Rules of Inferencepg. 63 - 69

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Argument

Definition• An argument is a sequence of statements that

ends with a conclusion. Valid mean that the conclusion must follow from the truth of the preceding statements or premises.

IF premise-1, ….., premise-n THEN conclusion

• An argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false. If the premises are all true, then the conclusion is also true.

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Example

• If you have a current password, then you can log onto the network

• You have a current password• Therefore,• You can log onto the network

p qp q

• The symbol , read “therefore”• Construct the truth table

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Argument

• If Socrates is a human being, then Socrates is mortal

• Socrates is a human being• Therefore,• Socrates is mortal

p qp q

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Testing the Validity of an Argument• Identify the premises and conclusion of the

argument• Construct a truth table showing the truth values

of all the premises and the conclusion• Find the rows (called critical rows) in which all

the premises are true• In each critical row, determine whether the

conclusion of the argument is also true.– If in each critical row the conclusion is also true, then

the argument form is valid– If there is at least one critical row in which the

conclusion is false, the argument form is invalid

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Valid or Invalid Argument ?

p ( q r)~rp q

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Valid or Invalid Argument ?p q ~rq p rp r

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Modus Ponensp qp q

Modus ponens: method of affirming

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Modus Tolensp q~q ~p

Construct the truth table

Modus tolens: method of denying

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Basic Rules of Inference

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ExampleIt is below freezing nowTherefore, It is either below freezing or raining now

pp q

Valid argument : addition rule

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ExampleIt is below freezing and raining nowTherefore, It is below freezing now

p qp

Valid argument : simplification rule

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ExampleIf it rains today, then we will not have a

barbecue todayIf we do not have a barbecue today, then

we will have a barbecue tomorrowTherefore,If it rains today, then we will have a

barbecue tomorrow

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ExampleIf it rains today, then we will not have a barbecue

todayIf we do not have a barbecue today, then we will have

a barbecue tomorrowTherefore,If it rains today, then we will have a barbecue

tomorrow

p qq r

p r

Valid argument : hypothetical syllogism

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Rule of Inference and Arguments

To show whether an argument is valid, when there are many premises in an argument:

- Construct truth table (not efficient)

- Use several rules of inference

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ExampleIt is not sunny this afternoon and it is colder

than yesterday.

We will go swimming only if it is sunny

If we do not go swimming, then we will take a canoe trip

If we take a canoe trip, then we will be home by sunset

Conclusion:

We will be home by sunset

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ExampleIt is not sunny this afternoon and it is colder

than yesterday.

We will go swimming only if it is sunny

If we do not go swimming, then we will take a canoe trip

If we take a canoe trip, then we will be home by sunset

Conclusion:

We will be home by sunset

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Solution

p: it is sunny this afternoon

q: it is colder than yesterday.

r: we will go swimming

s: we will take a canoe trip

t: we will be home by sunset

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Solution~p q

r p

~r s

s t

Conclusion:

t ?

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Solution1. ~p q2. ~p Simplication3. r p4. ~r Modus tollens5. ~r s6. s Modus ponens7. s t8. t Modus ponens

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ExampleIf you send me an e-mail message, then I will

finish writing the program

If you do not send me an e-mail message, then I will go to sleep early

If I go to sleep early, then I will wake up feeling refreshed

Conclusion:

If I do not finish writing the program, then I will wake up feeling refreshed.

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Solution

p: you send me an e-mail message

q: I will finish writing the program

r: I will go to sleep early

s: I will wake up feeling refreshed

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Solutionp q

~p r

r s

Conclusion:

~q s?

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Solution1. p q

2. ~q ~p Contrapositive

3. ~p r

4. ~q r Hypothetical syllogism

5. r s

6. ~q s Hypothetical syllogism

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ResolutionUsed to automate the task of reasoning

and proving theorems.

((p q) p r)) (q r)

It is a tautology.

Construct the truth table, if you don’t believe it.

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ExampleJasmine is skiing or it is not snowing.It is snowing or Bart is playing hockey.

Conclusion: Jasmine is skiing or Bart is playing hockey.

p: it is snowingq: Jasmine is skiing r: Bart is playing hockey

((p q) p r)) (q r)

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Solve this problemIf George does not have eight legs, then

he is not an insect.

George is an insect.

Therefore

George has eight legs.

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Solve this problemRudy works hard.

If Rudy works hard, then he is a dull boy.

If Rudy is a dull boy, then he will not get the job.

Conclusion:

Rudy will not get the job

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Solve this problemIf it does not rain or it is not foggy, then

the sailing race will be held and the lifesaving demonstration will go on.

If the sailing race is held, then the trophy will be awarded.

The trophy was not awarded

Conclusion:

It rained

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