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# Discrete Mathematics - Propositions

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Logical operators, laws of logic, rules of inference.

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• reorganized the rules of inference parts and fixed some problems

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• improved the examples to be simpler and more consistent

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### Discrete Mathematics - Propositions

1. 1. Discrete Mathematics Propositions H. Turgut Uyar Ay¸seg¨ul Gen¸cata Yayımlı Emre Harmancı 2001-2016
3. 3. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
4. 4. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
5. 5. Proposition Deﬁnition proposition (or statement): a declarative sentence that is either true or false law of the excluded middle: a proposition cannot be partially true or partially false law of contradiction: a proposition cannot be both true and false
6. 6. Proposition Deﬁnition proposition (or statement): a declarative sentence that is either true or false law of the excluded middle: a proposition cannot be partially true or partially false law of contradiction: a proposition cannot be both true and false
7. 7. Proposition Examples propositions The Moon revolves around the Earth. Elephants can ﬂy. 3 + 8 = 11 not propositions What time is it? Exterminate! x < 43
8. 8. Proposition Examples propositions The Moon revolves around the Earth. Elephants can ﬂy. 3 + 8 = 11 not propositions What time is it? Exterminate! x < 43
9. 9. Propositional Variable propositional variable: a name that represents the proposition examples p1: The Moon revolves around the Earth. (T) p2: Elephants can ﬂy. (F) p3: 3 + 8 = 11 (T)
10. 10. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
11. 11. Compound Propositions compound propositions are obtained by applying logical operators truth table: a table that lists the truth value of the compound proposition for all possible values of its variables
12. 12. Compound Propositions compound propositions are obtained by applying logical operators truth table: a table that lists the truth value of the compound proposition for all possible values of its variables
13. 13. Negation (NOT) ¬p p ¬p T F F T examples ¬p1: The Moon does not revolve around the Earth. ¬T : F ¬p2: Elephants cannot ﬂy. ¬F : T
14. 14. Negation (NOT) ¬p p ¬p T F F T examples ¬p1: The Moon does not revolve around the Earth. ¬T : F ¬p2: Elephants cannot ﬂy. ¬F : T
15. 15. Conjunction (AND) p ∧ q p q p ∧ q T T T T F F F T F F F F examples p1 ∧ p2: The Moon revolves around the Earth and elephants can ﬂy. T ∧ F : F p1 ∧ p3: The Moon revolves around the Earth and 3 + 8 = 11. T ∧ T : T
16. 16. Conjunction (AND) p ∧ q p q p ∧ q T T T T F F F T F F F F examples p1 ∧ p2: The Moon revolves around the Earth and elephants can ﬂy. T ∧ F : F p1 ∧ p3: The Moon revolves around the Earth and 3 + 8 = 11. T ∧ T : T
17. 17. Disjunction (OR) p ∨ q p q p ∨ q T T T T F T F T T F F F example p1 ∨ p2: The Moon revolves around the Earth or elephants can ﬂy. T ∨ F : T
18. 18. Disjunction (OR) p ∨ q p q p ∨ q T T T T F T F T T F F F example p1 ∨ p2: The Moon revolves around the Earth or elephants can ﬂy. T ∨ F : T
19. 19. Exclusive Disjunction (XOR) p q p q p q T T F T F T F T T F F F examples p1 p2: Either the Moon revolves around the Earth or elephants can ﬂy. T F : T p1 p3: Either the Moon revolves around the Earth or 3 + 8 = 11. T T : F
20. 20. Exclusive Disjunction (XOR) p q p q p q T T F T F T F T T F F F examples p1 p2: Either the Moon revolves around the Earth or elephants can ﬂy. T F : T p1 p3: Either the Moon revolves around the Earth or 3 + 8 = 11. T T : F
21. 21. Implication (IF) p → q p q p → q T T T T F F F T T F F T also called conditional if p then q p is suﬃcient for q q is necessary for p p: hypothesis q: conclusion
22. 22. Implication Examples p4: 3 < 8, p5: 3 < 14, p6: 3 < 2, p7: 8 < 6 p4 → p5: if 3 < 8, then 3 < 14 T → T : T p4 → p6: if 3 < 8, then 3 < 2 T → F : F p6 → p4: if 3 < 2, then 3 < 8 F → T : T p6 → p7: if 3 < 2, then 8 < 6 F → F : T
23. 23. Implication Examples p4: 3 < 8, p5: 3 < 14, p6: 3 < 2, p7: 8 < 6 p4 → p5: if 3 < 8, then 3 < 14 T → T : T p4 → p6: if 3 < 8, then 3 < 2 T → F : F p6 → p4: if 3 < 2, then 3 < 8 F → T : T p6 → p7: if 3 < 2, then 8 < 6 F → F : T
24. 24. Implication Examples p4: 3 < 8, p5: 3 < 14, p6: 3 < 2, p7: 8 < 6 p4 → p5: if 3 < 8, then 3 < 14 T → T : T p4 → p6: if 3 < 8, then 3 < 2 T → F : F p6 → p4: if 3 < 2, then 3 < 8 F → T : T p6 → p7: if 3 < 2, then 8 < 6 F → F : T
25. 25. Implication Examples p4: 3 < 8, p5: 3 < 14, p6: 3 < 2, p7: 8 < 6 p4 → p5: if 3 < 8, then 3 < 14 T → T : T p4 → p6: if 3 < 8, then 3 < 2 T → F : F p6 → p4: if 3 < 2, then 3 < 8 F → T : T p6 → p7: if 3 < 2, then 8 < 6 F → F : T
26. 26. Implication Examples p4: 3 < 8, p5: 3 < 14, p6: 3 < 2, p7: 8 < 6 p4 → p5: if 3 < 8, then 3 < 14 T → T : T p4 → p6: if 3 < 8, then 3 < 2 T → F : F p6 → p4: if 3 < 2, then 3 < 8 F → T : T p6 → p7: if 3 < 2, then 8 < 6 F → F : T
27. 27. Implication Example ”If I weigh over 70 kg, then I will exercise.” p: I weigh over 70 kg. q: I exercise. when is this claim false? p → q p q p → q T T T T F F F T T F F T
28. 28. Implication Example ”If I weigh over 70 kg, then I will exercise.” p: I weigh over 70 kg. q: I exercise. when is this claim false? p → q p q p → q T T T T F F F T T F F T
29. 29. Implication Example ”If I weigh over 70 kg, then I will exercise.” p: I weigh over 70 kg. q: I exercise. when is this claim false? p → q p q p → q T T T T F F F T T F F T
30. 30. Biconditional (IFF) p ↔ q p q p ↔ q T T T T F F F T F F F T p if and only if q p is necessary and suﬃcient for q
31. 31. Example mother tells child: ”If you do your homework, you can play computer games.” h: The child does her homework. p: The child plays computer games. what does the mother mean? h → p ¬h → ¬p h ↔ p
32. 32. Example mother tells child: ”If you do your homework, you can play computer games.” h: The child does her homework. p: The child plays computer games. what does the mother mean? h → p ¬h → ¬p h ↔ p
33. 33. Example mother tells child: ”If you do your homework, you can play computer games.” h: The child does her homework. p: The child plays computer games. what does the mother mean? h → p ¬h → ¬p h ↔ p
34. 34. Example mother tells child: ”If you do your homework, you can play computer games.” h: The child does her homework. p: The child plays computer games. what does the mother mean? h → p ¬h → ¬p h ↔ p
35. 35. Example mother tells child: ”If you do your homework, you can play computer games.” h: The child does her homework. p: The child plays computer games. what does the mother mean? h → p ¬h → ¬p h ↔ p
36. 36. Well-Formed Formula syntax which rules will be used to form compound propositions? a formula that obeys these rules: well-formed formula (WFF) semantics interpretation: calculating the value of a compound proposition by assigning values to its variables truth table: all interpretations of a proposition
37. 37. Well-Formed Formula syntax which rules will be used to form compound propositions? a formula that obeys these rules: well-formed formula (WFF) semantics interpretation: calculating the value of a compound proposition by assigning values to its variables truth table: all interpretations of a proposition
38. 38. Formula Examples not well-formed ∨p p ∧ ¬ p ¬ ∧ q
39. 39. Operator Precedence 1 ¬ 2 ∧ 3 ∨ 4 → 5 ↔ parentheses are used to change the order of calculation implication associates from the right: p → q → r means p → (q → r)
40. 40. Precedence Examples s: Phyllis goes out for a walk. t: The Moon is out. u: It is snowing. what do the following WFFs mean? t ∧ ¬u → s t → (¬u → s) ¬(s ↔ (u ∨ t)) ¬s ↔ u ∨ t
41. 41. Precedence Examples s: Phyllis goes out for a walk. t: The Moon is out. u: It is snowing. what do the following WFFs mean? t ∧ ¬u → s t → (¬u → s) ¬(s ↔ (u ∨ t)) ¬s ↔ u ∨ t
42. 42. Precedence Examples s: Phyllis goes out for a walk. t: The Moon is out. u: It is snowing. what do the following WFFs mean? t ∧ ¬u → s t → (¬u → s) ¬(s ↔ (u ∨ t)) ¬s ↔ u ∨ t
43. 43. Precedence Examples s: Phyllis goes out for a walk. t: The Moon is out. u: It is snowing. what do the following WFFs mean? t ∧ ¬u → s t → (¬u → s) ¬(s ↔ (u ∨ t)) ¬s ↔ u ∨ t
44. 44. Precedence Examples s: Phyllis goes out for a walk. t: The Moon is out. u: It is snowing. what do the following WFFs mean? t ∧ ¬u → s t → (¬u → s) ¬(s ↔ (u ∨ t)) ¬s ↔ u ∨ t
45. 45. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
46. 46. Metalanguage target language: the language being worked on metalanguage: the language used when talking about the properties of the target language
47. 47. Metalanguage target language: the language being worked on metalanguage: the language used when talking about the properties of the target language
48. 48. Metalanguage Examples a native Turkish speaker learning English target language: English metalanguage: Turkish a student learning programming target language: C, Python, Java, . . . metalanguage: English, Turkish, . . .
49. 49. Metalanguage Examples a native Turkish speaker learning English target language: English metalanguage: Turkish a student learning programming target language: C, Python, Java, . . . metalanguage: English, Turkish, . . .
50. 50. Formula Properties WFF is true for all interpretations: tautology WFF is false for all interpretations: contradiction these are concepts of the metalanguage
51. 51. Formula Properties WFF is true for all interpretations: tautology WFF is false for all interpretations: contradiction these are concepts of the metalanguage
52. 52. Tautology Example p ∧ (p → q) → q p q p → q p ∧ A B → q (A) (B) T T T T T T F F F T F T T F T F F T F T
53. 53. Contradiction Example p ∧ (¬p ∧ q) p q ¬p ¬p ∧ q p ∧ A (A) T T F F F T F F F F F T T T F F F T F F
54. 54. Logical Implication and Equivalence if P → Q is a tautology, then P logically implies Q: P ⇒ Q if P ↔ Q is a tautology, then P and Q are logically equivalent: P ⇔ Q
55. 55. Logical Implication and Equivalence if P → Q is a tautology, then P logically implies Q: P ⇒ Q if P ↔ Q is a tautology, then P and Q are logically equivalent: P ⇔ Q
56. 56. Logical Implication Example p ∧ (p → q) ⇒ q p ∧ (p → q) → q p q p → q A ∧ p B → q (A) (B) T T T T T T F F F T F T T F T F F T F T
57. 57. Logical Equivalence Example ¬p ⇔ p → F ¬p ↔ p → F p ¬p p → F ¬p ↔ A (A) T F F T F T T T
58. 58. Logical Equivalence Example p → q ⇔ ¬p ∨ q (p → q) ↔ (¬p ∨ q) p q p → q ¬p ¬p ∨ q A ↔ B (A) (B) T T T F T T T F F F F T F T T T T T F F T T T T
59. 59. Logical Equivalence Example implication: p → q contrapositive: ¬q → ¬p converse: q → p inverse: ¬p → ¬q p → q ⇔ ¬q → ¬p (p → q) ↔ (¬q → ¬p) p q p → q ¬q ¬p ¬q → ¬p A ↔ B (A) (B) T T T F F T T T F F T F F T F T T F T T T F F T T T T T
60. 60. Logical Equivalence Example implication: p → q contrapositive: ¬q → ¬p converse: q → p inverse: ¬p → ¬q p → q ⇔ ¬q → ¬p (p → q) ↔ (¬q → ¬p) p q p → q ¬q ¬p ¬q → ¬p A ↔ B (A) (B) T T T F F T T T F F T F F T F T T F T T T F F T T T T T
61. 61. Metalogic P1, P2, . . . , Pn Q There is a proof which infers the conclusion Q from the assumptions P1, P2, . . . , Pn. P1, P2, . . . , Pn Q Q must be true if P1, P2, . . . , Pn are all true.
62. 62. Metalogic P1, P2, . . . , Pn Q There is a proof which infers the conclusion Q from the assumptions P1, P2, . . . , Pn. P1, P2, . . . , Pn Q Q must be true if P1, P2, . . . , Pn are all true.
63. 63. Formal Systems a formal system is consistent if for all WFFs P and Q: if P Q then P Q if every provable proposition is actually true a formal system is complete if for all WFFs P and Q: if P Q then P Q if every true proposition can be proven
64. 64. Formal Systems a formal system is consistent if for all WFFs P and Q: if P Q then P Q if every provable proposition is actually true a formal system is complete if for all WFFs P and Q: if P Q then P Q if every true proposition can be proven
65. 65. G¨odel’s Theorem propositional logic is consistent and complete Theorem (G¨odel’s Theorem) Any logical system that is powerful enough to express arithmetic must be either inconsistent or incomplete. liar’s paradox: “This statement is false.”
66. 66. G¨odel’s Theorem propositional logic is consistent and complete Theorem (G¨odel’s Theorem) Any logical system that is powerful enough to express arithmetic must be either inconsistent or incomplete. liar’s paradox: “This statement is false.”
67. 67. G¨odel’s Theorem propositional logic is consistent and complete Theorem (G¨odel’s Theorem) Any logical system that is powerful enough to express arithmetic must be either inconsistent or incomplete. liar’s paradox: “This statement is false.”
68. 68. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
69. 69. Propositional Calculus 1 semantic approach: truth tables too complicated when the number of primitive statements grow 2 syntactic approach: rules of inference obtain new propositions from known propositions using logical implications 3 axiomatic approach: Boolean algebra substitute logically equivalent formulas for one another
70. 70. Propositional Calculus 1 semantic approach: truth tables too complicated when the number of primitive statements grow 2 syntactic approach: rules of inference obtain new propositions from known propositions using logical implications 3 axiomatic approach: Boolean algebra substitute logically equivalent formulas for one another
71. 71. Propositional Calculus 1 semantic approach: truth tables too complicated when the number of primitive statements grow 2 syntactic approach: rules of inference obtain new propositions from known propositions using logical implications 3 axiomatic approach: Boolean algebra substitute logically equivalent formulas for one another
72. 72. Laws of Logic Double Negation (DN) ¬(¬p) ⇔ p Commutativity (Co) p ∧ q ⇔ q ∧ p p ∨ q ⇔ q ∨ p Associativity (As) (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r) (p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) Idempotence (Ip) p ∧ p ⇔ p p ∨ p ⇔ p Inverse (In) p ∧ ¬p ⇔ F p ∨ ¬p ⇔ T
73. 73. Laws of Logic Double Negation (DN) ¬(¬p) ⇔ p Commutativity (Co) p ∧ q ⇔ q ∧ p p ∨ q ⇔ q ∨ p Associativity (As) (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r) (p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) Idempotence (Ip) p ∧ p ⇔ p p ∨ p ⇔ p Inverse (In) p ∧ ¬p ⇔ F p ∨ ¬p ⇔ T
74. 74. Laws of Logic Double Negation (DN) ¬(¬p) ⇔ p Commutativity (Co) p ∧ q ⇔ q ∧ p p ∨ q ⇔ q ∨ p Associativity (As) (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r) (p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) Idempotence (Ip) p ∧ p ⇔ p p ∨ p ⇔ p Inverse (In) p ∧ ¬p ⇔ F p ∨ ¬p ⇔ T
75. 75. Laws of Logic Double Negation (DN) ¬(¬p) ⇔ p Commutativity (Co) p ∧ q ⇔ q ∧ p p ∨ q ⇔ q ∨ p Associativity (As) (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r) (p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) Idempotence (Ip) p ∧ p ⇔ p p ∨ p ⇔ p Inverse (In) p ∧ ¬p ⇔ F p ∨ ¬p ⇔ T
76. 76. Laws of Logic Double Negation (DN) ¬(¬p) ⇔ p Commutativity (Co) p ∧ q ⇔ q ∧ p p ∨ q ⇔ q ∨ p Associativity (As) (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r) (p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) Idempotence (Ip) p ∧ p ⇔ p p ∨ p ⇔ p Inverse (In) p ∧ ¬p ⇔ F p ∨ ¬p ⇔ T
77. 77. Laws of Logic Identity (Id) p ∧ T ⇔ p p ∨ F ⇔ p Domination (Do) p ∧ F ⇔ F p ∨ T ⇔ T Distributivity (Di) p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) Absorption (Ab) p ∧ (p ∨ q) ⇔ p p ∨ (p ∧ q) ⇔ p DeMorgan’s Laws (DM) ¬(p ∧ q) ⇔ ¬p ∨ ¬q ¬(p ∨ q) ⇔ ¬p ∧ ¬q
78. 78. Laws of Logic Identity (Id) p ∧ T ⇔ p p ∨ F ⇔ p Domination (Do) p ∧ F ⇔ F p ∨ T ⇔ T Distributivity (Di) p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) Absorption (Ab) p ∧ (p ∨ q) ⇔ p p ∨ (p ∧ q) ⇔ p DeMorgan’s Laws (DM) ¬(p ∧ q) ⇔ ¬p ∨ ¬q ¬(p ∨ q) ⇔ ¬p ∧ ¬q
79. 79. Laws of Logic Identity (Id) p ∧ T ⇔ p p ∨ F ⇔ p Domination (Do) p ∧ F ⇔ F p ∨ T ⇔ T Distributivity (Di) p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) Absorption (Ab) p ∧ (p ∨ q) ⇔ p p ∨ (p ∧ q) ⇔ p DeMorgan’s Laws (DM) ¬(p ∧ q) ⇔ ¬p ∨ ¬q ¬(p ∨ q) ⇔ ¬p ∧ ¬q
80. 80. Laws of Logic Identity (Id) p ∧ T ⇔ p p ∨ F ⇔ p Domination (Do) p ∧ F ⇔ F p ∨ T ⇔ T Distributivity (Di) p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) Absorption (Ab) p ∧ (p ∨ q) ⇔ p p ∨ (p ∧ q) ⇔ p DeMorgan’s Laws (DM) ¬(p ∧ q) ⇔ ¬p ∨ ¬q ¬(p ∨ q) ⇔ ¬p ∧ ¬q
81. 81. Laws of Logic Identity (Id) p ∧ T ⇔ p p ∨ F ⇔ p Domination (Do) p ∧ F ⇔ F p ∨ T ⇔ T Distributivity (Di) p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r) Absorption (Ab) p ∧ (p ∨ q) ⇔ p p ∨ (p ∧ q) ⇔ p DeMorgan’s Laws (DM) ¬(p ∧ q) ⇔ ¬p ∨ ¬q ¬(p ∨ q) ⇔ ¬p ∧ ¬q
82. 82. Equivalence Example p → q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
83. 83. Equivalence Example p → q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
84. 84. Equivalence Example p → q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
85. 85. Equivalence Example p → q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
86. 86. Equivalence Example p → q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
87. 87. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
88. 88. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
89. 89. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
90. 90. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
91. 91. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
92. 92. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
93. 93. Equivalence Example ¬(¬((p ∨ q) ∧ r) ∨ ¬q) ⇔ ¬¬((p ∨ q) ∧ r) ∧ ¬¬q DM ⇔ ((p ∨ q) ∧ r) ∧ q DN ⇔ (p ∨ q) ∧ (r ∧ q) As ⇔ (p ∨ q) ∧ (q ∧ r) Co ⇔ ((p ∨ q) ∧ q) ∧ r As ⇔ q ∧ r Ab
94. 94. Duality dual of s: sd replace: ∧ with ∨, ∨ with ∧, T with F, F with T principle of duality: if s ⇔ t then sd ⇔ td example s : (p ∧ ¬q) ∨ (r ∧ T) sd : (p ∨ ¬q) ∧ (r ∨ F)
95. 95. Duality dual of s: sd replace: ∧ with ∨, ∨ with ∧, T with F, F with T principle of duality: if s ⇔ t then sd ⇔ td example s : (p ∧ ¬q) ∨ (r ∧ T) sd : (p ∨ ¬q) ∧ (r ∨ F)
96. 96. Duality dual of s: sd replace: ∧ with ∨, ∨ with ∧, T with F, F with T principle of duality: if s ⇔ t then sd ⇔ td example s : (p ∧ ¬q) ∨ (r ∧ T) sd : (p ∨ ¬q) ∧ (r ∨ F)
97. 97. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
98. 98. Inference establish the validity of an argument starting from a set of propositions which are assumed or proven to be true notation p1 p2 . . . pn ∴ q p1 ∧ p2 ∧ · · · ∧ pn ⇒ q
99. 99. Inference establish the validity of an argument starting from a set of propositions which are assumed or proven to be true notation p1 p2 . . . pn ∴ q p1 ∧ p2 ∧ · · · ∧ pn ⇒ q
100. 100. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
101. 101. Trivial Rules Identity (ID) p ∴ p Contradiction (CTR) F ∴ p
102. 102. Trivial Rules Identity (ID) p ∴ p Contradiction (CTR) F ∴ p
103. 103. Basic Rules OR Introduction (OrI) p ∴ p ∨ q AND Elimination (AndE) p ∧ q ∴ p AND Introduction (AndI) p q ∴ p ∧ q
104. 104. Basic Rules OR Introduction (OrI) p ∴ p ∨ q AND Elimination (AndE) p ∧ q ∴ p AND Introduction (AndI) p q ∴ p ∧ q
105. 105. Basic Rules OR Introduction (OrI) p ∴ p ∨ q AND Elimination (AndE) p ∧ q ∴ p AND Introduction (AndI) p q ∴ p ∧ q
106. 106. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
107. 107. Modus Ponens Implication Elimination (ImpE) p → q p ∴ q example If Lydia wins the lottery, she will buy a car. Lydia has won the lottery. Therefore, Lydia will buy a car.
108. 108. Modus Ponens Implication Elimination (ImpE) p → q p ∴ q example If Lydia wins the lottery, she will buy a car. Lydia has won the lottery. Therefore, Lydia will buy a car.
109. 109. Modus Tollens Modus Tollens (MT) p → q ¬q ∴ ¬p example If Lydia wins the lottery, she will buy a car. Lydia did not buy a car. Therefore, Lydia did not win the lottery.
110. 110. Modus Tollens Modus Tollens (MT) p → q ¬q ∴ ¬p example If Lydia wins the lottery, she will buy a car. Lydia did not buy a car. Therefore, Lydia did not win the lottery.
111. 111. Modus Tollens p → q ¬q ∴ ¬p 1. p → q A 2. ¬q → ¬p EQ : 1 3. ¬q A 4. ¬p ImpE : 2, 3
112. 112. Modus Tollens p → q ¬q ∴ ¬p 1. p → q A 2. ¬q → ¬p EQ : 1 3. ¬q A 4. ¬p ImpE : 2, 3
113. 113. Modus Tollens p → q ¬q ∴ ¬p 1. p → q A 2. ¬q → ¬p EQ : 1 3. ¬q A 4. ¬p ImpE : 2, 3
114. 114. Modus Tollens p → q ¬q ∴ ¬p 1. p → q A 2. ¬q → ¬p EQ : 1 3. ¬q A 4. ¬p ImpE : 2, 3
115. 115. Modus Tollens p → q ¬q ∴ ¬p 1. p → q A 2. ¬q → ¬p EQ : 1 3. ¬q A 4. ¬p ImpE : 2, 3
116. 116. Fallacies p → q q ∴ p (p → q) ∧ q p p : F, q : T (F → T) ∧ T → F : F example If Lydia wins the lottery, she will buy a car. Lydia has bought a car. Therefore, Lydia has won the lottery.
117. 117. Fallacies p → q q ∴ p (p → q) ∧ q p p : F, q : T (F → T) ∧ T → F : F example If Lydia wins the lottery, she will buy a car. Lydia has bought a car. Therefore, Lydia has won the lottery.
118. 118. Fallacies p → q ¬p ∴ ¬q (p → q) ∧ ¬p ¬q p : F, q : T (F → T) ∧ T → F : F example If Lydia wins the lottery, she will buy a car. Lydia has not won the lottery. Therefore, Lydia will not buy a car.
119. 119. Fallacies p → q ¬p ∴ ¬q (p → q) ∧ ¬p ¬q p : F, q : T (F → T) ∧ T → F : F example If Lydia wins the lottery, she will buy a car. Lydia has not won the lottery. Therefore, Lydia will not buy a car.
120. 120. Topics 1 Propositions Introduction Logical Operators Metalanguage Laws of Logic 2 Rules of Inference Introduction Basic Rules Modus Ponens Provisional Assumptions
121. 121. Implication Introduction Implication Introduction (ImpI) p q ∴ p → q if it can be shown that q is true assuming p is true then p → q is true without assuming p is true p is a provisional assumption (PA) provisional assumptions have to be discharged
122. 122. Implication Introduction Implication Introduction (ImpI) p q ∴ p → q if it can be shown that q is true assuming p is true then p → q is true without assuming p is true p is a provisional assumption (PA) provisional assumptions have to be discharged
123. 123. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
124. 124. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
125. 125. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
126. 126. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
127. 127. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
128. 128. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
129. 129. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
130. 130. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
131. 131. Implication Introduction Example p → q ¬q ∴ ¬p 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. ¬q A 5. q → F EQ : 4 6. F ImpE : 5, 3 7. p → F ImpI : 1, 6 8. ¬p EQ : 7
132. 132. OR Elimination OR Elimination (OrE) p ∨ q p r q r ∴ r p and q are provisional assumptions
133. 133. Disjunctive Syllogism Disjunctive Syllogism (DS) p ∨ q ¬p ∴ q example Bart’s wallet is either in his pocket or on his desk. Bart’s wallet is not in his pocket. Therefore, Bart’s wallet is on his desk.
134. 134. Disjunctive Syllogism Disjunctive Syllogism (DS) p ∨ q ¬p ∴ q example Bart’s wallet is either in his pocket or on his desk. Bart’s wallet is not in his pocket. Therefore, Bart’s wallet is on his desk.
135. 135. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
136. 136. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
137. 137. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
138. 138. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
139. 139. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
140. 140. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
141. 141. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
142. 142. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
143. 143. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
144. 144. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
145. 145. Disjunctive Syllogism p ∨ q ¬p ∴ q applying OrE: p ∨ q p q q q ∴ q 1. p ∨ q A 2. ¬p A 3. p → F EQ : 2 4a1. p PA 4a2. F ImpE : 3, 4a1 4a. q CTR : 4a2 4b1. q PA 4b. q ID : 4b1 5. q OrE : 1, 4a, 4b
146. 146. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
147. 147. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
148. 148. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
149. 149. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
150. 150. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
151. 151. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
152. 152. Hypothetical Syllogism Hypothetical Syllogism (HS) p → q q → r ∴ p → r 1. p PA 2. p → q A 3. q ImpE : 2, 1 4. q → r A 5. r ImpE : 4, 3 6. p → r ImpI : 1, 5
153. 153. Hypotetical Syllogism Example Spock to Lieutenant Decker: It would be a suicide to attack the enemy ship now. Someone who attempts suicide is not psychologically ﬁt to command the Enterprise. Therefore, I am obliged to relieve you from duty. p: Decker attacks the enemy ship. q: Decker attempts suicide. r: Decker is not psychologically ﬁt to command the Enterprise. s: Spock relieves Decker from duty.
154. 154. Hypotetical Syllogism Example Spock to Lieutenant Decker: It would be a suicide to attack the enemy ship now. Someone who attempts suicide is not psychologically ﬁt to command the Enterprise. Therefore, I am obliged to relieve you from duty. p: Decker attacks the enemy ship. q: Decker attempts suicide. r: Decker is not psychologically ﬁt to command the Enterprise. s: Spock relieves Decker from duty.
155. 155. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
156. 156. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
157. 157. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
158. 158. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
159. 159. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
160. 160. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
161. 161. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
162. 162. Hypotetical Syllogism Example p p → q q → r r → s ∴ s 1. p → q A 2. q → r A 3. p → r HS : 1, 2 4. r → s A 5. p → s HS : 3, 4 6. p A 7. s ImpE : 5, 6
163. 163. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
164. 164. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
165. 165. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
166. 166. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
167. 167. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
168. 168. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
169. 169. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
170. 170. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
171. 171. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
172. 172. Inference Examples p → r r → s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. ¬u A 2. u ∨ ¬x A 3. ¬x DS : 2, 1 4. x ∨ ¬s A 5. ¬s DS : 4, 3 6. r → s A 7. ¬r MT : 6, 5 8. p → r A 9. ¬p MT : 8, 7
173. 173. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
174. 174. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
175. 175. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
176. 176. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
177. 177. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
178. 178. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
179. 179. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
180. 180. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
181. 181. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
182. 182. Inference Examples (¬p ∨ ¬q) → (r ∧ s) r → x ¬x ∴ p 1. ¬x A 2. r → x A 3. ¬r MT : 2, 1 4. ¬r ∨ ¬s OrI : 3 5. ¬(r ∧ s) DM : 4 6. (¬p ∨ ¬q) → (r ∧ s) A 7. ¬(¬p ∨ ¬q) MT : 6, 5 8. p ∧ q DM : 7 9. p AndE : 8
183. 183. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
184. 184. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
185. 185. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
186. 186. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
187. 187. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
188. 188. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
189. 189. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
190. 190. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
191. 191. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
192. 192. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
193. 193. Inference Examples p → (q ∨ r) s → ¬r q → ¬p p s ∴ F 1. p A 2. q → ¬p A 3. ¬q MT : 2, 1 4. s A 5. s → ¬r A 6. ¬r ImpE : 5, 4 7. p → (q ∨ r) A 8. q ∨ r ImpE : 7, 1 9. q DS : 8, 6 10. q ∧ ¬q : F AndI : 9, 3
194. 194. Inference Examples If there is a chance of rain or her red headband is missing, then Lois will not mow her lawn. Whenever the temperature is over 20, there is no chance for rain. Today the temperature is 22 and Lois is wearing her red headband. Therefore, Lois will mow her lawn. p: There is a chance of rain. q: Lois’ red headband is lost. r: Lois mows her lawn. s: The temperature is over 20.
195. 195. Inference Examples If there is a chance of rain or her red headband is missing, then Lois will not mow her lawn. Whenever the temperature is over 20, there is no chance for rain. Today the temperature is 22 and Lois is wearing her red headband. Therefore, Lois will mow her lawn. p: There is a chance of rain. q: Lois’ red headband is lost. r: Lois mows her lawn. s: The temperature is over 20.
196. 196. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
197. 197. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
198. 198. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
199. 199. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
200. 200. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
201. 201. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
202. 202. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
203. 203. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
204. 204. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
205. 205. Inference Examples (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴ r 1. s ∧ ¬q A 2. s AndE : 1 3. s → ¬p A 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 8. (p ∨ q) → ¬r A 9. ? 7, 8
206. 206. References Required Reading: Grimaldi Chapter 2: Fundamentals of Logic 2.1. Basic Connectives and Truth Tables 2.2. Logical Equivalence: The Laws of Logic 2.3. Logical Implication: Rules of Inference Supplementary Reading: O’Donnell, Hall, Page Chapter 6: Propositional Logic