Propositional Logic

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# Propositional Logic - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Propositional Logic' - geraldine

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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
• 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

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

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

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

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

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.

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

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

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

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