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Second Thoughts, 4th ed.

Wanda Teays

McGraw-Hill Higher Ed.©2010. Wanda Teays

All rights reserved.

CHAPTER TENPatterns of Deductive Reasoning: Rules of

Inference

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The Rules of InferenceThe rules of inference are valid argument

forms.A familiarity with the rules of inference give us the

tools to evaluate arguments and draw inferences. This allows us to greatly expand our reasoning

capacity. Once we learn the rules of inference, we can spot poorly reasoned arguments, as well as strong ones!

Having a facility with logic gives us the techniques to examine and evaluate the many kinds of arguments we confront.

This does not help us develop moral fiber, but it does help us develop mental dexterity.

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ValidityAn argument is valid if the conclusion follows directly

from the premises and could not be false if the premises were assumed true.

This does not mean the premises have to be true! Repeat: A valid argument does not have to have true premises, even if that seems counterintuitive.

But it does mean that, if we assume they were true, the conclusion would necessarily be true as well.

FOR EXAMPLE: Rattlesnakes love to stretch out in the sun.All creatures that like to stretch out in the sun are lazy

lumps.Therefore, rattlesnakes are lazy lumps.

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Modus Ponens & Modus TollensModus ponens is the name of any argument in the following form:

If A then B.A is true.

Thus, B is also true.

 FOR EXAMPLE: If you buy the jumbo popcorn, then you will need a large drink.You bought the jumbo popcorn.So you will need a large drink.

Modus tollensis the name of any argument in the following form:

If A then B.B is not the case.

Therefore, A is not the case either.

FOR EXAMPLE:If you get the bon bons, you won’t need popcorn.Charlie needed popcorn.So he did not get the bon bons.

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More Valid Argument Forms: Hypothetical Syllogism & the Disjunctive Syllogism

The hypothetical syllogism consists of three “if.. .then” claims, where the middle term links the first term to the third. The form is:

If A then B. If B then C.

So, if A then C.

FOR EXAMPLE:If you sing while driving, you’ll be more attentive.If you are more attentive, you’ll be a better driver.Therefore, if you sing while driving, you’ll be a better driver.

The disjunctive syllogismstarts with an “either..or” claim and one of the disjuncts falls out, so you are left with the other one. The form is:

Either A or B.|A is not the case.(or B is not the case)So, B is the case. (or A is the case)

FOR EXAMPLE:Either Jamal will get rid of his clunker or he’ll fix it up.Jamal did not fix up his clunker.So he got rid of it.

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More Rules of Inference:Conjunction and Simplification

Conjunction is the rule that two claims that are each true are true in combination. The form is this:A is true. B is also true.

So, both A and B are true.

FOR EXAMPLE:Ice is tricky to walk on.Mud sticks to your shoes.So, ice is tricky to walk on and mud sticks to your shoes.

Simplificationis the opposite of conjunction. If you know two things together are true, you know each one is true. The form is this:A and B are true.

So, A [or B] is true.

FOR EXAMPLE:Angie likes both skiing and jazz dancing.Therefore, Angie likes skiing [also: Angie likes jazz dancing.]

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Logical AdditionLogical Addition is NOT a move using mathematics! Logical Addition allows us to expand by going from one thing we know to saying either that OR anything else. The form is this:

A is true.Therefore, either A or B.

FOR EXAMPLE:Randy found it hard to drive in the blizzard.Therefore, either Randy found it hard to drive in the blizzard or he was fooling me about how he got the dent in his fender.

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The Two DilemmasCONSTRUCTIVE DILEMMA

DESTRUCTIVE DILEMMA

The constructive dilemma is a valid argument with this form: If A then B, and, if C then D.Either A or B. Therefore, either C or D.

FOR EXAMPLE:If I eat the popcorn, I’ll feel guilty, but if I eat carrots, I’ll want popcorn.Either I ate popcorn or I ate carrots.Therefore, either I felt guilty or I wanted popcorn.

The destructive dilemma is a valid argument with this form:If A then B, and, if C then D.Either not B or not D. Thus, either not A or not C.

FOR EXAMPLE:If I eat the pretzel, I’ll need a drink, but if I eat ice cream, I’ll want chocolates.Either I won’t need a drink or I’ll not want chocolates.Thus, either I didn’t eat the pretzel or I didn’t eat the ice cream.

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Did you notice?

Did you notice anything when you were staring at the two dilemmas? Did little bells ring in your brain?Hopefully, yes!The constructive dilemma is like a compound modus ponens!The destructive dilemma is like a compound modus tollens!

Remember: All these rules of inference are valid argument forms. That means any argument in these various forms is a VALID argument!

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Our last Rule of Inference: Absorption

Absorption is the “sponge rule” —It goes like this: If you have a conditional (“if..then”) claim, you can repeat the antecedent in the consequent for extra emphasis. The form is this:

If A then BTherefore, if A then, A and B.

FOR EXAMPLE:If Carlos stays out in the sun, he’ll need a hat. Therefore, if Carlos stays out in the sun, he’ll stay

out in the sun and need a hat.

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Modus Ponens vs. The Fallacy of Affirming the Consequent

Modus Ponens is a valid argument form:

If A then B.A is true.

Therefore B is true.

FOR EXAMPLE:If Jasper eats his corn, he’ll want some peas.Jasper ate his corn; therefore, he’ll want some peas.

The Fallacy of Affirming the Consequent is one of the formal fallacies. The form is:

If A then B.B is true.

Therefore, A is true.

FOR EXAMPLE:If Dan runs out of gas, his car won’t start.Dan’s car didn’t start.Therefore, he ran out of gas.

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Modus Tollens vs. The Fallacy of Denying the Antecedent

Modus Tollens is a valid argument form:

If A then B.B is not true.

So, A is not true.

FOR EXAMPLE:If the traffic is bad, Jim will be late to the movie.Jim wasn’t late to the movie.Therefore, the traffic wasn’t bad.

The Fallacy of Denying the Antecedent is one of the formal fallacies. The form is:

If A then B.A is not true.

So, B is not true.

FOR EXAMPLE:If she goes snorkling, Anita will wear sun lotion.Anita didn’t go snorkling.So, she didn’t wear sun lotion.