reasoning strategies (course 3)

8/8/2019 Reasoning Strategies (Course 3)

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1.1: Reasoning Strategies

Analyzing Arguments


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I. INDUCTIVE ARGUMENT

An inductive argument makes a caseforageneral conclusionfrom more specific premises.

In other words, specific premises are used to draw

a general conclusion.

(A general conclusion is induced.) specific premi

ses

general conclusion


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Example: Inductive Argument

Premise: Sparrows are birds that fly.

Premise: Eagles are birds that fly.

Premise: Hawks are birds that fly.

Premise: Larks are birds that fly.

Conclusion: All birds fly.


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Notes on Inductive Arguments

1. An inductive argument cannot proveits conclusion true.

2. An inductive argument can be evaluated only

in terms of its strength.3. The strength of an inductive argument is a

measure of how well the premises support

the conclusion. Clearly, this is subjective (apersonal judgment).


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II. DEDUCTIVE ARGUMENT

A deductive argument makes a caseforaspecific conclusionfrom more general premises.

In other words, general premises are used to form

a specific conclusion.

(The specific conclusion is deduced.)

generalpremise

s

specificconclusio

n


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Example: DeductiveArgument

Premise: All college professors have doctoral

degrees.

Premise: Ms. Gomez is a college professor.

Conclusion: Ms. Gomez has a doctoral degree.


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more Examples

All men are mortal (major premise),

Socrates is a man (minor premise),

Therefore Socrates is mortal.

All men are purple-skinned.

Socrates is a man.

Therefore Socrates is purple-skinned.


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Notes on Deductive Arguments

1. A Deductive Argument can be evaluated in terms ofits Validity and Soundness.2. A Deductive Argument is Valid if its conclusion

follows necessarily from its premises. Validity is

concerned only with the logical structure of theargument. It has nothing to do with the truth of thepremises or the conclusion.

3. A Deductive Argument is Sound if

a) it is validand

b) its premises are all true.4. A Sound Deductive Argument provides definitiveproof that its conclusion is true. (However, thisoften involves personal judgment.)


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Testing The Validity of an

Argument with a Venn Diagram1. Draw a Venn diagram that represents all the

information contained in the premises.

2. Check to see whether the Venn diagram also

contains the conclusion. If it does, then the

argument is valid. Otherwise, the argument

is not valid.


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A Venn Diagram Test of Validity

1-D

All politicians are married.

Senator Harris is a politician.

Therefore, Senator Harris is married.


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A Venn Diagram Test of Validity

1-D

All fish live inthe water.

Whales arenot fish.

Therefore,whales donot live in thewater.


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CONDITIONALDEDUCTIVE ARGUMENTS

A Conditional Deductive Argument has aconditional statement for its first premise.

There are four basic conditional arguments:

1. Affirming the Hypothesis.

2. Affirming the Conclusion.

3. Denying the Hypothesis.4. Denying the Conclusion.


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1. Affirming The Hypothesis

Structure: Ifp, then q.

p is true

q is true

Validity: Valid

This argument is also known asModus Ponens

.

Example:Premise: If one gets a college degree, then one can get a good job.

Premise: Marilyn has a college degree.

Conclusion: Marilyn can get a good job.


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2. Affirming The Conclusion


q is true

p is true

Validity: InvalidConverse Fallacy


Premise: Marilyn gets a good job.

Conclusion: Marilyn has a college degree.


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3. Denying The Hypothesis


p is not true

q is not trueValidity: InvalidInverse Fallacy


Premise: Marilyn does not have a college degree.

Conclusion: Marilyn cannot get a good job.


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4. Denying The Conclusion


q is not true

p is not true

Validity: Valid

This argument is also known asModus Tollens.


Premise: Marilyn does not have a good job.

Conclusion: Marilyn does not have a college degree.


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Examples:If I go to the game, then Ill eat a hotdog.

If I eat a hotdog, then Ill get sick.

So, if I go to the game, Ill get sick.

We agreed that if you shop, I make dinner.

We also agreed that if you take out thetrash,

I make dinner. Therefore, if you shop, you

should take out the trash.


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Deductive Arguments With AChain Of Conditionals

1. Structure: Ifp, then q.

Ifq, then r.

Ifp, then r.

Validity:

2. Structure: Ifp, then q.

Ifr, then q.

Ifp, then r.

Validity:

Valid

Invalid


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Mathematics relies heavily on

proofs. A mathematical proof is adeductive argument that

demonstrates the truth of a certain

claim or theorem. A theorem isproven if it is supported by a validand sound proof. Although

mathematical proofs useDeduction, theorems are oftendiscovered by Induction.

1 D


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Inductive Counterexample

1-D

42 4 11 23 (prime)4

52 5 11 31 (prime)5

32 3 11 17 (prime)3

22 2 11 13 (prime)212 1 11 11 (prime)1

02 0 11 11 (prime)0

n2 n 11n

Consider the following algebraic expression:n

2 n

11It appears that

n2 n 11

will always equal a

prime numberwhen n 0.

Or does it?

How about n= 11?

112 11 + 11 = 121(a non-prime counterexample)

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