Lecture Notes 6 CS1502

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Lecture Notes 6CS1502

• Formal Proofs in Propositional Logic

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Reiteration

• P P

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Conjunctive Elimination
1. P?Q . . . P Q
? Elimination 1
? Elimination 1
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Conjunctive Introduction

• 1. P . . .2. Q . . .P ? Q

? Introduction 1,2
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Proof

• 1. Cube(b) ? Tet(d) 2. Large(d) 3.
  Tet(d) ? Elim 1 4.
  Tet(d) Large(d) ? Intro 3, 2

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Disjunction Introduction
1. P . . .P ? Q
? Introduction 1
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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
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Disjunctive EliminationProof by cases

• P ? Q . . . P
  S Q SS

? Elimination
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(A B) v C C ? B
Prove
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Negative Elimination

• 1. ? ?P . . .P

? Elimination 1
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Bottom Introduction
1. P10. ?P...?
? Introduction 1, 10
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Negation Introduction

• 7. P . . . 15.
  ? ? P

? Introduction 7-15
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Negation IntroductionProof by contradiction

• 10. ? P . . . 22.
  ? 23. ? ? P

? Introduction 10-22
? Elimination 23
24. P
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Prove
Note This is a resolution step, something we are
covering later
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Reminder Equivalences

• Two FOL sentences P and Q mean the same thing
  (are logically equivalent, 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)
• DeMorgans Laws (P Q) ? P v Q
• (P v Q) ? P
  Q

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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