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Copyright 2008, Scott Gray1 Propositional Logic 9) Or

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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

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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

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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

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

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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 V P v (K v V) The first is problematic when doing wedge inference (well get to this later) The second two are equivalent

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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

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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 isnt widely used: going from more specificity to less. However, it is still valid.

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Copyright 2008, Scott Gray 9 Wedge In Example (F v A) G F G 1(1) (F v A) G A 2(2)FPA 2(3)F v A2 vI 1,2(4)G1,3 O 1(5)F G2-4 I

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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

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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)APA 2(3)B v A2 vI (4)A (B v A)2-3 I 5(5)BPA 5(6)B v A5 vI (7)B (B v A)5-6 I 1(8)B v A1,4,7 vO

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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

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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 dont understand something or cant solve a problem

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