Transcript 9) Or

Propositional Logic
9) Or
Copyright 2008, Scott Gray
1
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
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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
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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
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OR Symbolization Guidance,
cont.
□ Symbolization of exclusive disjunctions:
□ (A v B) & ~(A & B)
A ↔ ~B
~A ↔ B
Copyright 2008, Scott Gray
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Wedge is Associative
□ Consider: I will eat either a pickle or kimchi
or pickled veggies.
□ How do you symbolize this?
□ PvKvV
(P v K) v V
P 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
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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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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.
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Wedge In Example
(F v A) → G ∴ F → G
1
(1) (F v A) → G
2
(2) F
2
(3) F v A
1,2
(4) G
1
(5) F → G
Copyright 2008, Scott Gray
A
PA
2 vI
1,3 →O
2-4 →I
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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
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Wedge Out Example:
Proof of Commutivity
AvB∴BvA
1
(1)
2
(2)
2
(3)
(4)
5
(5)
5
(6)
(7)
1
(8)
AvB
A
A
PA
BvA
2 vI
A → (B v A)
2-3 →I
B
PA
BvA
5 vI
B → (B v A) 5-6 →I
BvA
1,4,7 vO
Copyright 2008, Scott Gray
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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
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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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