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# Discrete Mathematics - Propositions

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Logical connectives, laws of logic, rules of inference.

Logical connectives, laws of logic, rules of inference.

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• 1. Discrete Mathematics Propositions H. Turgut Uyar Ay¸eg¨l Gen¸ata Yayımlı s u c 2001-2014 Emre Harmancı
• 3. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 4. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 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. 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. Proposition Examples Example (propositions) Example (not propositions) The Moon revolves around the Earth. What time is it? Elephants can ﬂy. x < 43 3 + 8 = 11 Exterminate!
• 8. Proposition Examples Example (propositions) Example (not propositions) The Moon revolves around the Earth. What time is it? Elephants can ﬂy. x < 43 3 + 8 = 11 Exterminate!
• 9. Propositional Variable Deﬁnition propositional variable: a name that represents the proposition Example p1 : The Moon revolves around the Earth. (T ) p2 : Elephants can ﬂy. (F ) p3 : 3 + 8 = 11 (T )
• 10. Propositional Variable Deﬁnition propositional variable: a name that represents the proposition Example p1 : The Moon revolves around the Earth. (T ) p2 : Elephants can ﬂy. (F ) p3 : 3 + 8 = 11 (T )
• 11. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 12. Compound Propositions compound propositions are obtained by negating a proposition, or by combining two or more propositions using logical connectives truth table: a table that lists the truth value of the compound proposition for all possible values of its proposition variables
• 13. Compound Propositions compound propositions are obtained by negating a proposition, or by combining two or more propositions using logical connectives truth table: a table that lists the truth value of the compound proposition for all possible values of its proposition variables
• 14. Negation (NOT) Example Table: ¬p p T F ¬p F T ¬p1 : The Moon does not revolve around the Earth. ¬T : F ¬p2 : Elephants cannot ﬂy. ¬F : T
• 15. Negation (NOT) Example Table: ¬p p T F ¬p F T ¬p1 : The Moon does not revolve around the Earth. ¬T : F ¬p2 : Elephants cannot ﬂy. ¬F : T
• 16. Conjunction (AND) Table: p ∧ q p T T F F q T F T F p∧q T F F F Example p1 ∧ p2 : The Moon revolves around the Earth and elephants can ﬂy. T ∧F :F
• 17. Conjunction (AND) Table: p ∧ q p T T F F q T F T F p∧q T F F F Example p1 ∧ p2 : The Moon revolves around the Earth and elephants can ﬂy. T ∧F :F
• 18. Disjunction (OR) Table: p ∨ q p T T F F q T F T F p∨q T T T F Example p1 ∨ p2 : The Moon revolves around the Earth or elephants can ﬂy. T ∨F :T
• 19. Disjunction (OR) Table: p ∨ q p T T F F q T F T F p∨q T T T F Example p1 ∨ p2 : The Moon revolves around the Earth or elephants can ﬂy. T ∨F :T
• 20. Exclusive Disjunction (XOR) Table: p p T T F F q T F T F q p Example q F T T F p1 p2 : Either the Moon revolves around the Earth or elephants can ﬂy. T F :T
• 21. Exclusive Disjunction (XOR) Table: p p T T F F q T F T F q p Example q F T T F p1 p2 : Either the Moon revolves around the Earth or elephants can ﬂy. T F :T
• 22. Implication (IF) Table: p → q p T T F F q T F T F p→q T F T T if p then q p is suﬃcient for q q is necessary for p p: hypothesis q: conclusion ¬p ∨ q
• 23. Implication (IF) Table: p → q p T T F F q T F T F p→q T F T T if p then q p is suﬃcient for q q is necessary for p p: hypothesis q: conclusion ¬p ∨ q
• 24. Implication Examples Example p4 : 3 < 8, p5 : 3 < 14, p6 : 3 < 2, p7 : 8 < 6 p4 → p5 : if 3 < 8, then 3 < 14 T →T :T p6 → p4 : if 3 < 2, then 3 < 8 F →T :T p4 → p6 : if 3 < 8, then 3 < 2 T →F :F p6 → p7 : if 3 < 2, then 8 < 6 F →F :T
• 25. Implication Examples Example p4 : 3 < 8, p5 : 3 < 14, p6 : 3 < 2, p7 : 8 < 6 p4 → p5 : if 3 < 8, then 3 < 14 T →T :T p6 → p4 : if 3 < 2, then 3 < 8 F →T :T p4 → p6 : if 3 < 8, then 3 < 2 T →F :F p6 → p7 : if 3 < 2, then 8 < 6 F →F :T
• 26. Implication Examples Example p4 : 3 < 8, p5 : 3 < 14, p6 : 3 < 2, p7 : 8 < 6 p4 → p5 : if 3 < 8, then 3 < 14 T →T :T p6 → p4 : if 3 < 2, then 3 < 8 F →T :T p4 → p6 : if 3 < 8, then 3 < 2 T →F :F p6 → p7 : if 3 < 2, then 8 < 6 F →F :T
• 27. Implication Examples Example p4 : 3 < 8, p5 : 3 < 14, p6 : 3 < 2, p7 : 8 < 6 p4 → p5 : if 3 < 8, then 3 < 14 T →T :T p6 → p4 : if 3 < 2, then 3 < 8 F →T :T p4 → p6 : if 3 < 8, then 3 < 2 T →F :F p6 → p7 : if 3 < 2, then 8 < 6 F →F :T
• 28. Implication Examples Example p4 : 3 < 8, p5 : 3 < 14, p6 : 3 < 2, p7 : 8 < 6 p4 → p5 : if 3 < 8, then 3 < 14 T →T :T p6 → p4 : if 3 < 2, then 3 < 8 F →T :T p4 → p6 : if 3 < 8, then 3 < 2 T →F :F p6 → p7 : if 3 < 2, then 8 < 6 F →F :T
• 29. Implication Examples Example ” I weigh over 70 kg, then I will exercise.” If Table: p → q p: I weigh over 70 kg. q: I exercise. when is this claim false? p T T F F q T F T F p→q T F T T
• 30. Implication Examples Example ” I weigh over 70 kg, then I will exercise.” If Table: p → q p: I weigh over 70 kg. q: I exercise. when is this claim false? p T T F F q T F T F p→q T F T T
• 31. Implication Examples Example ” I weigh over 70 kg, then I will exercise.” If Table: p → q p: I weigh over 70 kg. q: I exercise. when is this claim false? p T T F F q T F T F p→q T F T T
• 32. Biconditional (IFF) Table: p ↔ q p T T F F q T F T F p↔q T F F T p if and only if q p is necessary and suﬃcient for q (p → q) ∧ (q → p) ¬(p q)
• 33. Biconditional (IFF) Table: p ↔ q p T T F F q T F T F p↔q T F F T p if and only if q p is necessary and suﬃcient for q (p → q) ∧ (q → p) ¬(p q)
• 34. Example Example The parent tells the child: ” you do your homework, you can play computer games.” If s: The child does her homework. t: The child plays computer games. what does the parent mean? s→t ¬s → ¬t s↔t
• 35. Example Example The parent tells the child: ” you do your homework, you can play computer games.” If s: The child does her homework. t: The child plays computer games. what does the parent mean? s→t ¬s → ¬t s↔t
• 36. Example Example The parent tells the child: ” you do your homework, you can play computer games.” If s: The child does her homework. t: The child plays computer games. what does the parent mean? s→t ¬s → ¬t s↔t
• 37. Example Example The parent tells the child: ” you do your homework, you can play computer games.” If s: The child does her homework. t: The child plays computer games. what does the parent mean? s→t ¬s → ¬t s↔t
• 38. Example Example The parent tells the child: ” you do your homework, you can play computer games.” If s: The child does her homework. t: The child plays computer games. what does the parent mean? s→t ¬s → ¬t s↔t
• 39. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 40. Propositional Logic Language 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 propositional variables truth table: all interpretations of a proposition
• 41. Propositional Logic Language 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 propositional variables truth table: all interpretations of a proposition
• 42. Formula Examples Example (not well-formed) ∨p p∧¬ p¬ ∧ q
• 43. 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 )
• 44. Precedence Examples Example 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. Precedence Examples Example 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
• 46. Precedence Examples Example 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
• 47. Precedence Examples Example 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
• 48. Precedence Examples Example 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
• 49. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 50. Metalanguage Deﬁnition target language: the language being worked on Deﬁnition metalanguage: the language used when talking about the properties of the target language
• 51. Metalanguage Deﬁnition target language: the language being worked on Deﬁnition metalanguage: the language used when talking about the properties of the target language
• 52. Metalanguage Examples Example a native Turkish speaker learning English target language: English metalanguage: Turkish Example a student learning programming target language: C, Python, Java, . . . metalanguage: English, Turkish, . . .
• 53. Metalanguage Examples Example a native Turkish speaker learning English target language: English metalanguage: Turkish Example a student learning programming target language: C, Python, Java, . . . metalanguage: English, Turkish, . . .
• 54. Formula Properties a formula that is true for all interpretations is a tautology a formula that is false for all interpretations is a contradiction a formula that is true for some interpretations is valid these are concepts of the metalanguage
• 55. Formula Properties a formula that is true for all interpretations is a tautology a formula that is false for all interpretations is a contradiction a formula that is true for some interpretations is valid these are concepts of the metalanguage
• 56. Tautology Example Example Table: p ∧ (p → q) → q p q T T F F T F T F p→q (A) T F T T p∧A (B) T F F F B→q T T T T
• 57. Contradiction Example Example Table: p ∧ (¬p ∧ q) p q ¬p T T F F T F T F F F T T ¬p ∧ q (A) F F T F p∧A F F F F
• 58. 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.
• 59. 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.
• 60. 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
• 61. 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
• 62. G¨del’s Theorem o propositional logic is consistent and complete G¨del’s Theorem o Any logical system that is powerful enough to express arithmetic must be either inconsistent or incomplete. liar’s paradox: “This statement is false.”
• 63. G¨del’s Theorem o propositional logic is consistent and complete G¨del’s Theorem o Any logical system that is powerful enough to express arithmetic must be either inconsistent or incomplete. liar’s paradox: “This statement is false.”
• 64. G¨del’s Theorem o propositional logic is consistent and complete G¨del’s Theorem o Any logical system that is powerful enough to express arithmetic must be either inconsistent or incomplete. liar’s paradox: “This statement is false.”
• 65. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 66. Propositional Calculus 1 semantic approach: truth tables too complicated when the number of primitive statements grow 2 axiomatic approach: Boolean algebra substituting logically equivalent formulas for one another 3 syntactic approach: rules of inference obtaining new propositions from existing propositions using logical implications
• 67. Propositional Calculus 1 semantic approach: truth tables too complicated when the number of primitive statements grow 2 axiomatic approach: Boolean algebra substituting logically equivalent formulas for one another 3 syntactic approach: rules of inference obtaining new propositions from existing propositions using logical implications
• 68. Propositional Calculus 1 semantic approach: truth tables too complicated when the number of primitive statements grow 2 axiomatic approach: Boolean algebra substituting logically equivalent formulas for one another 3 syntactic approach: rules of inference obtaining new propositions from existing propositions using logical implications
• 69. Truth Table Example p→q contrapositive: ¬q → ¬p converse: q → p inverse: ¬p → ¬q Example p T T F F q T F T F p→q T F T T ¬q → ¬p T F T T q→p T T F T ¬p → ¬q T T F T an implication and its contrapositive are equivalent the converse and the inverse of an implication are equivalent
• 70. Truth Table Example p→q contrapositive: ¬q → ¬p converse: q → p inverse: ¬p → ¬q Example p T T F F q T F T F p→q T F T T ¬q → ¬p T F T T q→p T T F T ¬p → ¬q T T F T an implication and its contrapositive are equivalent the converse and the inverse of an implication are equivalent
• 71. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 72. Logical Equivalence Deﬁnition if P ↔ Q is a tautology, then P and Q are logically equivalent: P⇔Q
• 73. Logical Equivalence Example Example ¬p ⇔ p → F Table: ¬p ↔ p → F p ¬p T F F T p→F (A) F T ¬p ↔ A T T
• 74. Logical Equivalence Example Example p → q ⇔ ¬p ∨ q Table: (p → q) ↔ (¬p ∨ q) p q T T F F T F T F p→q (A) T F T T ¬p F F T T ¬p ∨ q (B) T F T T A↔B T T T T
• 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. 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. 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
• 78. 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
• 79. 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
• 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. 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. 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
• 83. 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
• 84. 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
• 85. Equivalence Example Example p→q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
• 86. Equivalence Example Example p→q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
• 87. Equivalence Example Example p→q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
• 88. Equivalence Example Example p→q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
• 89. Equivalence Example Example p→q ⇔ ¬p ∨ q ⇔ q ∨ ¬p Co ⇔ ¬¬q ∨ ¬p DN ⇔ ¬q → ¬p
• 90. Equivalence Example 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. Equivalence Example 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. Equivalence Example 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. Equivalence Example 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. Equivalence Example 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
• 95. Equivalence Example 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
• 96. Equivalence Example 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
• 97. Duality Deﬁnition If s contains no logical connectives other than ∧ and ∨, then the dual of s, denoted s d , is the statement obtained from s by replacing each occurrence of ∧ by ∨, ∨ by ∧, T by F , and F by T . Example (dual proposition) s : (p ∧ ¬q) ∨ (r ∧ T ) d s : (p ∨ ¬q) ∧ (r ∨ F )
• 98. Duality Deﬁnition If s contains no logical connectives other than ∧ and ∨, then the dual of s, denoted s d , is the statement obtained from s by replacing each occurrence of ∧ by ∨, ∨ by ∧, T by F , and F by T . Example (dual proposition) s : (p ∧ ¬q) ∨ (r ∧ T ) d s : (p ∨ ¬q) ∧ (r ∨ F )
• 99. Principle of Duality principle of duality Let s and t be statements that contain no logical connectives other than ∧ and ∨. If s ⇔ t then s d ⇔ t d .
• 100. Topics 1 Propositions Introduction Compound Propositions Propositional Logic Language Metalanguage 2 Propositional Calculus Introduction Laws of Logic Rules of Inference
• 101. Rules of Inference Deﬁnition if P → Q is a tautology, then P logically implies Q: P⇒Q
• 102. Logical Implication Example Example p ∧ (p → q) ⇒ q Table: p ∧ (p → q) → q p q T T F F T F T F p→q (A) T F T T p∧A (B) T F F F B→q T T T T
• 103. Inference establishing 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
• 104. Inference establishing 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
• 105. Trivial Rules Identity (ID) Contradiction (CTR) p ∴p F ∴p
• 106. Trivial Rules Identity (ID) Contradiction (CTR) p ∴p F ∴p
• 107. Basic Rules OR Introduction (OrI) p ∴p∨q AND Elimination (AndE) p∧q ∴p AND Introduction (AndI) p q ∴p∧q
• 108. Basic Rules OR Introduction (OrI) p ∴p∨q AND Elimination (AndE) p∧q ∴p AND Introduction (AndI) p q ∴p∧q
• 109. Basic Rules OR Introduction (OrI) p ∴p∨q AND Elimination (AndE) p∧q ∴p AND Introduction (AndI) p q ∴p∧q
• 110. Implication Elimination Modus Ponens (ImpE) p→q p ∴q Modus Tollens (MT) p→q ¬q ∴ ¬p
• 111. Implication Elimination Modus Ponens (ImpE) p→q p ∴q Modus Tollens (MT) p→q ¬q ∴ ¬p
• 112. Implication Elimination Example Example (Modus Ponens) If Lydia wins the lottery, she will buy a car. Lydia has won the lottery. Therefore, Lydia will buy a car. Example (Modus Tollens) If Lydia wins the lottery, she will buy a car. Lydia did not buy a car. Therefore, Lydia did not win the lottery.
• 113. Implication Elimination Example Example (Modus Ponens) If Lydia wins the lottery, she will buy a car. Lydia has won the lottery. Therefore, Lydia will buy a car. Example (Modus Tollens) If Lydia wins the lottery, she will buy a car. Lydia did not buy a car. Therefore, Lydia did not win the lottery.
• 114. Fallacies aﬃrming the conclusion p→q q ∴p (p → q) ∧ q denying the hypothesis p→q ¬p ∴ ¬q p: (F → T ) ∧ T → F : F (p → q) ∧ ¬p ¬q: (F → T ) ∧ T → F : F
• 115. Fallacies aﬃrming the conclusion p→q q ∴p (p → q) ∧ q denying the hypothesis p→q ¬p ∴ ¬q p: (F → T ) ∧ T → F : F (p → q) ∧ ¬p ¬q: (F → T ) ∧ T → F : F
• 116. Fallacy Examples Example (aﬃrming the conclusion) If Lydia wins the lottery, she will buy a car. Lydia has bought a car. Therefore, Lydia has won the lottery. Example (denying the hypothesis) If Lydia wins the lottery, she will buy a car. Lydia has not won the lottery. Therefore, Lydia will not buy a car.
• 117. Fallacy Examples Example (aﬃrming the conclusion) If Lydia wins the lottery, she will buy a car. Lydia has bought a car. Therefore, Lydia has won the lottery. Example (denying the hypothesis) If Lydia wins the lottery, she will buy a car. Lydia has not won the lottery. Therefore, Lydia will not buy a car.
• 118. Provisional Assumptions 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) PAs have to be discharged OR Elimination (OrE) p∨q p r q r ∴ r p and q are PAs
• 119. Provisional Assumptions 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) PAs have to be discharged OR Elimination (OrE) p∨q p r q r ∴ r p and q are PAs
• 120. Provisional Assumptions 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) PAs have to be discharged OR Elimination (OrE) p∨q p r q r ∴ r p and q are PAs
• 121. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 122. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 123. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 124. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 125. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 126. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 127. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 128. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 129. Implication Introduction Example Example (Modus Tollens) 1. p PA 2. p → q 3. p→q ¬q ∴ ¬p A q ImpE : 1, 2 4. ¬q A 5. q → F EQ : 4 6. ImpE : 3, 5 F 7. p → F 8. ¬p ImpI : 1, 6 EQ : 7
• 130. 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.
• 131. 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.
• 132. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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
• 133. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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
• 134. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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
• 135. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Disjunctive Syllogism Example 1. p∨q A 2. ¬p A 3. p → F p∨q ¬p ∴q 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. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 143. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 144. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 145. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 146. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 147. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 148. Hypothetical Syllogism 1. Hypothetical Syllogism (HS) p→q q→r ∴p→r p PA 2. p → q A 3. q ImpE : 1, 2 4. q → r A 5. ImpE : 3, 4 r 6. p → r ImpI : 1, 5
• 149. Hypotetical Syllogism Example Example (Star Trek) 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.
• 150. Hypotetical Syllogism Example Example (Star Trek) 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.
• 151. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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
• 152. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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
• 153. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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
• 154. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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
• 155. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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. Hypotetical Syllogism Example Example 1. p → q A p p→q q→r r →s ∴s 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. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬r ¬p A MT : 6, 5 A MT : 8, 7
• 160. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s ¬r 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬p A MT : 6, 5 A MT : 8, 7
• 161. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s ¬r 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬p A MT : 6, 5 A MT : 8, 7
• 162. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s ¬r 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬p A MT : 6, 5 A MT : 8, 7
• 163. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s ¬r 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬p A MT : 6, 5 A MT : 8, 7
• 164. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s ¬r 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬p A MT : 6, 5 A MT : 8, 7
• 165. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬r ¬p A MT : 6, 5 A MT : 8, 7
• 166. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬r ¬p A MT : 6, 5 A MT : 8, 7
• 167. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬r ¬p A MT : 6, 5 A MT : 8, 7
• 168. Inference Examples Example p→r r →s x ∨ ¬s u ∨ ¬x ¬u ∴ ¬p 1. u ∨ ¬x A 6. r → s 2. ¬u A 7. 3. ¬x DS : 1, 2 8. p → r A 9. 4. x ∨ ¬s 5. ¬s DS : 4, 3 ¬r ¬p A MT : 6, 5 A MT : 8, 7
• 169. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 170. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 171. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 172. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 173. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 174. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 175. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 176. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 177. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 178. Inference Examples Example (¬p ∨ ¬q) → (r ∧ s) r →x ¬x ∴p 1. r →x A 6. (¬p ∨ ¬q) → (r ∧ s) A 2. ¬x A 7. ¬(¬p ∨ ¬q) 3. ¬r MT : 1, 2 8. p∧q OrI : 3 9. p 4. ¬r ∨ ¬s 5. ¬(r ∧ s) DM : 4 MT : 6, 5 DM : 7 AndE : 8
• 179. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 180. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 181. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 182. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 183. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 184. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 185. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 186. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 187. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 188. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 189. Inference Examples Example 1. A 2. p → (q ∨ r ) s → ¬r q → ¬p p s ∴F q → ¬p p A 3. ¬q 4. s A 5. s → ¬r A 6. ¬r MT : 1, 2 ImpE : 5, 4 7. p → (q ∨ r ) A 8. q∨r 9. q 10. q ∧ ¬q : F ImpE : 7, 2 DS : 8, 6 AndI : 9, 3
• 190. Inference Examples Example 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. ‰ ‰
• 191. Inference Examples Example p: There is a chance of rain. q: Lois’ red headband is lost. r : Lois mows her lawn. ‰ s: The temperature is over 20 .
• 192. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 193. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 194. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 195. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 196. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 197. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 198. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 199. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 200. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 201. Inference Examples Example 1. 2. s 3. (p ∨ q) → ¬r s → ¬p s ∧ ¬q ∴r s ∧ ¬q s → ¬p 4. ¬p ImpE : 3, 2 5. ¬q AndE : 1 6. ¬p ∧ ¬q AndI : 4, 5 7. ¬(p ∨ q) DM : 6 A AndE : 1 A 8. (p ∨ q) → ¬r A 9. 7, 8 ?
• 202. 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