1. Muhammad Arief download dari http://arief.ismy.web.id
Rules of Inference pg
Muhammad Arief
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2. IF premise-1, ….., premise-n THEN conclusion
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.
Kita hanya memperhatikan dimana semua premises benar, jadi (kalau bisa) tidak perlu membuat truth table.

3. 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  q p q
The symbol , read “therefore”
Construct the truth table

4. Argument If Socrates is a human being, then Socrates is mortal
Therefore,
Socrates is mortal
p  q p q

5. 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

6. Valid or Invalid Argument ?
p  ( q  r) ~r
p  q

7. Valid or Invalid Argument ?
p  q  ~r
q  p Ù r
p  r

8. Modus Ponens p  q p  q Modus ponens: method of affirming

9. Construct the truth table
Modus Tolens
p  q ~q
 ~p
Construct the truth table
Modus tolens: method of denying

10. Basic Rules of Inference

11. Example It is below freezing now Therefore,
It is either below freezing or raining now p
p  q
Valid argument : addition rule

12. Example It is below freezing and raining now Therefore,
It is below freezing now
p Ù q p
Valid argument : simplification rule

13. Example If it rains today, then we will not have a barbecue today
If we do not have a barbecue today, then we will have a barbecue tomorrow
Therefore,
If it rains today, then we will have a barbecue tomorrow

14. Example If it rains today, then we will not have a barbecue today
If we do not have a barbecue today, then we will have a barbecue tomorrow
Therefore,
If it rains today, then we will have a barbecue tomorrow
p  q
q  r
 p  r
Valid argument : hypothetical syllogism

15. 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

16. Example
It 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

17. Example
It 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

18. 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

19. Solution ~p Ù q r  p ~r  s s  t Conclusion: t ?

20. Solution ~p Ù q ~p Simplication r  p ~r Modus tollens ~r  s
s Modus ponens
s  t
t Modus ponens

21. Example
If you send me an message, then I will finish writing the program
If you do not send me an 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.

22. 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

23. Solution p  q ~p  r r  s Conclusion: ~q  s?

24. Solution p  q ~q  ~p Contrapositive ~p  r
~q  r Hypothetical syllogism
r  s
~q  s Hypothetical syllogism
The contrapositive of p  q is ~q  ~p
A conditional statement is logically equivalent to its contrapositive.

25. ((p  q) Ù (~ p  r))  (q  r)
Resolution
Used to automate the task of reasoning and proving theorems.
((p  q) Ù (~ p  r))  (q  r)
It is a tautology.
Construct the truth table, if you don’t believe it.

26. ((p  q) Ù (~ p  r))  (q  r)
Example
Jasmine 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 snowing
q: Jasmine is skiing
r: Bart is playing hockey
((p  q) Ù (~ p  r))  (q  r)

27. Solve this problem
If George does not have eight legs, then he is not an insect.
George is an insect.
Therefore
George has eight legs.

28. Solve this problem Rudy 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

29. Solve this problem
If 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