~ v Contraositive: ~ ~ Inverse: ~ ~ Converse: Å Review ( Ú ) Ù ~( Ù ) ~( Ù ) ~ Ú ~ ~( Ú ) ~ Ù ~ is sufficient for is necessary for ~ ~ 1
Valid and Invalid Arguments CS 231 Dianna Xu 2
Associative Law: (Ù)Ùr Ù(Ùr) (Ú)Úr Ú(Úr) Distributive Law: Review Ù(Úr) (Ù)Ú(Ùr) Ú(Ùr) (Ú)Ù(Úr) Absortion Law: Ú ( Ù ) Ù ( Ú ) 3
Definitions An argument is a seuence of statements (statement forms). All statements in an argument excet for the last one, are called remises. (assumtions, hyotheses) The final statement is the conclusion. A valid argument means the conclusion is true if the remises are all true, with all combinations of variable truth values. 4
Examles All Greeks are human and all humans are mortal; therefore, all Greeks are mortal. Some men are athletes and some athletes are rich; therefore, some men are rich. Some men are swimmers and some swimmers are fish; therefore, some men are fish. 5
Modus Ponens \ 6
Modus Ponens examle Assume you are given the following two statements: you are in this class If you are in this class, you are a student Let = you are in this class Let = you are a student \ By Modus Ponens, you can conclude that you are a student. 7
Modus Ponens Consider ( Ù ( )) Ù( ) (Ù( )) T T T T T T F F F T F T T F T F F T F T \ 8
Modus Tollens Assume that we know: ~ and Recall that ~ ~ Thus, we know ~ and ~ ~ We can conclude ~ ~ ~ 9
Modus Tollens examle Assume you are given the following two statements: you are not a student if you are in this class, you are a student Let = you are in this class Let = you are a student ~ ~ By Modus Tollens, you can conclude that you are not in this class 10
Generalization & Secialization Generalization: If you know that is true, then Ú will ALWAYS be true \ Ú Secialization: If Ù is true, then will ALWAYS be true \ Ù 11
Examle of roof We have the hyotheses: r s t 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 tri If we take a canoe tri, then we will be home by sunset Does this imly that we will be home by sunset? ~ Ù r ~r s s t t 12
Examle of roof 1. ~ Ù 1 st hyothesis 2. ~ Secialization using ste 1 3. r 2 nd hyothesis 4. ~r Modus tollens using stes 2 & 3 5. ~r s 3 rd hyothesis 6. s Modus onens using stes 4 & 5 7. s t 4 th hyothesis 8. t Modus onens using stes 6 & 7 ~ Ù \ \ ~ 13
More rules of inference Conjunction: if and are true searately, then Ù is true \ Ù Elimination: If Ú is true, and is false, then must be true ~ ~ Transitivity: If is true, and r is true, then r must be true r \ r 14
Even more rules of inference Proof by division into cases: if at least one of or is true, then r must be true Contradiction rule: If ~ c is true, we can conclude (via the contra-ositive) Ú r r \ r ~ c Resolution: If Ú is true, and ~Úr is true, then Úr must be true Not in the textbook ~ r r 15
Examle of roof Given the hyotheses: If it does not rain or if 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 trohy will be awarded The trohy was not awarded Can you conclude: It rained? (~r Ú ~f) (s Ù l) s t ~t r 16
Examle of roof 1. ~t 3 rd hyothesis 2. s t 2 nd hyothesis 3. ~s Modus tollens using stes 1 & 2 4. (~rú~f) (sùl) 1 st hyothesis 5. ~(sùl) ~ (~rú~f) Contraositive of ste 4 6. (~sú~l) (rùf) DeMorgan s law and double negation law 7. ~sú~l Generalization using ste 3 8. rùf Modus onens using stes 6 & 7 9. r Secialization using ste 8 \ Ù \ Ú \ ~ ~ 17
Fallacy of the converse Modus Badus Fallacy of affirming the conclusion Consider the following: ~ ~ Is this true? \ Ù( ) (Ù( )) T T T T T T F F F T F T T T F F F T F T Not a valid rule! 18
Modus Badus examle Assume you are given the following two statements: you are a student If you are in this class, you are a student Let = you are in this class Let = you are a student \ It is clearly wrong to conclude that if you are a student, you must be in this class 19
Fallacy of the inverse Modus Badus Fallacy of denying the hyothesis Consider the following: Is this true? ~ ~ ~Ù( )) (~Ù( )) ~ T T T F T T F F F T F T T T F F F T T T Not a valid rule! 20
Modus Badus examle Assume you are given the following two statements: you are not in this class if you are in this class, you are a student Let = you are in this class Let = you are a student ~ ~ You CANNOT conclude that you are not a student just because you are not taking Discrete Math 21