Deductive Arguments

A deductive argument is one whose premises are claimed to provide conclusive grounds for the truth of its conclusion.

A deductive argument is one whose premises are claimed to provide conclusive grounds for the truth of its conclusion.

A deductive argument is one whose premises are claimed to provide conclusive grounds for the truth of its conclusion.

Types of Propositions

Disjunctive or Alternative Proposition

Either A or B.

A and B are called the disjuncts or alternatives.

Hypothetical or Conditional Proposition

If A, then B.

A is called the antecedent.B is called the consequent.

Categorical Proposition

All A is B. (Universal Affirmative or A)

No A is B. (Universal Negative or E)

Categorical Proposition

Some A is B. (Particular Affirmative or I)Some A is not B. (Particular Negative or O)

Deductive Arguments

Disjunctive Syllogism

1st Premise: Either A or B.

2nd Premise: Not A (or Not B).

Conclusion: B (or A).Note: A and B are called the alternatives or the disjuncts.

Disjunctive Syllogism

Either interest rates go up or inflation gets worse. Since interest rates have not gone up, we can be sure that inflation is getting worse.

Disjunctive SyllogismThe premise “Either A or B”

asserts that at least one of the two disjuncts is true.

“Or” means “and/or” (inclusive).

Note that it is possible that both disjuncts are true.

That one disjunct is true does not mean that the other is false.

Either A or B.A.So, not B.

Fallacy of Affirming a Disjunct

Pure Hypothetical Syllogism

1st Premise: If A, then B.

2nd Premise: If B, then C.

Conclusion: If A, then C.

Pure Hypothetical SyllogismIf more prisons are built,

public education will get worse due to lack of funding. If public education gets worse due to lack of funding, there will be more criminals. As a result, if more prisons are built, there will be more criminals.

Mixed Hypothetical Syllogism: Modus Ponens

1st Premise: If A, then B.

2nd Premise: A.Conclusion: B.

Mixed Hypothetical Syllogism: Modus Ponens

If Republicans favor free market economy, then they should oppose farm subsidies. Republicans favor free market economy. So they should oppose farm subsidies.

Fallacy of Affirming the Consequent

“If A, then B” claims that if A happens, then B will follow. It does not say that if B happens, A preceded it.

If A, then B.B.So, A.

Fallacy of Affirming the Consequent

If Barack Obama is a U.S. president, then he is a U.S. citizen. Barack Obama is a U.S. citizen, so he is a U.S. President.

Mixed Hypothetical Syllogism: Modus Tollens

1st Premise: If A, then B.

2nd Premise: Not B.Conclusion: Not A.

Mixed Hypothetical Syllogism: Modus Tollens

If it is raining, then there are clouds in the sky. There are no clouds in the sky. It is not raining.

Fallacy of Denying the Antecedent

“If A, then B” claims that if A happens, then B will follow. It does not say that if A does not happen, B will not happen either.

If A, then B.Not A.So, not B.

Fallacy of Denying the Antecedent

If there is smoke, there is a fire.There is no smoke. So, there is no fire.

Categorical SyllogismBARBARA All M are P. All S are M.

So, all S are P.

BAROCOAll P are M.

Some S are not M. So, some S are not

P.

BOCARDOSome M are not P.

All M are S. So, some S are not

P. CAMENESAll P are M. No M are S.

So, no S are P.

CAMESTRESAll P are M. No S are M.

So, no S are P.

CELARENTNo M are P. All S are M.

So, no S are P. CESARE

No P are M. All S are M.

So, no S are P.

DARIIAll M are P.

Some S are M. So, some S are P.

DATISIAll M are P.

Some M are S. So, some S are P.

Categorical SyllogismDISAMIS

Some M are P. All M are S.

So, some S are P.

DIMARISSome P are M.

All M are S. So, some S are P.

FERIOSome P are M.

All M are S. So, some S are P.

FESTINONo P are M.

Some S are M. So, some S are not

P.

FRESISONNo P are M.

Some M are S. So, some S are not

P.

FERISONNo M are P.

Some M are S. So, some S are not

P.

EXERCISES

Determine the form of each argument. Then, say whether it is valid or invalid.

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Smith is the fireman or Smith is the engineer. Smith is not the fireman. So Smith is the engineer.

[Example]

Smith is the fireman or Smith is the engineer. Smith is not the fireman. So Smith is the engineer.

Answer: Disjunctive Syllogism; Valid

[Example]

If Mr. Jones lived in Chicago, then Jones is the brakeman. Mr. Jones lives in Chicago. So Jones is the brakeman.

[1]

If Mr. Jones is the brakeman’s next-door neighbor, then $20,000 is exactly divisible by 3. But $20,000 is not exactly divisible by 3. So, Mr. Jones is not the brakeman’s next-door neighbor.

[2]

“J.J.,” I replied, “if it was any of your business, I would have invited you. It is not, and so I did not.”

[3]

If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.

[4]

I can’t have anything more to do with the operation. If I did, I’d have to lie to the Ambassador. And I can’t do that.

[5]

If the one-eyed prisoner does not know the color of his hat, then the blind prisoner cannot have on a red hat. The one-eyed prisoner does not know the color of his hat. So the blind prisoner cannot have on a red hat.

[6]

I have already said that he must have gone to King’s Pyland or to Capleton. He is not at King’s Pyland, So he is at Capleton.

[7]

If then, it is agreed that things are either the result of coincidence or for an end, and these cannot be the result of coincidence or spontaneity, it follows that they must be for an end.

[8]

A theoryless position is possible only if there are no theories of evidence. But there are theories of evidence. So, a theoryless position is impossible.

[9]

If the first native denied being a politician, then the second native told the truth. If the second native told the truth, then the second native is not a politician. So if the first native denied being a politician, then the second native is not a politician.

[10]

We should be against big corporations only if we are against their stockholders. We are not against the stockholders. So we should not be against big corporations.

[11]