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Discrete Math

Discrete Math. Rules of Inference. Arguments. An argument is a sequence of propositions called premises that end with a proposition called a conclusion

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Discrete Math

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  1. Discrete Math Rules of Inference

  2. Arguments • An argument is a sequence of propositions called premises that end with a proposition called a conclusion • An argument is valid if the conclusion logically follows from the truth of the premises. That is, if the truth of all its premises implies its conclusion

  3. Argument form • An argument form is a sequence of compound propositions involving propositional variables. • An argument form is valid no matter which propositions are substituted for the propositional variables in its premises

  4. Argument form and argument • Argument form • If P(x)→Q(x) and Q(x)→R(x) Then it follows that P(x)→R(x) • Arguments • If it is raining then it is cloudy and if it is cloudy we eat fish then it follows that if it is raining we eat fish • If we finish the assignment we get a good grade on the assignment and it we get a good grade on the assignment we get a good grade on the course then it follows that if we get a good grade on the assignment we get a good grade on the course

  5. Rules of inference • In propositional logic we can use true tables to establish if a conclusion is true based on it premises. • This can be complicated and tedious • Instead we will state and use some commonly needed rules of inference. • These rules of inference can be proven using truth tables, then directly applied to help prove that more complex arguments lead to their hypothesized conclusions

  6. Definition • If p and q are arbitrary propositions such that p→q is a tautology then we say that p logically implies q • Commonly used logical implications are called rules of inference

  7. Modus Ponens • (P ^ (P→Q) ) →Q • A Tautology • Argument Form : If P and P→Q then Q

  8. Modus Ponens • (P ^ (P→Q) ) →Q • A Tautology • Argument Form: If P and P→Q then Q • P : I am hungry • Q : I will eat lunch • ARGUMENT: I am hungry. If I am hungry I will eat lunch. Therefore I will eat lunch

  9. Your turn • (P ^ (P→Q) ) →Q • OR P P→Q Q • A Tautology • Argument Form: If P and P→Q then Q • Make your own premises and argument as an example for Modus ponens

  10. Modus Tollens • (¬Q ^ (P→Q) ) →¬P • A Tautology • Argument Form: If ¬Q and P→Q then ¬P

  11. Modus Tollens • (¬Q ^ (P→Q) ) →¬P • A Tautology • Argument Form: If ¬Q and P→Q then ¬P • P : I am hungry • Q : I will eat lunch • ARGUMENT: I will not eat lunch. If I am hungry then I eat lunch. Therefore I am not hungry

  12. Your turn • (¬Q ^ (P→Q) ) →¬P • OR ¬Q P→Q ¬P • A Tautology • Argument Form: If ¬Q and P→Q then ¬P • Make your own premises and argument as an example for Modus tollens

  13. Your turn • Let’s do the same thing for • Addition • P → (PvQ) • Hypothetical Syllogism • { (P→Q) ^ (Q→R) } → (P→R) • Simplification • ( P ^ Q )→ P

  14. Addition • P → (PvQ) • Argument Form : If P then PvQ • Is this a tautology?

  15. Addition • P → (PvQ) • Argument Form : If P then PvQ • Is this a tautology? YES

  16. Addition • P → (PvQ) • Write in the other form • Argument Form: If P then (PvQ) • P : I am hungry • Q : I will eat lunch • ARGUMENT:

  17. Addition • P → (PvQ) • P PvQ • Argument Form: If P then (PvQ) • P : I am hungry • Q : I will eat lunch • ARGUMENT: I am hungry so either I am hungry or I will eat lunch

  18. Simplification • ( P ^ Q ) → P • Is this a tautology? • Argument Form:

  19. Simplification • ( P ^ Q ) → P • Is this a tautology? Yes • Argument Form:

  20. Simplification • ( P ^ Q ) → P • Write argument in other form • Argument Form: If P and Q then P • P : I am hungry • Q : I will eat lunch • ARGUMENT:

  21. Simplification • ( P ^ Q ) → P P ^ Q P • Argument Form: If P and Q then P • P : I am hungry • Q : I will eat lunch • ARGUMENT: If I am hungry and I eat lunch then I am hungry

  22. Hypothetical Syllogism • { (P→Q) ^ (Q→R) } → (P→R) • A Tautology ? • Write in alternate form • Argument Form: If P→Q and Q→R then P→R • P : I am hungry • Q : I will eat lunch • R: I will be sleepy • ARGUMENT:

  23. Hypothetical Syllogism • { (P→Q) ^ (Q→R) } → (P→R) • Argument Form: If P→Q and Q→R then P→R

  24. Hypothetical Syllogism • { (P→Q) ^ (Q→R) } → (P→R) • Argument Form: If P→Q and Q→R then P→R

  25. Hypothetical Syllogism • { (P→Q) ^ (Q→R) } → (P→R) • A Tautology - YESP →Q Q→R P→R • Argument Form: If P→Q and Q→R then P→R • P : I am hungry • Q : I will eat lunch • R : I will be sleepy • ARGUMENT: If I am hungry I will eat lunch, If I eat lunch I will be sleepy therefore If I am hungry I will be sleepy

  26. Using rules of inference • Building arguments • State Hypotheses • Use arguments (based on the rules of inference) to show hypotheses lead to the conclusion

  27. Example • Begin with 3 propositions • p Amid reviews his discrete math notes • q Amid goes to movies • r Amid passes discrete math • Based on 3 premises below show the conclusion “Amid goes to movies” is true • If Amid reviews his discrete math notes then he will pass discrete math • If Amid does not go to movies then he will review his discrete math notes • Amid failed discrete math

  28. Example • Premises • If Amid reviews his discrete math notes then he will pass discrete math • p → r • If Amid does not go to movies then he will review his discrete math notes • ¬q → p • Amid failed discrete math • ¬r • Conclusion • q

  29. Truth table solution • { (P→R) ^ (¬Q→P) ^ ¬R } → Q

  30. Using rules of inference • Show { (P→R) ^ (¬Q→P) ^ ¬R } → Q • P→R hypothesis (premise) • ¬R →¬P rule 2 in table 7 • ¬R hypothesis (premise) • ¬P using 3, 2 and modus ponens • ¬Q→P hypothesis (premise) • ¬P →¬(¬Q) rule 2 in table 7 • ¬P → Q double negation • Q using 4, 7 and modus ponens

  31. Your turn • Rita is baking a cake • If Rita is baking a cake she is not practicing piano • If Rita is not practicing piano her father will not send her to Europe • Therefore Rita’s father will not send her to Europe • Start by defining the premises in terms of propositions

  32. Propositions • P Rita is baking a cake • Q Rita is practicing piano • R Rita’s father will not send her to Europe • P • P→¬Q • ¬Q→R

  33. Demonstrate the argument • P Premise • P→¬Q Premise • ¬Q Modus ponens • ¬Q→R Premise • R Modus ponens

  34. Now lets try a couple of more complex arguments

  35. Here are a set of arguments • p→q • q→(r ^ s) • ¬r v (¬t v u) • p ^ t Show that they lead to the conclusion u

  36. Demonstrating the argument (1) • p→q Premise • q→(r ^ s) Premise • p→(r ^ s) Hypothetical Syllogism (1,2) • p ^ t Premise • p Simplification (4) • r ^ s Modus ponens (3,5) • r Simplification (6) • ¬r v (¬t v u) Premise

  37. Demonstrating the argument (2) • ¬r v (¬t v u) Premise • (¬r v ¬t ) v u Associative (8) • ¬(r ^ t) v u DeMorgan’s (9) • t Simplification (4) • r ^ t Conjunction (7,11) • ¬(¬(r ^ t ) ) Double Negation (12) • u Disjunctive Syllogism (10, 13)

  38. Another set of arguments • ¬p v q → r • r → (s v t) • ¬s ^ ¬u • ¬u → ¬t Leads to the conclusion p

  39. Demonstrating the argument (1) • ¬s ^ ¬u Premise • ¬u Simplification (1) • ¬u → ¬t Premise • ¬t Modus ponens (2,3) • ¬s Simplification (1) • ¬s ^ ¬t Conjunction (4,5) • ¬(s v t) DeMorgan’s (6) • r → (s v t) Premise • ¬r Modus tollens (7,8)

  40. Demonstrating the argument (2) • ¬r Modus tollens (7,8) • ¬p v q → r Premise • ¬r→ ¬(¬p v q) Contrapositive (10) (logically equivalent) • ¬r→ (p ^ ¬q) DeMorgan’s (11) • p ^ ¬q Modus ponens (9,12) • p Simplification (13)

  41. Beware of common fallacies • Fallacy of Affirming the conclusion • If p → q and q then p (argument form is FALSE) • If go swimming then you will be wet. You are wet • Therefore you went swimming • This argument is not true. Perhaps you are wet because you just took a shower

  42. Beware of common fallacies • Fallacy of denying the hypothesis • If p → q and ¬p then ¬q (argument form is FALSE) • If go swimming then you will be wet. You are not wet. • Therefore you did not go swimming • This argument is not true. Perhaps you dried yourself already.

  43. Quantified statements • There are also rules of inference that apply only to quantified statements • Universal instantiation • ∀xP(x) then P(c) where c is any one member the universe of discourse • ∀xP(x) All people eat food, then person Jane eats food P(Jane) • Universal generalization • Show P(c) for c an arbitrary member of the universe of discourse is true, to demonstrate ∀xP(x). C must be an arbitrary element, not a specific element

  44. Quantified statements • There are also rules of inference that apply only to quantified statements • Existential instantiation • ∃xP(x) then P(c) there is at least one member c is the universe of discourse for which P(c) is true • There exists at least one coin that was minted in China, Let us call it myYuan. Then P(myYuan) is true. (we still have to demonstrate the conclusion is true) • Existential generalization • P(c) is true for some c in the universe of discourse then ∃xP(x) • The parrot that lives next door flies, therefore, there exists at least one parrot that files. 44

  45. Example:Universal Instantiaition All people who speak two languages can learn a third language. Tai speaks two languages • Premises • x universe of discourse, all people • P(x) x can speak 2 languages • Q(x) x can learn a third language • ∀x (P(x) →Q(x)) • P(Tai) Tai speaks two languages • Conclusion • Q(Tai) Tai can learn a third language

  46. Example:Universal Instantiaition • ∀x (P(x) → Q(x)) Premise • P(Tai) → Q(Tai) Universal instantiation (1) • P(Tai) Premise • Q(Tai) Modus Ponens (2,3)

  47. Another Example Since every penguin is a bird and every bird is an animal it follows that every penguin is an animal • Premises • x universe of discourse, all creatures • P(x) x is a penguin • Q(x) x is an bird • R(x) x is an animal • ∀ x (P(x) →Q(x)) • ∀ x (Q(x) →R(x)) • Conclusion • ∀ x (P(x) →R(x))

  48. Another example: demonstration • ∀x (P(x) → Q(x)) Premise • P(c) → Q(c) Universal instantiation (1) c is an arbitrary penguin • ∀x (Q(x) → R(x)) Premise • Q(c) → R(c) Universal instantiation (3) • P(c) → R(c) Hypothetical syllogism (2,4) • ∀x (P(x) → R(x)) Universal Generalization (5)

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