Today we’ll try to cover a number of things: 1. Learning philosophy/philosophy of statistics 2. Situating the broad issues within philosophy of science 3. Little bit of logic 4. Probability and random variables

1. Day #2: 6334
Today we’ll try to cover a number of things:
1.
2.
3.
4.
Learning philosophy/philosophy of statistics
Situating the broad issues within philosophy of science
Little bit of logic
Probability and random variables
(need a full list to get participants in Scholar!)
1

2.  Learning to do philosophy is very different from learning
philosophy as a museum of different views, what so and so
said, various “isms”
 How to teach “doing philosophy,” and how the student is to
recognize acquiring the skill, is not cut and dried—(no one can
really say, but I can tell when you’re “getting philosophical”)
 To teach you to be philosophers—even if it’s only in this
class—we’ve got to teach something that is unfamiliar and
maybe even painful for some. (I know we have some philosufferers here, who should know what I mean.)
2

3.  We won’t typically come out and “give the answer” in
wrestling with a philosophical issue, but may deliberately hold
back to encourage you to wrestle with it.
 On the other hand, Professor Spanos and I have been working
in this general area for a long time, and developing positions,
responses to challenges, etc.
 You also learn how to do philosophy by witnessing the
strongest arguments, and the effort that goes into carefully
engaging them.
 So there’s a balance here wherein we want to both give and
hold back, feed but make you struggle…
 So expect that.
3

4.  Everyone can see our publications, no secret there, and people
participating in this course (listed or added to Scholar) will also
get bits from my still-being-written book (on which I definitely
want your feedback).
 So you will be getting our strongest arguments, but also the
strongest arguments out there, and we’ll want you to work
through the arguments on your own.
 The most important thing—admittedly, also the most
unusual—is that anyone who wants to make progress in the
area of philosophy of statistics must be prepared to question
and challenge everything you read (even by the highest of the
high priests).
4

5.  We know it feels strange. (This does not typically hold for
other subjects.)
 In that first chapter of How To Tell What’s True About
Statistical Inference, I note that “a certain trepidation and
groupthink take over when it comes to philosophically tinged
notions such as evidence, inductive reasoning, objectivity,
rationality, truth”.
 “The general area of philosophy that deals with knowledge,
evidence, inference, and rationality is called epistemology.
 The epistemological standpoints of leaders, be they
philosophers or scientists, are too readily taken on faith as
canon by others, including researchers who would otherwise
cut through unclarity, ambiguity, and the fuzziness and
handwaving that often surround these claims”.
5

6.  Progress in mathematical statistics doesn’t require philosophy,
but too often “we don’t need to be philosophical about these
concepts or methods” means “I don’t want to examine them
very closely”.
 Writing philosophy (I gave you some pointers already on the
blog):
o Here, too, a slow examination of the question involved is
required.
o Less (much less!) is more.
o We guarantee you will find this kind of analytical writing
of value in your other fields.
o As a start, the first assignment will be make-believe for
next time. (e-mail)
6

7. PHILOSOPHY:
Understand and Justify Human Knowledge About the World
PHILOSOPHY OF SCIENCE:
How Do We Learn About the World In The Face of Uncertainty
and Error?
 Is there a scientific method?
 How do we obtain good evidence?
 How do we make reliable inferences from evidence?
 What makes an inquiry “scientific” or rational?
 Is there scientific progress?
7

8. Problems with answers from logic empiricist philosophers of
science:
Is there a scientific method?
 The “logics” for science are oversimple, open to paradoxes
 Standard canons are violated in actual science
 Scientific methods change with changing aims, values,
technologies, societies
How do we obtain good evidence?
 Empirical data are uncertain, finite, probabilistic
 Data are not just “given”, they have to be interpreted
introducing biases, theory-ladenness, value-ladenness
(scientific, social, ethical, economic)
8

9. How do we make reliable inferences from evidence?
 Inductivist: cannot justify “induction”
 logics of induction failed
 Falsificationist: deduction won’t teach anything new; cannot
pinpoint blame for a failed prediction
What makes an inquiry “scientific” or rational?
 None of the philosophical attempts to erect a demarcation for
science seem to work.
Is there scientific progress?
 no account of cumulative growth of knowledge
 old “paradigms” are swept away by new ones which are
“incommensurable” with the old
9

10. PHILOSOPHY:
Understand and Justify Human Knowledge About the World
PHILOSOPHY OF SCIENCE:
How Do We Learn About the World In The Face of Uncertainty and Error?
LOGICAL EMPIRICIST ATTEMPTS: 1930-60
“Armchair Philosophy of Science”
Inductive Logics: C(h,e)
Carnap
Logic of Falsification
Popper
Problems, Paradoxes
“Historicism in Philosophy of
KUHN (1962)
Science”
CRISIS IN PHILOSOPHY OF SCIENCE
POST-POSITIVISM
STS/ HPS
optimistic
pessimistic
Relativism
New Models of Rationality
“naturalistic turn”
THEORY CHANGE
MOVEMENT-1970s, 80s
Kuhn, Lakatos, Laudan
Search for More Adequate
Theories of Induction, Testing,
& Decision-Making
Postmodernism
anarchy
“Rational Reconstruction”
dadaism
BAYESIANS
irrationality
Inference as updating degrees of belief
social
Sc
constructivism
New Experimentalist Turn 1980s,1990s
Error Statistical Philosophy: Mayo 1996?
10

11. Little Bit of Logic
(Double purpose: both for arguing philosophically and for
understanding inductive/deductive methods)
Argument:
A group of statements, one of which (the conclusion) is claimed to
follow from one or more others (the premises), which are regarded
as supplying evidence for the truth of that one.
This is written:
P1, P2,…Pn/ ∴ C.
In a 2-value logic, any statement A is regarded as true or false.
11

12. A deductively valid argument: if the premises are all true then,
necessarily, the conclusion is true.
To use the “⊨” (double turnstile) symbol:
P1, P2,…Pn ⊨ C.
Note: Deductive validity is a matter of form—any argument with
the same form or pattern as a valid argument is also a valid
argument.
(Simple truth tables serve to determine validity)
12

13. EXAMPLES (listing premises followed by the conclusion)
Modus Ponens
If H then E
H_______
E
Modus Tollens
If H then E
Not-E____
Not-H
If (H) GTR, then (E) deflection effect.
(not-E) No light deflection observed.
 (not-H) GTR is false
(falsification)
These results depend on the English meaning of “if then” and of
“not.” In context, sentence meanings aren’t always so clear.
13

14. Disjunctive syllogism:
(1) Either the (A) experiment is flawed or (B) GTR
is false
(2) GTR is true (i.e., not-B).
Conclusion: Therefore, (A) experiment is flawed.
If either A or B are true, and not-B, then infer A.
Since if A were not true, you can’t also hold the two
premises true—without contradiction.
(e.g., soup or salad)
Either A or B. (disjunction)
Not-B
Therefore, A
So we have 3 valid forms.
14

15. Deductively Valid Argument (argument form):
Three equivalent definitions:
 An argument where if the premises are all true, then
necessarily, the conclusion is true. (i.e., if the conclusion is
false, then (necessarily) one of the premises is false.)
 An argument where it’s (logically) impossible for the premises
to be all true and the conclusion false. (i.e., to have the
conclusion false with the premises true leads to a logical
contradiction: A & not-A.)
 An argument that maps true premises into a true conclusion
with 100% reliability. (i.e., if the premises are all true, then
100% of the time the conclusion is true).
15

16. True or False?
If an argument is deductively valid, then its conclusion must be
true.
To detach the conclusion of a deductively valid argument as true,
the premises must be true.
Here’s an instance of the valid form modus tollens:
Example: Let A be: Bayesian methods are acceptable.
and let B stand for: Bayesian methods always give the same
answer as frequentist methods.
If A then B
Not-B
Not A.
16

17. So you may criticize a valid argument by showing one of its
premises is false.
(Deductively) Sound argument: deductively valid + premises are
true/approximately true.
17

18. Exercise: showing deductively valid arguments can have false
conclusions.
(1) (I did this above) Give an example of an argument that
follows the valid pattern of modus tollens that has a false
conclusion.
(2) Exercise (YOU do this one) Show that an argument
following the form of a disjunctive syllogism can have a false
conclusion.
Either A or B
Not-B
Therefore A.
18

19. Invalid Argument: Consider this argument:
If H then E
E________
H
(H) If everyone watched the SOTU speech last night, then (E)
you did.
(E)You did._________________________
So, (H) everyone did.
Affirming the consequent.
19

20. Invalid argument: An argument where it’s possible to have all
true premises and a false conclusion without contradiction.
20

21. Inductive argument. With an inductive argument, the conclusion
goes beyond the premises. So it’s logically possible for all the
premises to be true and the conclusion false. So an inductive
argument is invalid.
The premises might be experimental outcomes or data points:
E1, E2,…En (e.g., light deflection observations, drug reactions,
radiation levels in fish)
Even if all observed cases had a property, or followed a law,
there’s no logical contradiction in the falsity of the generalization:
H: All E’s are F
21

22. This is also true for a statistical generalization:
60% in this class watched SOTU
Thus, H: 60% of all people watched SOTU
H agrees with the data, but it’s possible to have such good
agreement even if H is false.
The problem of induction: how to justify such inductive
inferences.
It’s not clear how to characterize it much less to justify the move
(if it should be justified).
(Some would say we only use deduction, and we’ll talk about this
two classes from now.)
22

23. Inductively “good” argument (how should we define it?)
Can we try to parallel the definition of deduction validity?
(i)A deductively valid argument, if the premises are all true then,
necessarily, the conclusion is true.
E1, E2,…En ⊨ H.
(i)* An inductively good argument is one where if the premises are
true, then the conclusion is probable (?)
E1, E2,…En ⊨ probably H.
(i)* probabilism
Does this mean infer H is probably true: Pr(H) = high?
23

24. Try a second definition of deductively valid:
(ii) Deductive. The argument leads from true premises to a true
conclusion 100% of the time.
(ii)* Inductive. The argument leads from true (or approximately
true) premises to a true conclusion (1 – α)% of the time. (?)
(ii)* highly reliable method (reliabilism?)
(Differs from highly reliable conclusion)
24

25. High “reliability” of the method, low long-run error isn’t enough.
Remember the BP Deepwater Horizon oil spill of 2010?
The BP representatives claimed to have good (inductive) evidence
that
H: the cement seal is adequate (no gas should leak)
In fact they kept decreasing the pressure until H passed, rather
than performing a more severe test (as they were supposed to)
called "a cement bond log” (using acoustics)
25

26. Oil Exec: Our inference to H: the cement seal is fine
is highly reliable
Senator: But didn’t you just say no cement bond log was
performed, when it initially failed…?
Oil Exec: That’s what we did on April 20, but usually we do---I’m
giving the average.
(Imaginary): We use a randomizer that most of the time directs us
to run the gold-standard check on pressure, but with small
probability tells us to assume the pressure is fine, or keep
decreasing the pressure of the test till it passes….
26

27. Overall, this test might rarely err but that is irrelevant in appraising
the inference from the data on April 20.
The oil rep gives a highly misleading of the stringency of the
actual test that H managed to “pass”.
Passing this overall “test” made it too easy for H for pass, even if
false.
The long-run reliability of the rule is a necessary but not a
sufficient condition to infer H (with severity)
27

28. Severity demands considering, for the context at hand, what
error(s)* must be (statistically) ruled out in order the have
evidence for some claim H
(*construed as any erroneous claim or misunderstanding
empirical and theoretical)
And being able to evaluate a tools probative capacity to have
discerned the flaw, were it present.
An inductive argument is good if its conclusion has passed a
severe test (with the premises and background).
H agrees with the data, but it’s highly improbable to have such
good agreement even if H is false.
28

29. Error (Probability) Statistics
Error probabilities may be used to quantify probativeness or
severity of tests (for a given inference)
The logic of severe testing (or corroboration) is not probability
logic
It may well be that there’s a role for both, but that’s not to unify
them, or claim the differences don’t matter…
29

30. I began with a set of questions, and hope to show how they are
tackled in the error statistical philosophy.
30

31. Problems with answers from logic empiricist philosophers of
science:
Is there a scientific method?
The “logics” for science are oversimple, open to paradoxes
Standard canons are violated in actual science
Scientific methods change with changing aims, values,
technologies, societies
How do we obtain good evidence?
Empirical data are uncertain, finite, probabilistic
Data are not just “given”, they have to be interpreted introducing
biases, theory-ladenness, value-ladenness
(scientific, social, ethical, economic)
31

32. How do we make reliable inferences from evidence?
 Inductivist: cannot justify “induction”
 logics of induction failed
 Falsificationist: deduction won’t teach anything new; cannot
pinpoint blame for a failed prediction
What makes an inquiry “scientific” or rational?
 None of the philosophical attempts to erect a demarcation for
science seem to work.
Is there scientific progress?
 no account of cumulative growth of knowledge
 old “paradigms” are swept away by new ones which are
“incommensurable” with the old
32