Introductory Logic |
Test #2
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March 20, 2006
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Name: ________________________ |
R. Hammack
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Score: _________
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(a) | Your test grade will be dropped only if it is your lowest grade or you have an excused absence. |
Y ⊃ (L ∨ E) | Y = "Your test grade will be dropped." L = "It is your lowest grade." E = "You have an excused absence." |
(b) | If it does not rain soon, then the risk of forest fires will be great and cuation will be necessary. |
~R ⊃ (F • C) | R = "It rains soon." F = "The risk of forest fires will be great." C = "Cuation will be necessary." |
(c) | Oregon does not have a sales tax, but Virginia does. |
~O • V | O = "Oregon has a sales tax." V = "Virginia has a sales tax." |
(d) | If affirmative action programs are dropped, then if new programs are not created, then minority applicants will suffer. |
A ⊃ (~N ⊃ M) | A = "Affirmative action programs are dropped." N = "New programs are created." M = "Minority applicants will suffer." |
(e) | Yosemite and Kings Canyon restrict vehicle traffic unless Bryce and Zion do not. |
~(~B • ~Z) ⊃ (Y • K) OR (B ∨ Z) ⊃ (Y • K) OR (Y • K) ∨ (~B • ~Z) |
Y = "Yosemite restricts vehicle traffic" K = "Kings Canyonrestricts vehicle traffic" B = "Bryce restricts vehicle traffic" Z = "Zion restricts vehicle traffic" |
(a) |
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(b) |
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3. (24 points) Determine if the following pairs of statements are logically
equivalent, contradictory, consistent, or inconsistent.
(a) |
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(b) |
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(c) |
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(a) |
P
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⊃
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(
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R
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≡
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A
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)
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/
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A
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⊃
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(
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W
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•
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~
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R
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)
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/
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R
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≡
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(
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W
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∨
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K
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)
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/
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P
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•
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U
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/
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U
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⊃
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K
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T
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T
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T
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T
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T
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T
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T
|
|
?
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F
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F
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T
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T
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T
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?
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T
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T
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T
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T
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T
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T
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T
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T
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||||||||||
(b) |
M
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∨
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B
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/
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~
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B
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/
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M
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•
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A
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/
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B
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⊃
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M
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/
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A
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∨
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B
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T
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T
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F
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T
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F
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T
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T
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T
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F
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T
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T
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T
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T
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F
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|||||
(a) |
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M
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⊃
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(
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C
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∨
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D
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)
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/
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~
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X
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∨
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M
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/
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(
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D
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∨
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C
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)
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⊃
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X
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//
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M
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≡
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X
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T
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T
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F
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F
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F
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T
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F
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T
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T
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F
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F
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F
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T
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F
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T
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F
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F
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||||||
F
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T
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F
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T
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T
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F
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T
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T
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F
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F
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T
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There are two ways for the conclusion to be false, and a line is filled in
for each way. Notice that there is a contradiction on both lines, so the argument
is VALID
(b) |
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M
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⊃
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(
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L
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⊃
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K
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)
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/
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P
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⊃
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M
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/
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~
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S
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/
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S
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∨
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L
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//
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K
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⊃
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P
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?
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T
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T
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T
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T
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F
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T
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?
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T
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F
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F
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T
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T
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T
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F
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F
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||||||
Notice that when the table is filled out, the value of M cannot be determined. However, if you set M=T (or F), all the premises are true and the conclusion is false. Therefore the argument is INVALID.