Transcript Ling 21: Language and Thinking Lecture 4: Basic Logical Concepts ROAD MAP • After a brief side journey into language and the brain, including: – The.

Ling 21: Language and Thinking
Lecture 4:
Basic Logical Concepts
ROAD MAP
• After a brief side journey
into language and the
brain, including:
– The physiology of the
brain;
– The brain and language
disorders; and
– Sign language and the
brain,
• We now return to some
basic logical concepts
Previously on ‘Language & Thinking’,
We . . .
 Defined critical thinking;
 Identified traits of a critical
thinker;
 Identified some of the barriers
to critical thinking;
and
 Defined and analyzed
arguments in terms of their
component parts
(Premises & conclusions).
This chapter is
EXTREMELY IMPORTANT because . . .
• It forms the foundation of
EVERYTHING else that is to
follow in this course.
• If you don’t read and
understand this chapter, you will
not do well in this course.
• So,
– read the chapter,
– actively participate in the class and
– ASK QUESTIONS if you don’t think
you get it!
BASIC LOGICAL CONCEPTS
• Task: To distinguish good arguments from bad
• Two questions:
– Are the premises true?
– Do the premises provide good reasons to accept the
conclusion?
TWO ARGUMENT TYPES
• Deductive arguments
(try to) PROVE their conclusions
• Inductive arguments
(try to) show that their conclusions are
PLAUSIBLE or LIKELY
DEDUCTIVE ARGUMENTS
• Some pigs have wings.
All winged things sing.
Therefore, some pigs sing.
• Everyone has one and only one biological mother.
Full sisters have the same biological mother.
No one is her own biological mother.
Therefore, there is no one whose biological mother is
also her sister.
EXERCISE: Solve the mysteries, CT pages 54-55.
INDUCTIVE ARGUMENTS
• Every ruby discovered thus far has been red.
So, probably all rubies are red.
• Polls show that 87% of 5-year-olds believe in the tooth
fairy.
Marta is 5 years old.
Marta probably believed in the tooth fairy.
• Chemically, potassium chloride is very similar to
ordinary table salt (sodium chloride).
• Therefore, potassium chloride tastes like table salt.
THE DIFFERENCE
Key: deductive / inductive
• If the premises are true the conclusion is
necessarily / probably true.
• The premises provide conclusive / good evidence
for the conclusion.
• It is impossible / unlikely for the premises to be
true and the conclusion to be false.
• It is logically inconsistent / consistent to assert
the premises but deny the conclusion.
FOUR TESTS
• Four tests allow us to identify deductive /
inductive arguments
– The indicator word test
– The strict necessity test
– The common pattern test
– The principle of charity test
INDICATOR WORD TEST
Deduction
Induction
Certainly
Definitely
Absolutely
Conclusively
This entails that
Probably
Likely
Plausible
Reasonable
The odds are that
CAUTION!
-Arguments may not contain any indicator words.
Pleasure is not the same thing as happiness.
The occasional self-destructive behavior of the rich
and famous confirms this too vividly.
(Tom Morris)
-Arguers may use indicator words incorrectly.
(People very often overstate their cases.)
-In these cases, other tests must be used to determine
whether an argument is deductive or inductive.
The Strict Necessity Test
• An argument’s conclusion either follows with
strict logical necessity from its premises or it
does not.
• If an argument’s conclusion does follow with
strict logical necessity from its premises, the
argument should always be treated as deductive.
• if an arguments conclusion does not follow with
strict logical necessity from its premises, the
argument should normally be treated as
inductive.
The Strict Necessity Test
• Examples:
• Alan is a father. Therefore Alan is a male.
• Jill is a six-year-old. Therefore, Jill cannot run a
mile in one minute flat.
COMMON PATTERN TEST
• Modus ponens (affirming the antecedent)
– If A then B.
– A.
– Therefore B.
(A = antecedent; B = consequent)
This is a very common pattern of deductive reasoning.
Common Pattern Test
• Example (modus ponens)
• If we are in Paris, then we are in France.
-------A------------------B----------• We are in Paris.
--------A--------• Therefore, we are in France.
---------B-----------
PRINCIPLE OF CHARITY TEST
• When interpreting an unclear argument,
always give the speaker / writer the benefit of
the doubt.
– Fosters good will and mutual understanding in an
argument.
– Promotes the discovery of truth by insisting that
we confront arguments that we ourselves admit to
be the strongest and most plausible versions of
those arguments.
Exceptions to the Strict Necessity Test
• An argument in which the conclusion does not
follow necessarily from the premises should
be treated as deductive if either:
1. The language or context make clear that the arguer
intended to offer a logically conclusive argument,
but the argument is in fact not logically conclusive;
2. The argument has a pattern of reasoning that is
characteristically deductive, and nothing else about
the argument indicated clearly that the argument is
meant to be inductive.
Exceptions to the Strict Necessity Test
• Examples
1. Magellan’s ships sailed around
the world. It necessarily follows,
therefore, that the earth is a sphere.
(The arguer intended to offer a logically conclusive
argument, so it should be treated as deductive.)
2. If I’m Bill Gates, then I’m mortal. I’m not Bill Gates.
Therefore, I’m not mortal. (The argument has a
pattern of reasoning characteristic of deductive
arguments, so should be treated as deductive.)
SUMMARY: How to distinguish deductive
from inductive arguments
• If the conclusion follows necessarily from the premises =
deductive
• If the conclusion does not follow necessarily from the premises
= inductive, unless
– Language indicates it is deductive
– Argument has deductive pattern of reasoning
• If the argument has a pattern of reasoning that is
characteristically deductive = deductive, unless
– Clear evidence indicates it is intended to be inductive
• If the argument has a pattern of reasoning that is
characteristically inductive = inductive unless
– Clear evidence indicates it is intended to be deductive
• If the argument contains an indicator word
• If still in doubt: Principle of Charity
5 COMMON DEDUCTIVE PATTERNS
• Hypothetical syllogism
• Categorical syllogism
• Argument by elimination
• Argument based on mathematics
• Argument from definition
HYPOTHETICAL SYLLOGISM
• A syllogism is a three-line argument with two
premises, one of which is a conditional.
• Modes ponens is a syllogism.
• Other syllogisms are:
– Chain arguments
– Modus tollens (denying the consequent)
– Denying the antecedent
– Affirming the consequent
CHAIN ARGUMENT
If A then B.
• If B then C.
Therefore if A then C.
• If you are blue in the face then you are lying.
If you are lying then you can’t be my friend.
Therefore if you are blue in the face then you
can’t be my friend.
MODUS TOLLENS
• If A then B.
Not B.
Therefore not A.
• If we’re in Sacramento, we’re in California.
We’re not in California.
Therefore, we’re not in Sacramento.
• If you love me, you’ll come with me to Tibet.
You will not come with me to Tibet.
Therefore you do not love me.
DENYING THE ANTECEDENT***
• If A then B.
Not A.
Therefore not B.
*If Tiger Woods won this year’s Masters then he’s a great athlete.
Tiger Woods didn’t win this year’s Masters.
Therefore, Tiger Woods is not a great athlete.
*If Jack comes to the party, Jill will leave.
Jack did not come to the party.
Therefore Jill did not leave.
***Denying the antecedent is a fallacious deductive pattern
AFFIRMING THE CONSEQUENT***
• If A then B.
B.
Therefore A.
*If we are on Neptune then we are in the solar
system.
We are in the solar system.
Therefore we are on Neptune.
***Affirming the consequent is a fallacious deductive
pattern
Exercise: Identify the argument pattern (ex. 3.2, p. 65)
MODUS PONENS (affirming the antecedent): If A then B. A.
Therefore B.
CHAIN: If A then B. If B then C. Therefore if A then C.
MODUS TOLLENS: If A then B. Not B. Therefore not A.
*DENYING THE ANTECEDENT: If A then B. Not A. Therefore
not B.
*AFFIRMING THE CONSEQUENT: If A then B. B. Therefore A.
PRINCIPLE OF CHARITY
• Attribute an arguer the strongest argument possible.
Andy told me he ate at JB’s yesterday.
But JB’s was destroyed by a fire a month
ago.
It is certain therefore that Andy is either
lying or mistaken.
Caution – The Principle of Charity is a principle of
argument interpretation, not a principle of argument
repair.
CATEGORICAL SYLLOGISM
• A three-line argument in which each
statement begins with one of the words all,
some, or no.
– Some pigs have wings
All winged things sing.
Therefore some pigs sing.
ARGUMENT BY ELIMINATION
• Rules out various logical possibilities until only
a single possibility remains.
Either Dutch or Jack or Celia committed the murder.
If D or J committed the murder then the weapon was
a rope.
The weapon was not a rope.
Therefore neither D nor J committed the murder.
Therefore C committed the murder.
MATHEMATICS
• The conclusion depends largely or entirely on
mathematical calculation or measurement.
Light travels at a rate of 186,000 miles per second.
The sun is more than 94 million miles from earth.
Therefore it takes more than 8 minutes for the sun’s
light to reach earth.
Caution – not all arguments that make
use of numbers and mathematics are
deductive.
DEFINITION
• The conclusion follows from the definition of
some key word or phrase in the argument.
Josefina is a drummer.
Therefore Josefina is a
musician.
COMMON INDUCTIVE PATTERNS
• There are 6 common inductive patterns:
– Inductive generalization
– Predictive argument
– Argument from authority
– Causal argument
– Statistical argument
– Argument from analogy
INDUCTIVE GENERALIZATION
• A generalization attributes some characteristic to
all or most members of a given class.
• Information about some members of the class is
said to license the generalization.
All dinosaur bones discovered thus far have
been more than 65 million years old.
Therefore probably all dinosaur bones are
more than 65 million years old.
PREDICTIVE ARGUMENT
• A statement about what will (likely) happen in
the future is defended with reasons.
It has rained in Vancouver every February
since records have been kept.
Therefore it will probably rain in Vancouver
next February.
AUTHORITY, CAUSE, STATISTICS
• Argument from Authority
– The conclusion is supported by citing some
presumed authority or witness.
• Causal Argument
– Asserts or denies that something is the
cause of something else.
• Statistical Argument
– Rests on statistical evidence.
ANALOGY
• Common Pattern:
Two (or more) things are alike in one way. Therefore they
are probably alike in some further way.
As a man casts off worn-out garments and puts on others
that are new,
similarly, the soul, casting off worn-out bodies, enters into
others, which are new.
(Bhagavad-Gita)
Exercise: Determine whether arguments are deductive or
inductive (ex. 3.3, p. 71-72)
VALIDITY
• VALID arguments may have false premises and
false conclusions!
• At issue is the form. If the premises are true the
conclusion must be true.
All circles are squares.
All squares are triangles.
Therefore all circles are triangles.
All fruits are vegetables.
Spinach is a fruit.
Therefore spinach is a vegetable.
VALIDITY, CONT’D
• It is not enough that the conclusion happens
to be true. If the conclusion doesn’t follow
from the premises by strict logical necessity, a
deductive argument is invalid.
•
All pigs are animals.
Wilber is pink.
Therefore Wilber is a pig.
Exercise: What conclusions follow validly? (ex.
3.4, p. 73-74)
SOUNDNESS
• A deductive argument is sound if it is valid and
has true premises.
• A deductive argument with (at least) one
untrue premise, valid or invalid, is unsound.
Exercise: Determine whether arguments are
valid / sound (ex. 3.5 I & II, p. 81-82)
INDUCTIVE STRENGTH
• A ‘good’ deductive argument is valid.
• A ‘good’ inductive argument is strong.
• An inductive argument is strong if the
conclusion follows probably from the premises.
All recent US presidents have
been college graduates.
It is likely that the next US
president will be a college
graduate.
WEAKNESS
• An argument that is not strong is weak.
Most US presidents have been men. It is likely that the
next US president will be a woman.
• In a weak inductive argument, the conclusion does
not follow probably from the premises.
I dream about monsters. You dream about monsters.
Therefore everybody probably dreams about
monsters.
INDUCTIVE PROBABILITY
• The premises and conclusion do not have to be true –
The question is:
– If the premises were true, would the conclusion
follow?
• Deductive arguments are either 100% valid or 100%
invalid.
• Inductive arguments can be somewhat strong, strong,
very strong, depending on the degree of support the
premises provide for the conclusion.
According the National Weather Service, there is a 60% 70% - 90% chance of rain today.
It is likely that it will rain today.
INDUCTIVE ARGUMENTS
• A valid deductive argument with true premises is sound.
• A strong inductive argument with true premises is cogent.
• An inductive argument that is either weak or has at least one
false remise is uncogent.
- No US president has been a skateboarding champion.
Therefore the next US president will probably not be a
skateboarding champion. (Cogent)
- All previous US presidents have been rocket scientists.
Therefore the next US president will probably be a woman.
(Uncogent)
- All previous U.S. Presidents have been Democrats. Therefore
the next U.S. President will be a Democrat. (Uncogent)
INDUCTIVE ARGUMENTS
• Exercise: Determine whether arguments are
cogent or uncogent (ex. 3.5 III, p. 82-83)
Summary of Argument Types
Deductive
Valid
Sound Unsound
Inductive
Invalid
(all are
unsound)
Cogent
Strong
Weak
(all are
uncogent)
Uncogent
MEMORIZE THESE DIAGRAMS ! ! !
Culminating Activity
• Exercise 3.5 IV, Page 83:
Determine whether the arguments are deductive or
inductive. If the argument is deductive, determine
whether it is valid or invalid. If the argument is
inductive, determine whether it is strong or weak.