that seemed to go against

Adler’s theory, but Adler explained it even though he had never

met the child! Popper asked him how he could be so sure, Adler

replies, “Due to my thousand fold experience.” Popper thought

to himself (or said aloud), and now the experience is thousand

and one fold!

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The upshot is that all Adler’s observations were interpreted in

light of his theory.

In fact, one can use the theory to explain human actions that are

exactly opposite, e.g.,

(a) a man drowns a child,

(b) a man sacrifices his life to save a drowning child.

Both may be explained using the resources of either theory (e.g.,

inferiority complex or Oedipal complex.)

Freud:

In case (a), Freud might say the man suffered from repression,

while in (b) man had achieved sublimation.

Adler:

In case (a), the person suffered from an inferiority complex,

needs to show he can commit a crime; in (b), the behavior is due

to an inferiority complex, as the need to show that he died to

rescue the child.

There was no behavior that couldn’t be interpreted in terms of

either theory.

In contrast, Einstein’s theory was impressive because it had a

risky prediction: if the predicted light deflection effect was

observed to be absent, the theory would be refuted.

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Popper concluded (36):

1. It is easy to obtain confirmations, if we look for them.

2. Confirmations should only count if they are the result of risky

predictions, if without the theory, we should have expected an

event incompatible with the theory.

3. Every good theory is a prohibition, the more it forbids, the

better it is.

4. A theory that is not refutable by any conceivable event is not

scientific. Irrefutability is not a virtue.

5. Every genuine test of a theory is an attempt to falsify it or

refute it. There are degrees of testability.

6. Confirming evidence should not count except when it is the

result of a genuine test of the theory—it must be able to be

presented as a serious but unsuccessful attempt to falsify it

(corroborating evidence).

7. Some testable theories when found false are upheld by their

admirers, e.g., by introducing ad hoc some auxiliary

assumptions or reinterpreting it ad hoc so that it escapes

refutation. The price paid is to destroy or lower the scientific

status of the theory.

#6 (weakest form):

An accordance between data* and theory or hypothesis H

does not count as evidence for H if such a "passing result"

is guaranteed (or practically guaranteed), even if H is false.

*Data x does not falsify H, x fits H, H passes the test, etc.

Why? The upshot is “The criterion of the scientific status of a theory is

its FALSIFIABILITY.”

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(Note Popper is not talking about meaningfulness, only that

unfalsifiable theories are unscientific.)

But #2, and #6 are stronger…..they include a positive

side….how can Popper argue for them?

Before pondering this….let's look at the big picture of how

Popper claims to deal with Hume…

Side note: Popper always puts the weight on the theory to be

scientific and testable; whereas, I think he should put it on the

method or test.

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Popper’s attitude toward Hume’s Problem of Induction:

Hume was right. We can’t infer from Constant Conjunction to

the next case—we get an infinite regress when we try to justify

induction by appeal to experience. (See induction handout.)

Popper criticizes Hume on psychological (e.g., the puppies)

and logical grounds, p 43-44.

His logical criticism of Hume:

*Hume’s theory is that of repetition based on similarity.

* All we really have is repetition-for-us based on similarity-for-

us, we must respond to situations as if they were equivalent, take

them as similar, and interpret them as repetitions.

*Repetition can never be perfect, they are only similar, so they

are repetitions only form a certain point of view

* That means there must always be a point of view, a system of

expectations, anticipation, and interests BEFORE there can be

repetition.

We can’t explain recognizing cases as similar as due to earlier

repetitions because these earlier ones must also have been

recognized as repetitions.

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Two Options or Stances Given Hume’s Problem: p. 45

1. We obtain knowledge non-inductively.

2. We obtain knowledge by induction and thus by a logically

invalid and rationally unjustifiable procedure.

Hume chooses option 2 and concludes that knowledge is just

belief based on habit.

Popper wants to take the first option instead, top p. 46.

Instead of explaining our propensity to expect regularities as

the result of repetition, Popper proposes to explain repetition

for us as the result of our propensity to expect regularities and

to search for them.

We actively try to impose regularities upon the world: Theory

of trial and error, of conjectures and refutations.

(p.46—the idea that science proceeds from observation to

theory is so widely held that many thought Popper was crazy

to deny it. But as he says, one cannot “simply observe”.)

p. 51 Hume said our beliefs in regularities are irrational—he’s

right if he means we can’t prove them true or probable;

however, if belief includes our critical acceptance of scientific

theories, a tentative acceptance combined with an eagerness

to revise the theory if we design a test it cannot pass, then

(according to Popper) Hume was wrong, there’s nothing

irrational in accepting such a theory, in preferring it or in

relying on it for practical purposes.

Why? (assumed premise)

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Because no more rational course of action is open to us.

p. 51 “Assume we have made it our task to live in this

unknown world of ours; …”

nothing "safer" than accepting the "best tested" theory.

(Question: is this convincing? What’s so good about a

hypothesis that has passed a severe test, as Popper defines it?

How can Popper even know a test is severe given especially

that he thinks there are always infinitely many hypotheses?)

Another way Popperians like to put this (Popper's way of

avoiding the problem of induction):

Although the truth or approximate truth of H is unjustifiable,

it may be rational to (tentatively) accept H.

"One should be careful not to confuse the problem of the

reasonableness of the scientific procedure and the (tentative)

acceptance of the result of this procedure, i.e., the theory or

hypothesis—with the problem of the rationality…of the belief

that this procedure will succeed."

According to Popper: Although it is not reasonable to believe

that the method of conjecture and refutation will succeed, or

is likely to succeed (i.e., that it is reliable), the method of

conjecture and refutation is nevertheless, a reasonable

procedure.

Again, why?

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No better than the amoeba

p. 52: the critical attitude may be described as the conscious

attempt to make our theories, our conjectures, suffer in our

stead in the struggle for the survival of the fittest…a more

dogmatic attitude would eliminate theories by eliminating us.

Summary of Popper’s conclusions: p. 53-4

Since the problem of induction is viewed by Popper as a facet

of the problem of demarcation, the solution to the latter must

provide a solution to the former.

Because of the problem of induction a, b, c seem to clash—

some give up on empiricism, but Popper says there is no clash

if accepting a law or theory is only tentative.

We can preserve the principle of empiricism: the fate of a

theory is decided by observation and experiment, by the result

of tests.

So long as a theory stands up to the severest tests we can

design, it is accepted if it does not, it is rejected; but it is never

inferred from the evidence, only the falsity of the theory can

be inferred, and this inference is purely deductive.

(Question: Is it? To infer the conclusion one needs true

premises and determining truth demands other inductive

reasonings, not mere deduction.)

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Conjectures and refutations includes falsification (in case of a

failed prediction, although even Popper notes that we

shouldn’t give up the hypothesis too easily, a bit of

dogmatism may be good), and corroboration (tentatively

accepting a hypothesis or theory that has passed a stringent

test).

Critical rationalists: We may justify the rationality of

accepting, or preferring, or believing the “best tested” theory

T, without justifying T itself (as true, probably true)

Probabilism: Justifying claim H is showing it is true or

probable.

Popper rejects probabilism.

The Popperian suggestion is that it’s a matter of showing it

grew out of a rational method—(you couldn’t say it was a

reliable method, but at most that the method had stood up to

testing).

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Falsifiability is both too strong and too weak.

Too weak because H being logically falsifiable hardly

means you’ve done a good job probing its errors.

Too strong in that most interesting scientific theories aren’t

logically falsifiable:

-There are background assumptions and theories

required to arrive at any prediction (Duhem’s

problem).

-Most testable predictions are statistical.

Popper: Neyman-Pearson Fisher statistics

Carnap: Bayesian statistics

Popper knew all this, that doesn’t mean they weren’t

problems.

I have been reading the older Popper in writing my current

book, and there he explicitly refers to Fisherian rules of

rejection for statistical claims (buried in Popper’s LSD).

Want to be able to say:

Pr(d(x) > observed; H is false)= very low

For d some difference between x and what is expected

were H true.

Essentially a p-value or statistical significance level

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What about corroboration?

Popper sets out those “confirmation measures” alongside

Carnap.

But he declares that none of them measures corroboration

unless they may be seen as the result of severe testing

Doubts that severity can be formalized any more than the

rule of “total relevant evidence” among inductivists (i.e.,

probabilists)

Popper in a letter to me: I regret not having learned

statistics

In looking to statistics to solve induction, Bayes theory

philosophers generally looked to a Bayesian confirmation

theory (logic or subjective)

We saw last time: If H entails x, P(x |H) > P(x).

So x “confirms” H to a degree (non-subjectivists embrace

the Bayes-boost view, rather than a high posterior).

Chapter 4 of EGEK, Wesley Salmon’s approach:

P(H|x)/P(J| x) = P(H)/P(J)

In the case where H entails x and J entails x

[P(x |H)P(H)]/P(x)

[P(x |J)P(J)]/P(x)

and since P(x |H) = P(x |J) = 1

P(H| x)/P(J| x) = P(H)/P(J)

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Along with any background needed to get the entailments

Unfortunately, H and J are not exhaustive.

Salmon (a Reichenbach student) rejected subjective

accounts.

They were reluctant to appeal to statistical methods

because of their empirical assumptions.

Salmon changed his mind once I showed him simple tests

of assumptions.