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Lecture Notes 6 CS1502

Lecture Notes 6 CS1502. Formal Proofs in Propositional Logic. Reiteration. P P. Conjunctive Elimination. 1. PQ . . . P Q.  Elimination 1.  Elimination 1. Conjunctive Introduction. 1. P . . . 2. Q . . . P  Q.  Introduction 1,2. Proof.

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Lecture Notes 6 CS1502

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  1. Lecture Notes 6CS1502 Formal Proofs in Propositional Logic

  2. Reiteration • P P

  3. Conjunctive Elimination 1. PQ . . . P Q  Elimination 1  Elimination 1

  4. Conjunctive Introduction • 1. P . . .2. Q . . .P  Q  Introduction 1,2

  5. Proof 1. Cube(b)  Tet(d) 2. Large(d) 3. Tet(d)  Elim 14.Tet(d) ^Large(d)  Intro 3, 2

  6. Disjunction Introduction 1. P . . .P  Q  Introduction 1

  7. AB C (B  C)  D Prove: 1. AB 2. C 3. 4. 5. B  Elim: 1 B  C  Intro: 3, 2 (B  C)  D  Intro: 4

  8. Same conclusion Disjunctive EliminationProof by cases • P  Q . . . P … S Q … SS  Elimination

  9. 1. (A ^ B) v C 2. A ^ B 3. B ^ Elim: 2 4. C v B v intro: 3 5. C 6. C v B v Intro: 5 7. C v B v Elim: 1,2-4,5-6 (A ^ B) v C C  B Prove:

  10. Negative Elimination • 1.  P . . .P  Elimination 1

  11. Bottom Introduction 1. P…10. P...  Introduction 1, 10

  12. Negation Introduction • 7. P . . . 15.  P  Introduction 7-15

  13. Negation IntroductionProof by contradiction • 10.  P . . . 22.  23. P  Introduction 10-22  Elimination 23 24. P

  14. A  B B A 1. A  B 2. B 3. A 4. A 5.   Intro: 4, 3 6. B 7.   Intro: 6, 2 8.   Elim: 1, 4-5, 6-7 9. A  Intro: 3-8 10. A  Elim: 9 Prove: Note: This is a resolution step, something we are covering later

  15. Reminder: Equivalences • Two FOL sentences P and Q mean the same thing (are logicallyequivalent, written P  Q) iff they have the same truth value in all situations. If two sentences are logically equivalent, you can substitute one for the other. • Identity Laws: P ^ T  P; P v F  P • Domination Laws: P v T  T; P ^ F  F • Idempotent Laws: P v P  P; P ^ P  P • Double Negation: ~~P  P • Commutative Laws: P v Q  Q v P; P ^ Q  Q ^ P • Associative Laws: (P v Q) v R P v (Q v R) (P ^ Q) ^ R  P ^ (Q ^ R) • Distributive Laws: P v (Q ^ R)  (P v Q) ^ (P v R) P ^ (Q v R)  (P ^ Q) v (P ^ R) • DeMorgan’s Laws: ~(P ^ Q)  ~P v ~Q ~(P v Q)  ~P ^ ~Q

  16. 1. ~(P ^ Q) 2 . ~(~P v ~Q) 3 . ~P 4 . ~P v ~Q v intro: 3 5 . _|_ _|_ intro: 4,2 6 . ~~P ~ intro: 3-5 (p.by.cont) 7 . P ~ elim: 6 8 . ~Q 9 . ~P v ~Q v intro: 8 10. _|_ _|_ intro: 9,2 11. ~~Q ~ intro: 8-10 12. Q ~ elim: 11 13. P ^ Q ^ intro: 7,12 14. ~(P ^ Q) reit: 1 15. _|_ _|_ intro: 13,14 16. ~~(~P v ~Q) ~ intro: 2-15 17. ~P v ~Q ~ elim: 16

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