Inductive Logic
The shaky foundation of everything
we know.
The story so far…
-Informal fallacies.
• Woefully imprecise.
• Incomplete.
– Syllogistic logic.
• Rigid but complete.
• Fallacies identified by rule violations.
– Propositional Logic.
• Flexible and complete.
• Fallacies identified by form.
What we’ve learned
• What is a proof?
– Different methods but same result.
– Deducive logic gives absolute certainty, but few
surprises.
• You don’t get new knowledge.
– Are we willing to trade certainty for more
knowledge?
Induction vs. Deduction
If all Cats are Felines;
and if all Felines are Mammals;
You can conclude with certainty that
all Cats are Mammals…
• Yes but how do we confirm that all cats are
felines?
A Quick History of induction
• Unappreciated in Ancient Times
– Aristotle: Deduction = reasoning.
– Valid Deductive arguments generate truths from
truths without error.
– Induction = “invalid”
– How do you get the truths in the first place?
• Look it up in Aristotle.
A Quick History of induction 2.
• Renaissance: (re-birth)
– Happens because of rediscovery of ancient
knowledge.
• Christian / Islamic culture clashes.
• 1000 Ce Europeans rediscover Plato.
• 1200 CE Europeans rediscover Aristotle.
– emphasis on new discoveries.
– Observation seems important source of
knowledge.
Early Notables in Induction
• Francis Bacon (1561-1626): Emphasis on
Neutral observation and inductive reasoning.
• Isaac Newton (1642-1727): Dedicates several
chapters of Principia Mathematica to
Inductive Logic.
– “show your work” means demonstrating new
methods in logic and calculus.
Some general comments.
• Inductive arguments are never valid.
• Inductive arguments are strong or weak.
• Conclusions of inductive arguments are not
“proven” in the way we can prove the
conclusion of a deductive argument.
Scientific “proof”?
What do we mean when we say the
word “proof” in an inductive
context.
–“prove that this BPE bottle is perfectly
safe”
–“I’m not moving back in until you can
prove the bedbugs aren’t coming
back”
Inductive proof.
• What does “Proof” mean in this context?
– “Certainty” as in deductive logic?
– “Beyond reasonable doubt” as in law?
– “reasonably likely”?
– “within the margin of error”
logically naïve may have
unreasonable expectations.
What makes a strong inductive
argument?
• In Inductive arguments, conclusions are never
conclusive.
• Can be overturned by new evidence.
– Thus the stronger the conclusion,
the weaker the argument.
– Ex: (remember your direct inferences).
•
•
•
•
All schools need reform.
Some schools need reform.
All schools need minor reform.
All schools should be burned to the ground.
What makes a strong inductive
argument?
• Premises are always based on a sample never
the entire range of data.
• As the size of the sample increases, the
strength of the argument increases.
• As the diversity of the sample increases, the
strength of the argument increases.
– Math techniques of statistics models adequacy for
sample size and stratification.
Branches of Inductive Logic
• Induction by Enumeration.
• Statistical Induction.
• Arguments from analogy
• Scientific Reasoning…. (?).
– Which may be just an amalgam of the 3 previous.
Analogical Arguments
An analogical argument is an inductive
argument that uses an analogy to show that
because one case has a particular feature, the
other case should, too.
An analogy is a claim that compares two (or
more) things.
Analogies
Learning is like rowing upstream.
My love is like a red, red rose.
Life is a rollercoaster.
 Note: these are analogies, not analogical arguments.
Argument from Analogy…
Cats are like dogs. Since dogs make good
pets, cats probably make good pets.
P1: Cats are like dogs.
P2: Dogs make good pets.
 Cats make good pets.
Sometimes the Argument implies the analogy.
Both Brenda and Susan live in Orange County, have
three children in private schools, and drive BMWs.
Since Brenda is wealthy, Susan probably is, too.
P1:
P2:
P3:

Brenda and Susan live in Orange County.
Brenda and Susan have three children in private schools.
Brenda and Susan drive BMWs.
Susan is like Brenda.
P1: Susan is like Brenda.
P2: Brenda is wealthy.
 Susan is wealthy.
Evaluating Analogical Arguments
Analogical arguments may be strong or weak.
Consider evidence for the analogy
 Sample size
 The more instances in the sample, the stronger the argument.
 Quantity of similarities
 The more characteristics shared by the sample and target, the
stronger the argument.
Consider relevance of the analogy
Which argument is stronger?
Brenda and Susan live in Orange County. Since
Brenda is wealthy, Susan probably is, too.
Carla, Mary, Brenda, and Susan all live in
Orange County. Since Carla, Mary, and Brenda
are wealthy, Susan probably is, too.
Which argument is stronger?
Both Brenda and Susan live in Orange County,
have three children in private schools, and
drive BMWs. Since Brenda is wealthy, Susan
probably is, too.
Both Brenda and Susan live in Orange County.
Since Brenda is wealthy, Susan probably is,
too.
Is the analogy relevant?
Brenda and Susan have three children. Since
Brenda is wealthy, Susan probably is, too.
Assessing Analogical Arguments
• Evaluate its strength by considering the
evidence provided for the analogy and the
relevance of the analogy to the feature.
• Strong analogy and strong relevance makes a
strong analogical argument.
• Weak in either makes a weak argument.
Example of assessment
An "online affair" is just like an affair
in person because they both devalue
their primary partners. Since divorce
is the appropriate response to an
affair in person, it is also an
appropriate response to an online
affair.
Summary: Argument by analogy.
• The inference of properties based on likeness.
– Ex: Bill and Ted are alike in height, hair color, age
and taste.
– Bill likes expensive wines.
– Therefore Ted will appreciate this bottle of
Chateau La Tour I’m giving him for Xmas.
General Form of Analogy
• A & B are alike in respects p, q, r…
• A has property x
• B has property x.
• Strong Arg:
– Lots of likenesses.
– X relevant to p, q, r
Parts of an Analogical Argument
Orange = More is
better
Red = Less is better
Relevance
between orange
and Red is better.
• A & B are alike in
respects p, q, r…
• A has property x
• B has property x.
Target
Analogy
feature
Fallacy of False Analogy
• To falsely claim similarity where non-exists, or
where the similarity is irrelevant.
– The “Hitler” rebuttal: anytime someone in authority
acts in a way someone doesn’t like.
– “Hitler’s name can only be invoked in an argument
if:1. Someone invades Poland. 2. Someone fails out of
art school.3. Someone declares genetic superiority
over their opponents. 4. Someone starts a war that
ends in the death of over 30 million people. 5. While
discussing the plots of the first and third Indiana Jones
movies.” Blog discussion of Seinfeld’s “it always comes back to Hitler.”
False analogy (visual)
• “After invading, Nazis used pre-war lists of gun
owners to confiscate firearms, and many gun
owners simply disappeared. Following
confiscation, the Nazis were free to wreak their
evil on the disarmed populace, such as on
these helpless Jews from the Warsaw Ghetto.”
Dis-analogies
• Analogical arguments can be countered by
presenting dis-analogies.
• Dis-analogies are where there is commonality
in the basis of the analogy, but not in the
purported property.
Ex A and B are common in respects x, y , z.
A has property p
B does not have property p.
Example of dis-analogy
• Canada has strong gun control laws (relatively)
• But does not regularly round up and
“disappear” citizens.
The ease with which you can find disanalogies is some indication of how
strong the analogical argument is.
Counterexamples:
• Technical term for Counter-Examples is
“refutation by logical analogy”…
– Is this correct: doesn’t that constitute a proof?
– Hint: analogy is perfectly exact…
Inductive Generalizations
Recall: Fallacy of hasty generalization
• Committed when a single or few instances are
used as evidence that something is generally
true.
• All the ravens observed so far are black, thus
• All ravens are black.
• Is this hasty generalization?
Inductive Generalizations
 An inductive generalization is an inductive argument
that concludes that some, most, or all of a particular
group has some feature based on evidence that a
portion of that group has the feature.
 The conclusion of every inductive generalization is a
general claim.
 A general claim is a claim that makes a statement
about all, most, or many members of a group or set.
General Claims
All swans are white.
One-third of college students smoke
cigarettes.
Junk food is high in calories.
Every cat I’ve ever owned was a good pet.
I bet that all cats make good pets.
P: Every cat I’ve ever owned was a good pet.
 All cats make good pets.
Issue: Whether all cats make good pets.
Every cat I’ve ever owned was a good pet.
I bet that all cats make good pets.
P: Every cat I’ve ever owned was a good pet.
 All cats make good pets.
S: the cats I have owned
T: all cats
F: make good pets
P: 80% of the 1,126 respondents nationwide ,who were
randomly polled by telephone, opposed the military
policy toward gays of “Don’t Ask—Don’t
Tell.”_______________________________
 A large majority of Americans oppose the “Don’t
Ask—Don’t Tell” policy regarding gays serving in the
military.
S: the 1,1126 respondents polled in the survey
T: all Americans
F: oppose the “Don’t Ask—Don’t Tell” policy
regarding gays serving in the military
Evaluating Inductive Generalizations
Inductive Generalizations may be strong or
weak.
Consider how well the sample represents the
target.
What factors influence how well the sample
represents?
Evaluating Samples
A random sample is one in which all members
of the target are equally likely to be in the
sample.
Randomness aims to ensure that the diversity
of the target is represented in the sample.
When the sample misrepresents the target,
the argument is a biased generalization.
Fallacy of biased sample.
Fallacy of biased statistics.
Which argument is stronger?
A poll taken of students in the dormitories showed that
most of the respondents thought that availability of
campus parking was not a serious concern. Thus, it’s
likely that most students don’t think that parking on
campus is a problem.
A poll taken of students at the campus bookstore
showed that most of the respondents thought that
availability of campus parking was not a serious
concern. Thus, it’s likely that most students don’t think
that parking on campus is a problem.
Evaluating Sample Size
The larger the sample, the stronger the
argument.
When the sample is too small to offer even
minimal support for the conclusion, the
argument commits the fallacy of hasty
generalization.
Which argument is stronger?
A poll taken of 95 students at the campus bookstore showed
that most of the respondents thought that availability of
campus parking was not a serious concern. Thus, it’s likely
that most students don’t think that parking on campus is a
problem.
A poll taken of 250 students at the campus bookstore …
Which argument is stronger?
A poll taken of 95 students at the campus bookstore showed
that most of the respondents thought that e-texts for critical
thinking courses were a bad idea, so Prof Phil should not use
an e-text next term.
A poll taken of 95 students in Prof. Phil’s course…
Inductive Generalization Analysis.
When the argument is an
inductive generalization, then
evaluate its strength by considering
sample randomness and sample
size.
ShopperTrak provides shopper-traffic counting
technology and data analysis for retail
businesses. According to ShopperTrak’s Retail
Traffic Index (SRTI), shopping traffic rose by
1.1% in Manhattan last month. It’s likely that
shopping traffic across the United States rose
by approximately 1% last month.
This argument is an inductive generalization, and is
weak because it is biased and hasty. The sample is
not random, and there is only one instance in the
sample.
Wording of conclusion?
• Can’t I make an inductive argument stronger
by rewording the conclusion?
– Ex: 80% of Students in the back 3 rows will VW.
– Mr. Sleepy sits in the back.
– Mr. Sleepy will VW.
Better conclusion?
Mr. Sleepy has an 80% likely-hood of VW’ing.
Wording of conclusion?
• Can’t I make an inductive argument stronger
by rewording the conclusion?
– Ex: 80% of Students in the back 3 rows will VW.
– Mr. Sleepy sits in the back.
– Mr. Sleepy will VW.
Better conclusion?
Mr. Sleepy has an 80% likely-hood of VW’ing.
Wait: that’s a classic syllogism!
• All back sitters are those with 80% likely-hood of
VW’ing.
• All Mr. Sleepys are Back-Sitters.
• All Mr Sleepys are students with an 80% likelyhood of VW’ing.
• This argument is deductive in form (and valid).
• The intent of the original argument is
“disguised”.
• This translation violates “principle of Charity”.
Lesson
• Probabilistic calculations are not inductive
arguments.
Inductive arguments have as their conclusion general claims based on particular
premises.
There is a separate calculus of probabilities developed in
19th (Bayes) and 20th (Keynes) centuries.
Example Fallacy of Biased Statistics
• 90% of those who responded to our online
questionnaire indicated they found out about
our farmers market online.
• Should we drop conventional advertising
because only 10% of our customers will see it?
Classic Biased Statistics Error
• Gallup phone survey 1948.
Chicago Tribune Prints Election
Headline in advance.
Phone survey had 2 problems:
Owning a phone was not
ubiquitous: tended to be older
and economically advantaged.
Survey was conducted 2 weeks
previously.
Truman won.
Avoiding Bias Errors
• Basic strategies.
– Randomized sample
• Needs to be quite large. (5000 back pain patients, NYU)
– Stratified Sample
• Make sure no large groups missed.
• Permits a smaller overall sample.
– Time Lapse Sample
• Watch for trends that would change results.
Fallacy of inadequate sample size
• Inductive argument based on inadequate
sample size.
• Hasty Generalization.
Rush quotes: 'The ozone hole size and persistence
have developed similarly to the year 2000, with an
early rapid growth observed during August, a record
size observed in September and finally its
disappearance in mid-November,' said a statement
by the World Meteorological Organization."
Rush Concludes: So, it's totally normal!
Having a giant hole in the ozone 2 years in a row doesn’t make it normal!
Summary: Inductive Fallacies
• Fallacy of Biased Statistics.
– The sample used in the premises doesn’t
represent the population described in the
conclusion.
• Fallacy of inadequate sample size.
– AKA. Hasty Generalization.
• Fallacy of False Analogy.
– There is inadequate similarity between the two
analogues.
The problem of Induction.
• David Hume: 1711 - 1776
Historian (invented neutral history).
Charming: no successful diner without him!
Philosopher: (published A Treatise of Human
Nature 1739) and other subsequent works.
Never appointed to any academic position
because it was feared he was an atheist.
HE WAS! As published posthumously…
The problem of induction
• Every inductive argument relies on the
presumption that what is unexamined won’t
be relevantly different then what is already
examined.
Ex: I’ve seen 5 million crows on 5 continents and
they were all black.
Therefore all crows are black.
This presumes unseen crows are not going to be
different with respect to colour.
The Principle of the Uniformity of
Nature. “PUN”
•
•
•
•
It seems natural to accept this presumption.
But what justification does it have?
Can we deduce it deductively?
Can we justify it inductively?
– What fallacy would that be?
Hume’s Conclusion
• Every inductive argument has the hidden
presumption:
– Nature is uniform,
what is unobserved will be just like what we’ve
observed.
This presumption can only be justified by saying:
“this presumption has always worked in the past”
This is inductive reasoning.
So induction is question begging.
Humes Argument (detail)
• PUN = (Principle of Uniformity of Nature).
• PUN is tacit premise in all Inductive
arguments.
• PUN is justified either deductively or
inductively.
• PUN cannot be justified deductively.
• PUN must be justified inductively.
Justification of PUN is question Begging.
Why not deductive?
• Hume thinks knowledge is of two kinds:
1 Matters of fact and existence.
2 Relations of ideas.
Relations of ideas are trivial, true by definition.
2 + 2 = 4 is true, but trivial.
This is deductive logic.
The new
Problem of
induction
AKA: THE GRUE / BLEEN PROBLEM
• This seems like a good inductive argument:
– All emeralds observed to date are green.
– Therefore all emeralds are green.
But from the same premises…
• If we define Grue as Green when observed
and blue when not observed.
• Then this seems an equally good argument.
– All emeralds observed so far are green.
– Therefore all emeralds are Grue.
• It seems wrong to say this argument is not as
inductively strong as previous argument.
What’s wrong?
• The concept of “grue” seems phony.
• We think things in the world don’t change
colour when we are observing them.
– On what basis are we to believe this?
What’s wrong?
• The concept of “grue” seems phony.
• We think things in the world don’t change
colour when we are observing them.
– On what basis are we to believe this?
The principle of Uniformity of Nature!
Furthermore…
• Quantum Physics seems to predict that some
properties (motion and mass) do depend on
observation…
• The details are tricky…
F of B’ing the Q.
• We need a better way of understanding
Inductive logic.
• Modern Inductive logic focuses on the
hypothesis – confirmation model.
J.S. Mill (1806-1873)
•
•
•
•
•
•
•
Raised by his father as a Prodigy.
Read Greek and Latin by age 4.
Logic by age 12, Economics by age 16.
Nervous breakdown @ 20.
Thought Hume’s results were horrid.
Social Reformer: Utilitarianism theory of ethics.
Utilitarians, including Mill, were largely
responsible for important social reforms like
welfare and abolition (of slavery).
• Important writings: “On Liberty”, “Utilitarianism”.
Mill’s system of Logic 1843.
• Tried to provide complete system of logic
immune to Hume’s criticism.
• Deductive logic = Logic of consistency.
• Inductive logic = description of practices.
• Induction obviously works,
so lets treat it as a social behaviour
(observe what works).
Mill’s Methods: Study of causation.
Mill’s Methods.
•
•
•
•
•
Method of Agreement.
Method of Difference.
Joint Method of agreement and difference.
Method of Residues.
Method of Concomitant Variation.
– (co-variables)
Method of agreement.
• If there is only one common condition A for all
cases resulting in w, then A is a cause of w.
• Form:
– A B C D occur together with w x y z.
– A E F G occur together with w t u v.
– Therefore A is the cause of w.
Example: Method of agreement
• Suppose you want to find out what caused
illness of people attending Russian Folklorama
Pavilian?
• We suspect some sort of food based on
symptoms.
Patient/food
CAVIAR
VODKA
Syrniki
BEET JUICE
Ivan
Y
Y
Y
Y
Tanya
N
Y
N
Y
Vlad
Y
N
N
Y
Petr
Y
Y
N
Y
Veronica
Y
Y
Y
Y
Aleksey
N
N
N
Y
Alisa
Y
N
Y
Y
Boris
N
Y
Y
Y
Dimitri
N
Y
N
Y
Patient/food
CAVIAR
VODKA
Syrniki
BEET JUICE
Ivan
Y
Y
Y
Y
Tanya
N
Y
N
Y
Vlad
Y
N
N
Y
Petr
Y
Y
N
Y
Veronica
Y
Y
Y
Y
Aleksey
N
N
N
Y
Alisa
Y
N
Y
Y
Boris
N
Y
Y
Y
Dimitri
N
Y
N
Y
According to Method of agreement,
you are justified in thinking that the Beet Juice is the Cause.
Method of Difference
• If the only difference between cases is the
condition A and the result w, then A caused w.
• Form:
– A B C D occur together with w x y z.
– B C D occur together with x y z.
– Therefore, A the cause of w.
Example: Method of Difference
• Suppose we want to know what causes a
vibration in a complex machine?
• There are several different functions that are
operating jointly.
• These functions normally operated together.
• Procedure, one by one, turn off until knocking
vibration stops.
Example method of difference.
Operation/sound
Vibration stops.
Main Power
No
Infeed power
No
Outfeed power
No
Label applicator
No
Leak Tester
No
Palletizer
Yes.
More complex example
• Often, variables cannot be so easily operated.
• Example: group of teens who sometimes do
really stupid things.
• By definition of problem, you can’t isolate
each teen, since you are looking for what
happens when in a group.
Group Behaviour
Night
Steve
Eric
Tom
Braden
Billy
Dan
Problem?
Mon
Y
Y
Y
Y
N
Y
Y
Tues
Y
Y
Y
Y
Y
N
Y
Wed
N
Y
Y
Y
Y
Y
Y
Thurs
N
Y
N
N
Y
Y
Y
Fri
Y
N
Y
Y
Y
Y
N
Group Behaviour Example
Night
Steve
Eric
Tom
Braden
Billy
Dan
Problem?
Mon
Y
Y
Y
Y
N
Y
Y
Tues
Y
Y
Y
Y
Y
Y
Y
Wed
y
Y
Y
Y
Y
Y
Y
Thurs
y
Y
N
N
Y
Y
Y
Fri
Y
N
Y
Y
Y
Y
n
Its Eric!
Joint Method of Agreement and
Difference.
• If A is present in otherwise diverse cases exhibiting a
result x, and is absent in otherwise similar cases not
resulting in x, then A caused x.
• Form:
–
–
–
–
A B C occur together with x y z.
A D E occur together with x t v;
B, C occur together with just y and z.
Therefore, A is causally connected to x.
Example
• Suppose we want to find out what causes
students to VW.
• Possible causes are:
•
•
•
•
•
Text
Prof
Quizzes
Debt
Grades
Why VW?
Student
Text
Prof
Quiz
Debt
Grades
VW
Ted
Loves
Hates
99
High
Low
NO
Bill
Hates
Respects
45
High
Low
Yes
Susan
Loves
Respects
88
High
High
No
Sally
Hates
Hates
79
Low
High
No
Homer
Loves
hates
48
Low
High
Yes
Why VW?
Student
Text
Prof
Quiz
Debt
Grades
VW
Ted
Loves
Hates
99
High
Low
NO
Bill
Hates
Respects
45
High
Low
Yes
Susan
Loves
Respects
88
High
High
No
Sally
Hates
Hates
79
Low
High
No
Homer
Loves
hates
48
Low
High
Yes
According to the joint method.
Quiz marks are low in all the cases of VW. (agreement).
Quiz marks are high in all the negative cases of VW (difference).
Example of Joint Method.
We used Mass spectrometry-based proteomics (MSP) to identify and quantify
thousands of proteins from healthy and collapsing bee colonies.
Co-occurrence of these microbes consistently marked CCD in (1) bees from
commercial apiaries sampled across the U.S. in 2006–2007, (2) bees
sequentially sampled as the disorder progressed in an observation hive
colony in 2008, and (3) bees from a recurrence of CCD in Florida in 2009.
The pathogen pairing was not observed in samples from colonies with no
history of CCD, namely bees from Australia and a large, non-migratory
beekeeping business in Montana. Laboratory cage trials with a strain of IIV
type 6 and Nosema ceranae confirmed that co-infection with these two
pathogens was more lethal to bees than either pathogen alone.
“Iridovirus and Microsporidian Linked to Honey Bee Colony Decline.”
J.Bromenshenk et al. PlosOne 2010.
why not use Joint method always?
• Data doesn’t always present even distribution
of cases.
• EG: A hospital attempting to find the cause of
an outbreak of disease is not bombarded with
negative cases, so the method of agreement
works best.
Method of Residues.
• Isolate known causes from unknown causes to
discern the specific contribution x of a specific causal
factor A to a causal system.
• Form:
–
–
–
–
A B C occur together with x y z.
B is known to be the cause of y.
C is known to be the cause of z.
Therefore A is the cause of x.
Very simple example of Residues.
• How much honey is on the truck?
• The loaded truck weighs 8800Kg.
– Causal system.
• unLoaded truck weighs 4440 Kg.
– This is the “known cause”
• The honey weighs 4360 kg.
– This is the “residue”.
More complex example.
What causes the stomach ache?
You have noticed that when you go to the
beach, you tend to drink tequila, and come
home sunburnt and hung-over.
On your last beach visit you ate some shrimp,
drank tequila and came home sunburnt,
hung-over and with a terrible stomach ache.
You conclude the shrimp caused the stomach
ache. Why?
Limit to methods so far:
• All measurements are binary:
– Present or absent
– Positive or negative.
• Many measurements are multi-valued.
– More or less
– Few many.
– Often expressed mathematically.
• Remember the quiz grades (high low).
Method of Concomitant Variation
(co-variables)
• If changing the value of one causal factor A
changes the value of a resulting condition x,
then A is causally connected to x.
• Form:
– A B C occur together with x y z.
– (+/-A), B C occur together with (+/-x) y z.
– Change in A is the cause of the change in x.
Example of Co-variation
• Altitude and Mercury in vacuum tube.
– The higher the altitude the lower the mercury.
• Temperature and stopping distance.
– As temperature decreases, stopping distance
increases.
– So buy snow tires!
Problem for M. of C. Variation.
New Phenomena
• New Phenomena Tend to correlate highly with
each other.
• Radio licenses and Mental Defectives:
England 1930.
Problems:
• Part of the damage to the aircraft could be
attributed to its impact with the ground.
Another part was definitely due to the wind
shear that the plane experienced as it fell from
the sky. However, some of the damage cannot
be accounted for by either of these factors.
Investigators are examining this evidence closely
for evidence of explosives.
• Which of mill’s methods is this reasoning using?
Problem 2
• No college wrestler has died in fifteen years
until now. Why did Rocky die? He was using
creatine.
• Charles worked for two years at a hospital.
During this time, the number of deaths
increased dramatically.
Problem 3
• At first we could not determine the cause, but
then we noticed that there were more cases
of the infection when more monkeys from
Uganda were present.
Limitations of Mill’s Methods.
• 1st Problem: Very strong assumptions.
– Agreement assumes there is only one similarity
between perfectly diverse cases
– Difference assumes there is only one difference
between perfectly similar cases
– Residues & Concomitant Variation assume that all
other contributing causes are known
Limitations #2.
• Doesn’t tell us about relationships between
possible causes:
– Scientific drinker example.
Paddy and Mick: the scientific drinkers.
Day / Drink
Gin
Monday
Y
Tuesday
Wednesday
Rum
Whisky
Y
Y
Water
Leprechauns
Y
Y
Y
Y
Y
Y
Paddy and Mick: the scientific drinkers.
Day / Drink
Gin
Monday
Y
Tuesday
Rum
Whisky
Y
Wednesday
Y
Water
Leprechauns
Y
Y
Y
Y
Y
Y
It’s the water!
Nothing in this method finds that there is something common between
Gin, Rum and Whisky.
Limitations #3: Interactions
• Mill’s methods
presumes
there is no
interaction
between
causes.
How would you decide what
caused this crash?
Example interactions.
Accident /
factor
Turning Left
Alchohol
Drugs
Speed
Cell Phones
#1
Y
N
Y
N
N
#2
Y
N
N
N
N
#3
Y
Y
N
N
N
#4
Y
N
N
Y
N
#5
Y
N
N
N
Y
This list of accidents at a given intersection seem to indicate that it is turning left as
the cause of accidents.
We suspect that the other factors are probably contributory, even though they are not
identified as such by Mill’s methods.
Scientific Confirmation
• Considerations of the sort Hume and Mill
propose lead us to think of scientific theories
as “conjectures” or theories that may be
overturned by the next experiment.
• Some philosophers have tried to understand
this process in terms of deductive logic.
Sir Karl Popper
• Austrian/British
philosopher 19021994.
• Moved to New
Zealand in 1937.
• Moved to England
1946 (LSE).
• Guess why he moved
to NZ?
Falsification
• No theory is every proven true.
• Many theories are proven false.
• Copi text page 348:
– A hypothesis must be testable. (H1)
The logic of confirmation
• From your theory deduce some expected
observation under test conditions.
• Design an experiment to produce these
conditions.
• Does your experiment produce the expected
observational consequence?
• Either Yes or No.
Your argument
• Let T = theory is true.
• Let O = expected observation.
Confirmation
If T then O
O
Therefore T
Disconfirmation
If T then O
Not O
Therefore Not-T
Poppers’ argument
• Let T = theory is true.
• Let O = expected observation.
Confirmation
If T then O
O
Therefore T
Fallacy of A’ing the
Consequent
Disconfirmation
If T then O
Not O
Therefore Not-T
Valid
Modus Tollens
Popper concludes
• Scientific reasoning is deductive in nature.
• Scientific knowledge is negative in character.
• Theories are decisively falsified using
deductive knowledge
• Theories are never proven, they merely
survive until proven false.
• Does it matter how hard you try to prove it
false? Survival of the fittest!
Implications of falsification
• The goal of scientific rationalism is to construct
and test falsifiable theories.
• Copi’s H1: a hypothesis must be testable.
• Copi’s H2: if predictions based upon a hypothesis
are true, this tends to show that the hypothesis is
true. (page 348)
• Implies
– A theory is not “scientific” if it is not falsifiable.
• Evolution vs Creationism.
• Critique of Creationism as “non-falsifiable”
• Likewise for astrology, conspiracy theories, etc.
Quine and Duheme
Pierre Duheme
Fr. 1861-1916
William V. Quine
USA 1908-2000
Duheme
• Observed that Newton’s claim to have deduced his
laws of motion from ‘phenomena’ and work of
previous astronomers like Kepler cannot be true.
• Newton’s law of gravitation contradicts Kepler’s laws of
planetary motion because under Newton’s laws, the
planets would have gravitational interaction.
• Nothing can be deduced from something it contradicts.
• Newton is wrong about how he arrived at his
conclusions.
Duheme concludes
• Newton’s reasoning involved hidden
assumptions and ignored important contraindicating information.
• Newton was basically right.
• Science is not as rational as thought.
• Physical theories are proposed and tested
against background assumptions.
• Duheme proposes that ideas are tested as
part of theoretical groups.
Quine
• Generalized Duheme’s Critique.
– “all hypotheses are tested against the background
of all relevant assumptions.”
• Two Dogmas of Empiricism. 1953.
– “it is impossible to test a theory in isolation”.
– We assume things like:
• The meanings of words.
• Logic;
• related beliefs are true.
Example
• For example, to test Newton’s Law of Gravity
by observing the planets, you need to know
the mass and positions of the sun and planets.
• When predictions of Neptune’s orbit proved
inaccurate in the 19th century, instead of this
being a falsification of Newton’s Law, it falsified
the assumption that there was only 7 planets.
• If an experiment fails, something is false, but
what?
Implications
• Can a theory be tested in Isolation?
– Popper proposed experiments have this structure:
If T then O
~O,
therefore ~T (valid, MT)
Let T = Theory true
O = observational consequence
AH = Auxilary Hypotheses are
true.
Instead we should have:
If (T & AH) then O
~O
Therefore ~T v ~AH
Proof of validity
1.
2.
3.
4.
(T  AH)  O
~O / ~T v ~AH
~(T  AH)
1, 2 MT
~T v ~AH
3 DM
DeMorgan’s Theorm
~(T  AM)  (~T v ~AM)
~T v ~AH ?
• Falsification is much less useful if all you know
is that something is false.
– Popper suggests “survival of the fittest”
• Those theories which rely on the fewest AH and make
the boldest predictions are the best theories.
• Any theory can be “saved” by addition of
suitable auxiliary hypotheses.
– Ad Hoc Hypothesis: modifications made “just” to
fit the data.
Ad Hoc theory revision.
• “Ad Hoc”: Latin: “for this”
• An Ad Hoc committee is a struck for a specific
purpose, rather then part of a fundamental
structure.
• An “Ad Hoc” hypothesis is added to a theory for
the purpose of accounting for outlying data.
• In ordinary language Ad Hoc sometimes means
“made up as you go”.
• Ex: the bombers offensive strategy was totally ad
hoc this year.
Ex. Ad Hoc theory
• Creationism:
– The theory that the world was created quite
recently by a non-natural process.
– Geologic and Paleontological evidence that the
world is far older is often accounted for by Ad Hoc
hypothesis.
God buried the dinosaur bones to “exercise our intellect”.
Geo-strata are evidence of a supernatural flood, not the
slower process of erosion and deposition by natural
forces over millions of years.
Ad Hoc ≠Wrong.
• Einstein thought the universe was stationary, but
observed mass predicted the universe would
ultimately collapse.
• Einstein proposed “cosmological constant” as a
modification of theory of general relativity.
• Einstein later called this “his greatest blunder”.
• Current theory has accepted the cosmological
constant as having a positive value which explains
the accelerating expansion of the universe.
Occam’s razor
• William of Occam / Ockham /
Okham. 1288-1346ish
• Studied at Oxford
• Did not complete his degree.
• AKA: the worthy beginner.
• Prefer the explanation with
the fewest possible number of
causes, factors or variables.
Entia non
sunt
multiplicand
a sine
necessitate
Direct translation: Don’t multiply entities beyond necessity.
Occam’s Razor vs. Ad Hoc
• Ad Hoc hypothesis tend to make theories
more complex.
• This is a “rule of thumb” not a law of logic.
• Recall: informal vs. formal fallacies.
– Informal fallacies: imprecise advice.
– Formal fallacies: clear requirements of logic.
• Application of Occam’s razor requires
judgment.
Which is simpler?
1. All species on earth are descended from a
few common ancestors and have evolved by
genetic mutation to fit particular ecological
niches according by “survival of the fittest”.
– This theory requires gene theory, evolutionary
theory, continental drift, etc.
2. God made everything.
– This theory requires exactly one super-natural
being, possibly with a sense of humor.
Super-empirical virtues.
• Those properties of theories that make them
‘good theories’ without being formally
processed by logic.
– Simplicity (Occam's razor) (copi’s H4)
– explanatory power (copi’s H3)
– Consistency (H5)
– Fruitful (H6)
Copi’s H2: page 348
• H2: If predictions based upon a hypothesis
are true, this tends to show that the
hypothesis is true.
– Confirmation problem prevents us from claiming
anything stronger.
Weasel
Words:
Likely
Tends
Probably
Suggest
Indicate
Etc.
Explanatory power
• H3: A hypothesis is more probably true if it
has a broader explanatory scope, that is, if it
explains more phenomena than alternative
hypothesis.
– IF my theory is true, does it explain other things?
Theory: my car won’t start because my battery is
dead.
That explains why my radio won’t work, and the
lights don’t come on.
Comprehensiveness
• H5: A hypothesis is more probably true if it is
consistent with the best theoretical
explanations available.
– My car won’t start because my battery is dead is
consistent with theories of electricity, etc.
– My car won’t start because it is possessed by
demons:
• Not consistent with best theories.
Fruitfulness
• H6: A hypothesis is more probably true if it is
fruitful, that is, if it predicts previously
unknown phenomena.
– Einstein’s theory predicts light rays would be
affected by gravity.
– During solar eclipse of 1919, this was observed.
Super-empirical virtues.
• Those properties of theories that make them
‘good theories’ without being formally
processed by logic.
– Simplicity (Occam's razor) (copi’s H4)
– explanatory power (copi’s H3)
– Consistency (H5)
– Fruitful (H6)
– testability (H1) as per popper.
– confirmation (H2) “survival of the fittest”
How do we know these virtues
indicate theoretical strength?
• By experience?
• Past theories with
these properties have
tended to be accepted.
How do we know these virtues
indicate theoretical strength?
Fie! That’s
induction!
• By experience?
• Past theories with
these properties have
tended to be accepted.
Inference to the best explanation
• When you have two or more possible
theories, which one should you prefer?
– The one that exhibits the most possible virtues:
H1-H6.
– Possible problems:
1. Ties. One theory is strong on h1-h4, other h3-h6.
Are some virtues better then others?
2. quantification?
How much simplicity is enough?
3. Scientific disputes…
Everyone says their theory is the best!
Limitations Of Inference to the
Best Explanation.
• When you conclude X is the best theory, you
chose from only those theories you know.
• A better theory may still be unknown.
• Inference to best explanation doesn’t
guarantee correctness.
– all carefulness about inductive arguments apply
– Not a valid process, merely a strong one.
– Weasel words required!
Problems:
• My Astrologer predicted that I would hate my
critical thinking class, and I did!
• On this basis, is Astrology a good theory?
• Astrology presumes a causal relationship
between the planets and events in our lives.
• More likely: EVERYONE hates critical thinking!
CSI: Ye olde London
• Moriarty being the murderer would explain
his fingerprints on the weapon and the
victim's blood on his jacket. The assumption
that any of the other suspects has committed
the murder leaves us without an explanation
of these traces. Moriarty must be the
murderer.
– Explanatory power high.
More problems in Chapter 9.
• Media regularly report cases of “super-human”
strength in parents who act to save their children.
• “'Supermothers' and grandfather lift 1 ton
Renault Clio off trapped schoolboy”
www.dailymail.co.uk
• This seems inconsistent with what we know
about human anatomy, breaking strength of
bones etc.
• Better explanation, the suspension of the car did
most of the work.