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Chapter 10 Patterns of Deductive Reasoning: Rules of Inference
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valid arguments

the premises offer sufficient support for the conclusion. If we assume the premises are all true, then the conclusion must be true as well.

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Valid Argument Forms

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

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Rules of Inference

A sound argument is a valid argument that has true premises. That is, soundness goes one step beyond validity-it requires truth, as well as a good, solid structure.

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Modus ponens

a valid argument form that asserts, "if this, then that. This; therefore, that."

if we can affirm that this is true, then that must be true as well.

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Form of Modus Ponens

If A then B.

A is true.

Therefor, B is true as well.

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example of Modus Ponens

If that's an alligator, you better get out of the swamp.

That is an alligator.

So, you better get out of the swamp.

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Modus Tollens

valid argument form that means "mode that denies." The first premise is a conditional claim. the second premise denies the consequent by saying it did not happen. The conclusion is that the antecedent could not have happened either.

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Form of Modus Tollens

If A then B.

B is not true.

Therefore, A is not true either.

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Example of Modus Tollens

If that's not a hybrid that Louise is driving, it must be an electric car.

That's not an electric car.

Therefore, it must be a hybrid that Louise is driving.

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Hypothetical Syllogism

argument that is composed entirely of conditional hypothetical claims.

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Form of Hypothetical Syllogism

If A then B.

If B then C.

Therefore, if A then C.

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Example of Hypothetical Syllogism

If that's a cockroach in the kitchen, then either call an exterminator or set out poison.

If we either call an exterminator or set out poison, then we can solve the problem.

Therefore, if that's a cockroach in the kitchen, then we can solve the problem.

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Disjunctive Syllogism

a valid argument form that is an either/or claim in the first premise. The second premise denies the first. This forces the conclusion to be the remaining.

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Form of the Disjunctive Syllogism

Either A or B.

A is not the case (or B is not the case).

Therefore, Bis the case (or A is the case).

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Example of Disjunctive Syllogism

Either a wolf is howling or the wind is in the trees.

There's no wind tonight.

Therefore, it must be a wolf howling.

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Conjunction

very straightforward. Two claims that are each true are true in combination. The rule asserts that if we have two claims that we know to be true, then they are both true together.

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Form of Conjunction

A is true.

B is true.

Therefore, both A and B are true.

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Example of Conjunction

Omar was happy to see Lamu.

He was sorry he didnt see Amy.

Therefore, Omar was happy to see Lamu but sorry he didnt see Amy.

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Simplification

If both together are true, then it follows that each proposition is individually true.

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Form of Simplification

A and B are true together.

Therefore, A is true as well as B.

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Example of Simplification

The reviewer liked both Ghost Dog and Wings of Desire

Therefore, the reviewer liked Ghost Dog (or the reviewer liked Wings of Desire).

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If we know any one proposition is true, then a disjunction made up of this true proposition and any other proposition is necessarily true as well.

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A is true.

Therefore, either A or B is true.

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Ben's computer crashed. Therefore, Ben's computer crashed or the battery has died.

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Constructive Dilemma

two conditional propositions and in which one or the other antecedent is true. Consequently, either one or the other of the two consequences must also be true.

Like a compound modus ponen

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Form of Constructive Dilemma

If A then B, and if C then D

Either A or C,

Therefore, either B or D

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Destructive Dilemma

two conditional claims that either the first consequent is not true or the second consequent is not true. (like a compound Modus Tollens)

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Form of Destructive Dilemma

If A then B and if C then D

Either Bis not the case or D is not the case,

Therefore, either A is not the case or C is not the case.

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Absorption

start with a conditional claim. Infer that the antecedent can be repeated in the consequent. If this then that: therefore, if this then this and that.

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Form of Absorption

If A then B

Therefore, if A, Then both A and B

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Validity and Soundness

All the rules of inferences are valid. If we assume the premises are true, the conclusion will be forced to be true as well. The conclusion must be true whenever a valid argument has true premises. The evidence fully support the conclusion.

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Formal Fallacy

Are always invalid arguments, whether or not they have true premises. The error is with a misuse of the form, or structure.

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Two Key formal fallacies

Denying the antecedent

Affirming the consequent

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Two Valid Argument Forms

Modus ponens

modus tollens

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Fallacy of denying the antecedent

Asserts a causal relationship between the antecedent condition and the consequent. The conclusion does not automatically follow if the premises are true.

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Example of denying the antecedent

If George chases that skunk, he may be very sorry.

George did not chase the skunk, so he was not sorry.

Note: George could be sorry for other reasons, such as losing his wallet.

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Fallacy of affirming the consequent

faulty reasoning of the form. There could be a number of causal factors independently causing an event.

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Form of the fallacy of affirming the consequent

If A then B

B is the case,

Therefore, A is the case as well.

Note: A and B could be either positive or negative claims

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Example of fallacy of affirming the consequent

If the road is muddy, it will be hard to go hiking.

It was hard to go hiking,