Muhammad Arief download dari http://arief.ismy.web.id Rules of Inference pg. 63 - 69 Muhammad Arief download dari http://arief.ismy.web.id http://arief.ismy.web.id

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. http://arief.ismy.web.id

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 http://arief.ismy.web.id

Argument If Socrates is a human being, then Socrates is mortal Therefore, Socrates is mortal p  q p q http://arief.ismy.web.id

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 http://arief.ismy.web.id

Valid or Invalid Argument ? p  ( q  r) ~r p  q http://arief.ismy.web.id

Valid or Invalid Argument ? p  q  ~r q  p Ù r p  r http://arief.ismy.web.id

Modus Ponens p  q p  q Modus ponens: method of affirming http://arief.ismy.web.id

Construct the truth table Modus Tolens p  q ~q  ~p Construct the truth table Modus tolens: method of denying http://arief.ismy.web.id

Basic Rules of Inference http://arief.ismy.web.id

Example It is below freezing now Therefore, It is either below freezing or raining now p p  q Valid argument : addition rule http://arief.ismy.web.id

Example It is below freezing and raining now Therefore, It is below freezing now p Ù q p Valid argument : simplification rule http://arief.ismy.web.id

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 http://arief.ismy.web.id

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 http://arief.ismy.web.id

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 http://arief.ismy.web.id

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 http://arief.ismy.web.id

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 http://arief.ismy.web.id

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 http://arief.ismy.web.id

Solution ~p Ù q r  p ~r  s s  t Conclusion: t ? http://arief.ismy.web.id

Solution ~p Ù q ~p Simplication r  p ~r Modus tollens ~r  s s Modus ponens s  t t Modus ponens http://arief.ismy.web.id

Example If 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. http://arief.ismy.web.id

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 http://arief.ismy.web.id

Solution p  q ~p  r r  s Conclusion: ~q  s? http://arief.ismy.web.id

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. http://arief.ismy.web.id

((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. http://arief.ismy.web.id

((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) http://arief.ismy.web.id

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. http://arief.ismy.web.id

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 http://arief.ismy.web.id

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 http://arief.ismy.web.id