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Lecture 12

Lecture 12. fitch rules for conjunction fitch rules for disjunction. Formal proofs and boolean logic. Conjunction Elimination. From the truth table for ∧, we know that if a conjunction is true, then all of its conjuncts are true. Conjunction Elimination (∧ Elim )

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Lecture 12

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  1. Lecture 12 fitch rules for conjunction fitch rules for disjunction

  2. Formal proofs and boolean logic

  3. Conjunction Elimination • From the truth table for ∧, we know that if a conjunction is true, then all of its conjuncts are true. • Conjunction Elimination (∧Elim) P1 ∧…∧ Pi ∧…∧ Pn . . . Pi

  4. Conjunction introduction • If some sentences are true, then the conjunction of these sentences is true. • Conjunction Introduction (∧Intro): P1  Pn . . . P1∧…∧Pn Here ‘’ is used to refer to the sequence of sentences indicated by the horizontal ellipsis.

  5. Conjunction introduction and parentheses • If you apply ∧Intro to complex sentences, remember our rules about when it’s ok to omit parentheses in sentences with multiple connectives. • Correct: 1. A ∨ B 2. C 3. (A ∨ B)∧ C ∧Intro: 1,2 • Incorrect: 1. 1. A ∨ B 2. C 3. A ∨ B∧ C ∧Intro: 1,2

  6. Disjunction introduction • If a sentence is true, then we can disjoin it with any other sentences and the result will be true. • Disjunction Introduction (∨Intro): Pi . . . P1 ∨ … ∨ Pi ∨ … ∨ Pn

  7. Proof By Cases: A prelude to Disjunction Elimination • Proof By Cases: If at least one of a collection of claims is true, and if each of the claims implies the conclusion C, then C must be true. • Examples: • Our next president will be more conservative than Obama, since s/he will either be Clinton or Christie.But Clinton is more conservative than Obama, and so is Christie. • The murderer was either Professor Plum or Ms. Scarlett. Professor Plum wanted revenge. And so did Ms Scarlett. So the murderer wanted revenge.

  8. Disjunction elimination P1∨…∨Pn . . . P1 . . . S  Pn . . . S . . . S

  9. i>clicker question • Suppose you have the disjunction P∨Q∨R. You show S follows from P and from Q, but it does not follow from R. Now suppose that U follows from R. Which of the following can you conclude? • S, using disjunction elimination. • S∧R, using conjunction introduction and disjunction elimination. • S∨R, using disjunction introduction and disjunction elimination. • None of the above.

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