discreatemathpresentationrss-171107192105.pptx
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This document presents an introduction to rules of inference and logical arguments. It defines an argument as a sequence of statements ending with a conclusion, and premises as the statements preceding the conclusion. Two main valid rules of inference are then described: modus ponens, where if a conditional statement and its hypothesis are true then the conclusion must be true; and modus tollens, where if a conditional statement is true and the conclusion is false, then the hypothesis must be false. Examples are provided for each. Finally, two common fallacies - affirming the conclusion and denying the hypothesis - are explained.
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discreatemathpresentationrss-171107192105.pptx
- 1. WELCOME TO OUR PRESENTATION Team Members 1. Abdi Khaliq roll 12 2. Abdallah roll 11 3. Abdalla roll 48 Submitted to: lecturer dept of CSE DIU Shifat jahan setu
- 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 r If it rains, then it is cloudy. It rains. Therefore, it is cloudy. r is the proposition “it rains.” 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