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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

Propositional Logic

9) Or

Copyright 2008, Scott Gray

terminology reminder
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

exclusive inclusive or
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

or symbolization guidance
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

or symbolization guidance cont
OR Symbolization Guidance, cont.
  • Symbolization of exclusive disjunctions:
  • (A v B) & ~(A & B)A ↔ ~B~A ↔ B

Copyright 2008, Scott Gray

wedge is associative
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

wedge in
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

wedge in cont
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

wedge in example
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

wedge out
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

wedge out example proof of commutivity
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

wedge out strategy
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

assignments
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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