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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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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

- 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

- 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.

- Symbolization of exclusive disjunctions:
- (A v B) & ~(A & B)A ↔ ~B~A ↔ B

Copyright 2008, Scott Gray

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

- 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.

- 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

(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

- 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

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

- 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

- 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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