Critical Thinking 03 consistency

Consistency

A Standard of Critical Thinking

Review
Philosophy: the attempt to
answer, critically, the
epistemological, metaphysical,
and ethical questions.

An Issue Question: is a yes-orno question, and is the way to
formulate questions for the
disputed question method
(quaestio disputata).

Principle of Charity: for any
claim, give it the strongest
possible interpretation.

Principle of Sufficient
Reason: for or any claim,
give reason why it is true, or
not.
A virtue is an excellence, a
mean between two extremes
called vices.

Review
The Disputed Question Method

yes:

Reasons supporting
the yes answer or
refuting the no answer.
Which side
has the better
reasons?

the question
no:

Reasons supporting
the no answer or
refuting the yes
answer.

Review
Truth: when a claim matches
what is.

Ambiguous: A term with
more than one meaning.

Claims are candidates for
truths, such as beliefs
stated in language.

Vague: A term with an
unclear extension.

Review
Reasons supporting
the yes answer or

yes:
Does the thing designated
by the subject have the
property expressed by the
predicate?
no:

Which side
has the better
reasons?
Reasons supporting
the no answer or
refuting the yes
answer.

Review
Reasons supporting
the yes answer or

yes:
Do the things designated
by the nouns stand in the
relation expressed by the
verb.
no:

Which side
has the better
reasons?
Reasons supporting
the no answer or
refuting the yes
answer.

Review
to pursue the truth of a claim,
avoid matters of taste, fill in
the indices, and spell out the
ceteris paribus.

to evaluate the truth of a
claim determine whether the
thing designated by the
subject has the property
expressed by the predicate

to evaluate the truth of a
things designated by the
nouns stand in the relation
expressed by the verb.

Review
to determine the truth of a
thing designated by the
subject is in the extension of
the predicate.

to determine the truth of a
things designated by the
nouns are in the extension of
the verb.

to define a term give it one
clear meaning.

Review

yes:

Reasons supporting
the yes answer or

Are the things designated
by the nouns are in the
extension of the verb?
no:

Which side
has the better
reasons?
Reasons supporting
the no answer or
refuting the yes
answer.

Review
Normative definitions which
prescribe or stipulate how a
term ought to be used.

Descriptive definitions
which describe or designate
how a term is actually used.

Normative claims which
prescribe or stipulate how
things ought to be.

Descriptive claims which
describe or designate how
things actually are.

Law of Assumption—
assume anything at any
time.

Review
define key terms—key terms
are terms needed for the
claim to be evaluated as true
or false, they are terms
whose properties or relations
need to be specified or whose
extensions need to be
clarified.

define technical terms—
technical terms are terms
needed for critical thinking,
such as truth, charity, and
reason.

Logically Complex Truths

And Terms Indicative of Logical Structure

A Logically Simple
Truth

The man is cheating.
1

True

2

False

A Logically Simple
Truth

1

True

2

False

—He’s hiding cards

A Logically Simple
Truth

1

True

2

False

—He’s doing magic

A Logically Simple
Truth
Two logical
possibilities

1

True

2

False

Given that
we’ve filled in
the indices,
made the
ceteris paribus
explicit, and
defined key
terms.

A Logically Simple
Truth
Also called
States of
Affairs or
Possible
Worlds
1

True

2

False

A Logically Simple
Truth
How we
describe logical
possibilities

Logical
possibilities


State Descriptions

1

True


2

False

The man is not cheating.

Negation: Logical Complexity



1

True

1

False

2

False

2

True

it “flips” the truth value



1

True

1

False

2

False

2

True

Logical negation, often
expressed in English by ‘not’, is
true when the component claim
is false, false when the
component claim is true. It is
symbolized by ‘~’ and has the
logical form ~P.

~ “flips” the truth value
The Logical Form of Negation: ~P, -P, ¬ P
P

~P

1

True

1

False

2

False

2

True

double negation

Any even number of negations
cancel each other out.
No! They do not!

indicators for negations

-

a. not
b. It is not the case that…
c. n’t (the contraction)
negations are often suppressed in
common opposites such as on and off.

Fabricate a Truth
Test this out:
Take a true claim and make it false. Then
take a false claim and make it true.
What happens when you add not to a
true claim? What happens when you add
another not?

A Logically Simple
Truth
Two logical
possibilities

Derek is leaping.
1

True

2

False

Combining Logically Simple Truths

Derek is leaping.
1
2
3
4

Hansel is leaping.

True

True


Derek is leaping.

Hansel is leaping.

1

True

True

2

True

False

3
4


Derek is leaping.

Hansel is leaping.

1

True

True

2

True

False

3

False

True

4


Derek is leaping.

Hansel is leaping.

1

True

True

2

True

False

3

False

True

4

False

False

Four
possibilities

Derek is leaping.

Hansel is leaping.

1

True

True

2

True

False

3

False

True

4

False

False

Four states of
affairs (states)
or possible
worlds

Derek is leaping.

Hansel is leaping.

1

True

True

2

True

False

3

False

True

4

False

False

to calculate the number of possible worlds

raise two to the power of the number of
claims being evaluated

2

n

Four State
Descriptions

Derek is leaping. Hansel is leaping.

State Descriptions

1

True

True

Derek and Hansel are leaping

2

True

False

Derek leaps but Hansel doesn't

3

False

True

Hansel leaps but Derek doesn’t

4

False

False

Neither Derek nor Hansel leaps

Logical Possibilities
Place your bet

Here’s the bet:
In the next slide both Derek and
Hansel will be leaping.
Which way would you bet, and
why?

Who wins the bet?
In state #1 where both Derek and Hansel are leaping?

Derek is leaping.

AND Hansel is leaping.

1

True

?

True

2

True

False

3

False

True

4

False

False

Who wins the bet?
In state #1 where both Derek and Hansel are leaping?
Derek is leaping.


1

True

True

True

2

True

False

3

False

True

4

False

False

The complex claim is true when all the component claims
are true, as they are in state #1.

Who wins the bet?
The complex claim is true when all the component
claims are true, as they are in state #1.
Derek is leaping.


1

True

True

True

2

True

?

False

3

False

True

4

False

False

How about case #2 where Derek is leaping but Hansel is
not?

Who wins the bet?
The complex claim is false when one or the other
component claims are false…
Derek is leaping.


1

True

True

True

2

True

False

False

3

False

?

True

4

False

False

…as they are in state #2…

Who wins the bet?
The complex claim is false when one or the other
component claims are false…
Derek is leaping.


1

True

True

True

2

True

False

False

3

False

False

True

4

False

False

…or as they are in state #3, what about state #4?

Who wins the bet?
The complex claim is false when both component
claims are false.
Derek is leaping.


1

True

True

True

2

True

False

False

3

False

False

True

4

False

False

False

As they are in state #4.

Logical conjunction, often
expressed in English by ‘and’, is
true when the component
claims it joins are true,
otherwise it is false. It is
symbolized by ‘&’. It’s logical
form is P & Q.

Logical Form of Conjunctions
P AND Q
P&Q
P•Q
P⋀Q
P

&

Q

1

True

True

True

2

True

False

False

3

False

False

True

4

False

False

False

to prove a conjunction false

Prove that one of the component
claims is false.

indicators for conjunctions

a.
b.
c.
d.

and
but
yet
not both (includes negation)

Contradictions

Putting conjunction and negation together

Contradictions
A Necessity of Logic
Let’s define ‘this square’ as the thing depicted below at x-position 303 px
and y-position 318 px and of the dimensions 242 px by 242 px.

this square

Contradictions
Next let’s define ‘white’ as the color depicted below of the RGB values
Red = 255, Green = 254, Blue = 235.

white

Contradictions
Now pay attention to your mental processes as we make the following
claim:

Contradictions
Now pay attention to your mental processes as we make the following
claim:

>>This square is white and this square is not white.<<

Contradictions

What was your reaction?

Contradictions
Putting Negation and Conjunction Together
Which claim is not a contradiction?
Hockey is better than basketball but it is not
better than basketball.*

Jupiter is bigger than Mars and it is
not bigger than Mars.
Drinking milk is healthy and unhealthy.*
New York is and isn’t the largest city in the US.*

Same-sex schools are optimal and same-sex
schools are less than optimal.
Eleven is a prime number and
eleven is not a prime number.

Romeo and Juliette is a tragedy and it is not a tragedy.*

The child looks at the jellyfish and looks away from it*.

The jellyfish has tentacles—not!

The music is loud and the
Jacqui thinks black is more alluring than pink and
music is quiet.*
she doesn’t.

The Constitution of the United States was adopted on September 17, 1787 and
The Constitution of the United States was adopted on July 4, 1776.*

Contradictions
The Logic of a Contradiction

The square is white

&

The square is not white

1

True

?

False

2

False

?

True

Given that conjunctions are true when all component claims are
true, what is the truth value of this conjunction?

Contradictions
The Logical Form of a Contradiction: P & ~P

The square is white

&

The square is not white

1

True

False

False

2

False

False

True

Contradictions are false in all possible worlds.

Contradictions
The Logical Form of a Contradiction: P & ~P

P

&

~P

1

True

False

False

2

False

False

True

Contradictions are false in all possible worlds.

Contradiction, a special form of
conjunction in which a claim
and its negation are joined—
they are always false. The
logical form of a contradiction is
P & ~P.

Contradictions
The Logical Form of the Principle of Noncontradiction

~

(P

&

~P)

1

True

False

False

2

False

False

True

The Principle of Noncontradiction is true in all possible worlds.

Contradiction
An Emergent Property
Neither hydrogen nor
oxygen are wet at room
temperature—wetness
emerges as a property when
they are properly combined.
In a similar manner, being
false in all possible worlds
emerges when a claim and
its negation are properly
conjoined.

The Principle of Noncontradiction

~(P & ~P)

The Principle of
Noncontradiction, states that no
thing can, at the same time and
in the same manner, both have
and not have the same property.

The Principle of
Noncontradiction, (special) no
claim, adequately defined, can
be both true and not true.

The Principle of
Noncontradiction, no claim,
adequately defined, can be both
true and not true.
The Principle of
Noncontradiction ~(T & ~ T)

Consider…
Four Types of Truth

∏

Matters of Taste or Opinion
Matters of Convention
Matters of Fact

Matters of Necessity

=

3.141592...

π needs to be exact for a circle to be round

+

1

=

1

2

simple arithmetic is the way things are

~

(

P

&

P

)

Noncontradiction is needed for critical thinking

Consistency
A set of claims free from contradictions
Romeo and Juliette is a
tragedy.

The jellyfish has tentacles.

Eleven is a prime
The music is loud. number.

Drinking milk is healthy.

Jupiter is bigger than Mars.

Same-sex schools are optimal.
Hockey is better than basketball.

Jacqui thinks black is more alluring than pink.

New York is the largest city in the US.

The child looks at the jellyfish.

The Constitution of the United States was adopted on September 17, 1787

Consistency
A set of claims free from contradictions
Which claim causes the inconsistency?
Romeo and Juliette is a
tragedy.


Eleven is a prime
The music is loud. number.

Drinking milk is healthy.

Jupiter is bigger than Mars.

Same-sex schools are optimal.
Hockey is better than basketball.

The jellyfish has tentacles—not.
Jacqui thinks black is more alluring than pink.

New York is the largest city in the US.

The child looks at the jellyfish.

The Constitution of the United States was adopted on September 17, 1787

The Standard of Consistency—
accept only those beliefs which
are consistent with each other
and any accessible evidence.

Counterexamples

Using the Principle of Noncontradiction
to test definitions.

Proof by Counterexample
A Method for Reasoning with
Contradictions
Line of
Reasoning

An explanation showing
that the definition should
be true of a specific
example (thing or event).
Reject the
original
definition

Original definition.

Another Line of
Reasoning

Another explanation
showing that the
definition is not true of
the same example.

Counterexamples
Testing Definitions for Consistency
Definition: ‘father’ means the female parent

You know the definition is wrong, but
how can you prove it?

Extensions
Terms have extensions

being a vowel
M

X
N
D

C

L

Y

V

B

F
E

A
R
Q

O
W
P

S

I U
H

G
K
J

Z
T

True Definitions
The Subject and Predicate Have Identical Extensions
Even numbers are divisible by two without remainder

being divisible by two
without a remainder

being an even number
3

9
22

1
1
21

32

14
77

27
66

144

13
5

False Definitions
The Subject and Predicate Do Not Have Identical Extensions
a, e, i, o, and u are the only vowels

being a, e, i, o, and u

being vowels
B

L

M

F

V

G

C
X
N
D

E
A

O

I U

R
Q

K

Y
H

P

J

Z
T
S

W

Counterexamples
Definition: ‘father’ means the female parent
father

the female parent

List of fathers:

List of female parents:

Adam, Joseph, Martin,

Michelle, Mary, Hillary,
Sarah

Muhammad

Generate lists of things that fall under the term being defined
and the property used to define it — they should be identical—
stop when you show they are not.

False Definitions
a, e, i, o, and u are the only vowels

being a father

Joseph
Adam

Martin
Muhammad

being a female parent
Michelle
Hillary
Sarah

Mary

These lists are not identical, in fact, they have no overlap at
all, no members in common.

Counterexamples

So: ‘father’ does not mean the female parent
father
List of fathers:
Adam, Joseph, Martin,

Muhammad

the female parent
≠

List of female parents:

≠

Michelle, Mary,
Hillary, Sarah

Applied
By this
definition

Hillary should be the
father, because she is a
female parent.

Father means the female parent

By all other
sources

Reject: father
means the
female parent

Hillary is not the father
but the mother, which is
defined as the female
parent.

Counterexamples
Reject: ‘father’ means the female parent

Proof: by this definition, Hillary is a father
because she is the female parent, but she’s not a
father according to many other sources
(dictionaries and encyclopedias) which define
the female parent as the mother. This is a
contradiction, and I reject the definition in favor
of general usage.

Counterexamples
Reject: ‘father’ means the female parent

Alternate Proof: by this definition, Martin is
not a father because he is not the female parent,
but according to biology texts he is a father,
because he is the sperm donor to the offspring.
This definition is inconsistent with biological
terminology—so I reject the definition.

Counterexamples
Testing Complex Definitions for Consistency
Definition: Mammals have fur, mammary glands, and give live birth

Using the Principle of Noncontradiction,
prove that this claim is false.

Counterexamples
Definition: Mammals have fur, mammary glands, and give live birth

Mammals have fur
Mammals have mammary glands
Mammals give live birth

Break the definition into a series of claims which isolate each
property, to prepare to test each. Then take the one you will test.

Counterexamples
Definition: Mammals give live birth
mammal

giving live birth

List of mammals:

list of things which give live birth

cats, dogs, humans, platypuses

cats, dogs, humans

Generate lists of things that fall under the term being defined
and the property used to define it — stop when you determine
that they are not identical.

False Definitions

being a mammal

giving live birth
dogs

platypuses cats
humans

These do have significant overlap, but they are not identical.

Counterexamples

So: Some mammals do not give give live birth
mammal

giving live birth

List of mammals:

≠

list of things which give live birth

cats, dogs, humans, platypuses

≠

cats, dogs, humans

Applied
By this
definition

By research

Platypuses should give
live birth, as they have fur
and mammary glands.
Reject:
mammals give
live birth.
Platypuses lay eggs and
so do not give live birth.

Counterexamples
Reject: ‘mammals’ means gives live birth

Proof: by this definition, Platypuses should
give live birth, as they have fur and mammary
glands, but research has discovered that
Platypuses lay eggs and so do not give live
birth. This definition contradicts the evidence,
and I would revise the definition to be:
mammals have fur and mammary glands.

Counterexamples
Reject: ‘mammals’ means gives live birth

Alternate Proof: by this definition, Platypuses must
not be mammals as they lay eggs rather than give
live birth. But they are mammals insofar as they
have fur and mammary glands. This definition is
inconsistent with the rest of the taxonomical
systems, and I would revise the definition to be:
mammals have fur and mammary glands.

to evaluate by counterexample

Isolate the subject and predicate,
generate lists of things that fall
under each, stopping when you
determine that they are not
identical.

proof by counterexample

Choose an item that is not on both
lists, explain how the definition
says it should be, then explain why
it is not, indicate the inconsistency,
and reject or revise the definition.

Reductio ad absurdam

Using the Principle of Noncontradiction
to test claims.

A formal extension of reasoning
with contradictions
Line of
Reasoning
Original claim.

Another Line of
Reasoning

An explanation showing
how the thing designated
by the subject has the
property expressed by the
predicate.
Reject the
original claim.
Another explanation
showing how the thing
designated by the subject
does not have the property
expressed by the predicate.

Testing Claims for Truth
Claim: The jellyfish has no tentacles.

You know the claim is
wrong, but how can you
prove it?

True Claims
The Subject has the Property Expressed by Predicate.

The jellyfish has tentacles

Evidence:
the jellyfish
has tentacles

False Claims
The Subject is Inconsistent with the Property
Expressed by Predicate.

The jellyfish has no tentacles
Evidence:
the jellyfish
has no tentacles

Testing Claims for Truth
Claim: The jellyfish has no tentacles.
Evidence:
the jellyfish
has tentacles

A formal extension of reasoning
with contradictions
Line of
Reasoning

The claim is that the
jellyfish under
observation ought to have
no tentacles.

The jellyfish has no tentacles.

Another Line of
Reasoning

Reject the
original claim.

But observation shows that
the jellyfish in question has
many tentacles.

Testing claims for Consistency
Reject: the jellyfish has no tentacles

Proof: The claim is that the jellyfish under
observation ought to have no tentacles, perhaps
due to predation. But cursory examination
shows that the jellyfish has many tentacles
which appear healthy. As the claim contradicts
observation I reject it.

The Parts of a Reductio

The jellyfish under observation ought to have
no tentacles, perhaps due to predation. But
cursory examination shows that the jellyfish has
many tentacles which appear healthy. As the
claim contradicts observation I reject it.

The Parts of a Reductio
1. The jellyfish under
observation ought to
have no tentacles

2. due to predation

3. the jellyfish has no
tentacles

1.claim
2.reasons
3.conclusion
4.other reasons
5.other conclusion
6.contradiction
7.rejection

4. by cursory examination

5. the jellyfish has
many tentacles

6. the jellyfish has tentacles and the jellyfish has no tentacles
7. I reject it

The Logical Form of a Reductio
Reductio ad absurdam, Indirect Proof,
1.claim
2.reasons
3.conclusion
4.other reasons
5.other conclusion
6.contradiction
7.rejection

Indirect Proof, Proof by
Counterexample
Line of
Reasoning
1. Claim

2. reasons
3. conclusion

6. P & ~P

Another Line 4. other reasons
of Reasoning 5. other conclusion

7. Rejection


Some Cases

The Case of Sara Scatterleigh
Sara woke in a hurried blur. Her alarm did not go off. Her heart
pounded as she got out of bed, dragged a comb across her head,
found her way downstairs and drank a cup, looking up she
noticed she was late. She grabbed her coat and grabbed her pack,
out the door in seconds flat—her chemistry class started ten
minutes ago! And this teacher always took role.
Sara approached her class with apprehension, open the door and
march in like she’s not late? Or try to steal in. Still out of breath
from the jog to class, Sara opens the doors and marches right into
the large lecture hall, only she doesn’t recognize a single soul.
The teacher pauses his lecture to regard her, but it is not her
chemistry teacher! Confused, Sara retreats into the hall and looks
at the room number on the wall, it is the right room, she should be
late, her chemistry class meets on Tuesdays and Thursdays. With
a look of vacant frustration Sara draws out her phone to double
check the time, and only then notices the day, Monday.

The Case of Mrs. Riley
Most jurors were initially swayed by the Prosecutor’s claim
that the accused were guilty. This was based on the testimony
of Mrs. Riley, who positively identified the accused at the
scene of the crime at the time the crime was committed.
After all, Mrs. Riley seemed like an honest woman with no
bias against the accused. Further, she testified that saw them
from 100 feet away and was wearing her eye glasses—
because of this everyone assumed that she could see 100 feet.
However, on cross examination, the defense attorney, Vinny,
conducted an impromptu eye test by holding up two fingers
from a mere 50 feet away while she had her glasses on. Mrs.
Riley failed this test, thinking she saw four fingers instead of
two. So Mrs. Riley could not see 100 feet, because she could
not see even 50 feet. As this is a contradiction the accused
were found to be innocent.

The Case of Longfellow Deeds
In the case of one Longfellow Deeds it was claimed that Mr.
Deeds was not legally competent to manage his own affairs. An
attorney argued that Mr. Deeds suffered from what was then
called bipolar disorder (we now call it manic depression). To
show that Mr. Deeds was abnormal the attorney called many
witnesses, who claimed Mr. Deeds was ‘pixelated’, ‘crazy’,
‘cracked’, and ‘nuts’. Examples of his abnormality included
playing the tuba and running around naked in the park. However
by giving the proper context Mr. Deeds made it plain that playing
the tuba was as normal as doodling, filling in the ‘o’ s on a
printed page, or having a nervous tick. Also, he ran around naked
because he was drunk for the very first time—and behaving oddly
when you are drunk is fairly normal. Because Mr. Deeds
provided convincing counterexamples to the claims that his
behavior was abnormal the judge declared him legally competent
to manage his own affairs

The Case of the Reluctant Rubbernecker
A woman had just presented her paper to her local senator on the
negative effects traffic oscillation due to rubbernecking—in short
rubbernecking contributes to traffic jams. Her claim was that if
everyone knew that rubbernecking caused up to 60% of the delays due
to common accidents, they would do the right thing and there would be
fewer rubberneckers. She recommended a public service advertising
campaign, contending that once people knew the cause of such delays
they would not rubberneck. The presentation was a success and she
convinced the senator to back her plan. But as she drove home she
noticed traffic slowing to a crawl, then saw the cause—a horrific
accident. The woman felt the impulse to gawk, to be a rubbernecker.
But she kept her mind firmly on the fact that she knew rubbernecking
to be wrong and refused to contribute to a longer delay. She fought the
impulse but, in the end, she gave in to her impulse, slowed down to
gawk, and became another rubbernecker. Despite her knowledge and
her best effort at self-control, she knew her thesis was flawed.

Reductio: An Example in Neuroscience
While Studying the actions of motor cells in monkeys (called motor
cells because they are the first in the sequence that controls the muscles
that move the body) Vittorio Gallese was moving around the lab during
a lull in the day’s experiment. A monkey was sitting quietly, waiting
for her next assignment, when Vittorio reached for something (perhaps
ice cream) and heard a burst from the computer connected to the
electrodes in the monkey’s brain. It might have sounded like static but
to the ear of a neuroscientist meant the motor cells were firing. Vittorio
thought the reaction was strange—the monkey was sitting quietly, not
grasping anything, yet this neuron affiliated with grasping fired. No
one could imagine that motor cells could fire merely at the perception
of someone else’s actions. In light of the theory at the time this made
no sense. Cells in the brain that send signals to other cells that are
connected to muscles have no business firing when the monkey is
completely still, hands in lap, watching somebody else’s actions. And
yet they did.

The Case of the Mysterious Disease
In 1955 a mysterious illness infected nearly 300 of the staff of
the Royal Free Hospital, forcing it to close. Some tests showed
their muscles did not twitch or quiver uncontrollably, their
reflexes were normal, and their nervous systems were normal.
This led a few researchers to claim that it was merely mass
hysteria and the patients were otherwise healthy. However,
numerous studies showed that the group gave no indications of
mass hysteria and that they exhibited a pattern of symptoms
including severe exhaustion, memory loss, confusion, painful
lymph nodes, muscle pain, and unusual headaches. These
researchers claim that the disease was unnamed but real and
that the patients were sick. Since then, the balance of evidence
showed that the disease was real and it was subsequently
named myalgic encephalomyelitis (ME) or chronic fatigue
syndrome (CFS).

The Case of the Composite Soul
Against the claim that the soul is simple, Plato
tells of Leontius, the son of Aglaion, who saw
some corpses lying at the executioners feet.
Leontius had a strong urge to indulge his
morbid intrigue and gawk at the dead, but he
forced himself to show respect by not indulging
his morbid intrigue so he turned himself away.
For a while he fought the urge and covered his
face. But desire overcame him and he ran to the
corpses and looked, then rebuked himself for
this indignant act.

The Case of Renegade Mercury
Newton’s theory was remarkable in that it
described force correctly in terms of acceleration
and mass, explained gravity, and correctly
predicted the course of the planets. Newton’s
theory was right. Except for Mercury. Observation
showed Newton’s theory did not accurately predict
the orbit of Mercury. So the theory was wrong.
Some even postulated another planet, Vulcan, to
make the theory correspond with observation. But
there was no planet Vulcan, it wasn’t until Einstein
that a theory was discovered that could account for
Mercury’s orbit.

Fallacies

Some Relevant Fallacies

A Problem Reductio: Law’s Not Fair

Some claim our legal system is fair. They point
out that our legal system, when it functions
properly, gives out impartial sentences and so it
fair. However, our legal system is abstract, and so
is without color, and can’t be pale, so it is not fair
(look up fair, it means having a light complexion).
But this means our legal system is fair and unfair,
which is a contradiction. So I reject that our legal
system is fair.

A Problem Reductio
What is the problem?
Line of
Reasoning

Our legal system,
when it functions
properly, gives out
impartial sentences
and so it fair.

Our legal
system is
Our legal system is fair.
fair and
unfair
Our legal system is
abstract, and so is
Another Line
without color, and
of Reasoning can’t be pale, so it
is not fair.

Reject: our
legal
system is
fair

A Problem Reductio
Equivocation
Our legal system,
when it functions
properly, gives out
impartial sentences
and so it fair.

Our legal system is
abstract, and so is
without color, and
can’t be pale, so it
is not fair.

The problem is that the
argument uses the word fair in
two different ways—fair is
ambiguous, it has more than
one meaning. One meaning has
to do with being impartial, the
other has to do with having a
pale color. Using a word
ambiguously in an argument is
the fallacy of equivocation.

Equivocation, to use a term
ambiguously or vaguely in an
argument—it is a fallacy.

A Problem Reductio: Too Big to Fail

During a recent financial crisis many made the
claim that some banks were too big to fail and
needed to be bailed out with public funds to avoid
catastrophe. But banks did fail, both large banks
and small ones. Thus the claim is inconsistent with
the evidence. And so it must be false that some
banks were too big to fail.

to avoid equivocation

Define key terms by giving
them one (to disambiguate) clear
(to avoid vagueness) meaning.

A Problem Reductio
Line of
Reasoning

Some banks were
too big to fail and
needed to be bailed
out with public
funds.

Our legal system is fair.

The claim is
inconsistent

Banks did fail, both
Another Line
large banks and
of Reasoning small ones.

Reject:
some banks
are too big
to fail.

A Problem Reductio
Equivocation
Some banks were
too big to fail and
needed to be bailed
out with public
funds.

Banks did fail, both
large banks and
small ones.

The problem is that the argument
switches between a
prescriptive/normative claim
(banks ought not be allowed to fail
as they are too big and might
cause a catastrophe) and
descriptive claim (the observation
that even big banks did fail). This
is another form of the fallacy of
equivocation.

to avoid equivocation

Use the Principle of Charity to
settle on the best interpretation,
whether normative or
descriptive.

A Problem Reductio: Too Big to Fail
Socrates claimed that women could be leaders of
the ideal city state. He noted that women possess
the same capacities in terms of leadership that men
do. His pupils, however, noted that men and
women have very different capacities and that only
a fool would confuse women with men—they are
different! Because this line of reasoning leads to
the absurd conclusion that men and women are the
same and not, his pupils laughed at the notion that
women could be leaders.

A Problem Reductio
Line of
Reasoning

Women possess the
same capacities in
terms of leadership
that men do.

Men and
Reject:
women are
Women can be leaders.
women can
the same and
be leaders.
not.
Only a fool would
confuse women
Another Line
with men—they are
of Reasoning different with
different capacities!

A Problem Reductio
Reductio ad ridiculim
Women possess the
same capacities in
terms of leadership
that men do.

Only a fool would
confuse women
with men—they are
different with
different capacities!

The problem is that the
argument confuses ridicule
with reason. This is an
example of a reductio ad
ridiculum—a fallacy. This
example is based off an
argument given in Plato’s
Republic—an argument which
Plato is careful to refute.

Reductio ad ridiculum,
appealing to ridicule (making
fun of an opposing view) rather
than providing reasons against
it—it is a fallacy.

to avoid reductio ad ridiculums

Use the Principle of Sufficient
Reason and attempt to provide
reasons for each claim.

Assignment
What is the difference between
validity and soundness?

Soundness

How do you prove a conditional
false?

What does ‘if’ mean?

Critical Thinking 03 consistency

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