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Propositional Logic

Propositional Logic. 9) Or. Terminology Reminder. A statement containing statements joined by the connective “or” is called a disjunction . The statements separated by the “or” are called disjuncts . To symbolize “or” we use the wedge : v. Exclusive & Inclusive OR.

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Propositional Logic

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  1. Propositional Logic 9) Or Copyright 2008, Scott Gray

  2. Terminology Reminder • A statement containing statements joined by the connective “or” is called a disjunction. • The statements separated by the “or” are called disjuncts. • To symbolize “or” we use the wedge: v Copyright 2008, Scott Gray

  3. Exclusive & Inclusive OR • English or can carry a “both and” sense; this is the inclusive use of or. • The English or can also mean “exactly one of” sense; this is the exclusive use of or. • You must determine what is meant • The wedge operation is inclusive Copyright 2008, Scott Gray

  4. OR Symbolization Guidance • Most cases of exclusive “or” are commands; example: eat your food or go to bed • Descriptive use of “or” is generally inclusive: Bear is a dog or Coda has fleas • Translate a disjunction containing the phrase “but not both” as exclusive Copyright 2008, Scott Gray

  5. OR Symbolization Guidance, cont. • Symbolization of exclusive disjunctions: • (A v B) & ~(A & B)A ↔ ~B~A ↔ B Copyright 2008, Scott Gray

  6. Wedge is Associative • Consider: I will eat either a pickle or kimchi or pickled veggies. • How do you symbolize this? • P v K v V(P v K) v VP v (K v V) • The first is problematic when doing wedge inference (we’ll get to this later) • The second two are equivalent Copyright 2008, Scott Gray

  7. Wedge In • From a statement derive a disjunction which has that statement as one disjunct and any other statement as the other disjunct. • Wedge in is a “choice” rule. • The wedge in line depends on the disjunct with the existing statement Copyright 2008, Scott Gray

  8. Wedge In, cont. • Is this rule too free? Can the second disjunct really be anything? • Part of the difficulty some people have is that this is a pattern of reasoning which isn’t widely used: going from more specificity to less. • However, it is still valid. Copyright 2008, Scott Gray

  9. Wedge In Example (F v A) → G ∴ F → G 1 (1) (F v A) → G A 2 (2) F PA 2 (3) F v A 2 vI 1,2 (4) G 1,3 →O 1 (5) F → G 2-4 →I Copyright 2008, Scott Gray

  10. Wedge Out • If you have A v B, A → C, and B → C, derive C • The justification entry has 3 line numbers, those of the above items • The wedge in line has the same dependencies as the three above items Copyright 2008, Scott Gray

  11. Wedge Out Example:Proof of Commutivity A v B ∴ B v A 1 (1) A v B A 2 (2) A PA 2 (3) B v A 2 vI (4) A → (B v A) 2-3 →I 5 (5) B PA 5 (6) B v A 5 vI (7) B → (B v A) 5-6 →I 1 (8) B v A 1,4,7 vO Copyright 2008, Scott Gray

  12. Wedge Out Strategy • When you have an assumption that is a disjunction, say A v B, and goal line, say C • Try to PA A and derive C, then use arrow in to get A → C • Try to PA B and derive C, then use arrow in to get B → C • Use wedge out to get C Copyright 2008, Scott Gray

  13. Assignments • Read Chapter 7 • Do all of the exercises (you may skip the “challenge” ones) • Be sure to ask me questions if you don’t understand something or can’t solve a problem Copyright 2008, Scott Gray

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