Discrete Math Presentation(Rules of Inference)
- 1. WELCOME TO OUR PRESENTATION RULES OF INFERENCE GROUP MEMBER: 1.RUHUL 2.SOHAN 3.HRIDAY 4.SAJEDUL
- 2. OUTLINE • Definitions • Rules of Inference • Fallacies • Using Rules of Inference to Build Arguments • Rules of Inference and Quantifiers
- 3. DEFINITIONS By an argument, we mean a sequence of statements that ends with a conclusion .The conclusion is the last statement of the argument. The premises are the statements of the argument preceding the conclusion .By a valid argument, we mean that the conclusion must follow from the truth of the premises.
- 4. RULE OF INFERENCE Some tautologies are rules of inference. The general form of a rule of inference is (p1 ^ p2 ^ · · · ^ pn) c where pi are the premises and c is the conclusion.
- 5. MODUS PONENS p q p q If a conditional statement and the hypothesis of the conditional statement are both true, therefore the conclusion must also be true. The basis of the modus ponens is the tautology. (p ^ (p q)) q
- 6. EXAMPLE OF MODUS PONENS If it rains, then it is cloudy. It rains. Therefore, it is cloudy. r is the proposition “it rains.” r c c is the proposition “it is r c
- 7. MODUS PONENS P q P q p ^ (p q) (p ^ (p q)) q T T T T T T F F F T F T T F T F F T F T
- 8. MODUS TOLLENS p q ¬q ¬p This rule of inference is based on the contrapositive. The basis of the modus ponens is the tautology ((p q) ^ ¬q) ¬ p.
- 9. MODUS TOLLENS p q P q ¬q ¬p (p q) ^ ¬q ((p q) ^ ¬q) ¬p T T T F F F T T F F T F F T F T T F T F T F F T T T T T
- 10. THE ADDITION The rule of inference p p v q is the rule of addition. This rule comes from the tautology P (p v q)
- 11. THE SIMPLIFICATION The rule of inference p ^ q p is the rule of simplification. This rule comes from the tautology (p ^ q) p.
- 12. THE FALLACY OF AFFIRMING THE CONCLUSION The wrong “rule of inference” p q q p is denoted the fallacy of affirming the conclusion. The basis of this fallacy is the contingency (q ^ (p q)) p that is a misuse of the modus ponens and is not a tautology.
- 13. FALLACY OF AFFIRMING THE CONCLUSION p q P q q ^ (p q) (q ^ (p q)) p T T T T T T F F F T F T T T T F F T F F
- 14. THE FALLACY OF DENYING THE HYPOTHESIS The wrong “rule of inference” p q ¬p ¬q is denoted the fallacy of denying the hypothesis. The basis of this fallacy is the contingency (¬p ^ (p q)) ¬q that is a misuse of the modus tollens and is not a tautology.
- 15. FALLACY OF DENYING THE HYPOTHESIS p q p q ¬p (p q) ^ ¬p ¬q ((p q) ^ ¬p) ¬q T T T F F F T T F F F F T T F T T T T F F F F T T T T T
- 16. Thank You