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Rules of inference
The document discusses rules of inference in logic. It begins by defining an argument as having premises and a conclusion. Several common rules of inference are then outlined, including modus ponens, modus tollens, and disjunctive syllogism. The remainder of the document works through examples of arguments and tests their validity using the rules of inference. It symbolically represents the arguments and shows the step-by-step workings to determine if the conclusions follow logically from the premises.
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Rules of inference
- 1. Rules of Inference By, LakshmiR Asst. professor, Dept. of ISE
- 2. Argument: Premises |Hypothesis, Conclusion • Consider a set of proposition p1, p2, p3, … pn and q. (p1 ∧ p2 ∧ p3 ∧ … ∧ pn ) → q is called an argument p1, p2, p3, … pn – Premises or hypothesis of the argument q - conclusion Representation: Lakshmi R, Asst. Professor, Dept. Of ISE p1 p2 p3 pn q … Valid Argument: If p1, p2, p3, … pn are all true, then the conclusion is true.
- 3. Rules of Inference LakshmiR, Asst. Professor, Dept. Of ISE Rule of Conjunctive Simplification Rule of Disjunctive Amplification Rule of Conjunction Modus Ponens or Rule of Detachment Rule of Inference Logical Implication Name of the rule
- 4. Rules of Inference Lawof syllogism Modus Tollens Rule of Disjunctive syllogism Lakshmi R, Asst. Professor, Dept. Of ISE ¬ p →F Rule of Contradiction (¬ p →F) → p p
- 5. Rules of Inference LakshmiR, Asst. Professor, Dept. Of ISE 1. Test whether the following argument is valid If Sachin hits a century, then he gets a free car. Sachin hits a century Sachin gets a free car. Solution: Let p: Sachin hits a century q: Sachin gets a free car The given argument is symbolically represented as p → q p q In the view of Modus Ponens rule, this is a valid argument
- 6. Lakshmi R, Asst.Professor, Dept. Of ISE 2. Test whether the following argument is valid If Sachin hits a century, then he gets a free car. Sachin gets a free car Sachin has hit a century Solution: Let p: Sachin hits a century q: Sachin gets a free car The given argument is symbolically represented as p → q q p There is no rule of inference that asserts p is true. ⸫ It is not a valid argument.
- 7. Lakshmi R, Asst.Professor, Dept. Of ISE 3. Test whether the following argument is valid I will study calculus or I will not become a software engineer I will become a software engineer ⸫ I will study calculus Solution: Let p: I will study calculus q: I will become a software engineer The given argument can be symbolically represented as p ∨ ¬q q ⸫p ⇔ ¬q ∨ p q ⸫p ⇔ q → p q ⸫p In the view of Modus Ponens rule, this is a valid argument
- 8. Lakshmi R, Asst.Professor, Dept. Of ISE 4. Test the validity of the following argument Rita is driving. If Rita is driving, then she is not reading. If Rita is not reading, then she will not top the class. Therefore, Rita will not top the class. Solution: Let p: Rita is driving q: Rita is reading r: Rita will top the class The argument can be symbolically represented as: p p → ¬ q ¬ q → ¬r ⸫ ¬r
- 9. Steps 1. p →¬ q 2. ¬ q → ¬r 3. p → ¬r 4. p 5. ⸫ ¬r Reasons Premise Premise Step (1) and (2), law of syllogism Premise Step (3) and (4), Modus Ponens p p → ¬ q ¬ q → ¬r ⸫ ¬r Lakshmi R, Asst. Professor, Dept. Of ISE Therefore, the given argument is valid
- 10. Lakshmi R, Asst.Professor, Dept. Of ISE 5. Test the validity of the following argument ¬ p ↔ q q → r ¬r ⸫ p Solution: Steps 1. ¬ p ↔ q 2. (¬ p → q) ∧ (q → ¬ p ) 3. ¬ p → q 4. q → r 5. ¬p → r 6. ¬r 7. ¬(¬p) 8. ⸫ p Reasons Premise Step (1) , ¬ p ↔ q ⇔((¬ p → q) ∧ (q → ¬ p)) Step (2), rule of conjunctive simplification Premise Steps (3) and (4), law of syllogism Premise Steps (5) and (6), Modus Tollens Step (7), Law of double negation Therefore, the given argument is valid
- 11. Lakshmi R, Asst.Professor, Dept. Of ISE 6. Test the validity of the following argument p → r r → s t ∨ ¬s ¬t ∨ u ¬u ⸫ ¬ p
- 12. Lakshmi R, Asst.Professor, Dept. Of ISE Steps 1. p → r 2. r→ s 3. p → s 4. t ∨ ¬s 5. ¬s ∨ t 6. s → t 7. p → t 8. ¬t ∨ u 9. t → u 10. p → u 11. ¬u 12. ⸫ ¬ p Reasons Premise Premise Steps (1) and (2), law of syllogism Premise Step (4), commutative law Step (5) and the fact p → q ⇔ ¬p ∨ q Step (3) and (6), law of syllogism Premise Step (8) and the fact p → q ⇔ ¬p ∨ q Steps (7) and (9), law of syllogism Premise Steps (10) and (11), Modus Tollens Therefore, the given argument is valid p → r r → s t ∨ ¬s ¬t ∨ u ¬u ⸫ ¬ p
- 13. Lakshmi R, Asst.Professor, Dept. Of ISE 7. Test the validity of the following argument (¬ p ∨ ¬q) → (r ∧ s) r → t ¬t ⸫ p Steps 1. r → t 2. ¬t 3. ¬r 4. ¬r ∨ ¬s 5. ¬(r ∧ s) 6. (¬ p ∨ ¬q) → (r ∧ s) 7. ¬ (¬ p ∨ ¬q) 8. (p ∧ q) 9. ⸫ p Reasons Premise Premise Steps(1) and (2), Modus Tollens Step(3), rule of disjunctive amplification Step(4), DeMorgan’s Law Premise Step(6) and (5), Modus Tollens Step(7), Demorgan’s Law Step(8), rule of conjunctive simplification
- 14. Lakshmi R, Asst.Professor, Dept. Of ISE 8. Test the validity of the following argument a → b a ∨ (c ∧ d) ¬b ∧ ¬e ⸫ c Steps 1. ¬b ∧ ¬e 2. ¬b 3. a → b 4. ¬a 5. a ∨ (c ∧ d) 6. (c ∧ d) 7. ⸫ c Reasons Premise Step (1), law of conjunctive simplification Premise Step(2), (3), Modus Tollens Premise Step (4), (5), disjunctive syllogism Step(6), conjunctive simplification
- 15. Lakshmi R, Asst.Professor, Dept. Of ISE 8. Test the validity of the following argument Steps Reasons (t → e) ∧ (a → l) ⸫(t ∧ a) → (e ∧ l)