section_3.30_answers - x Dodec ( x ) x ( Dodec ( x ) ((...

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PHIL12A Section answers, 30 March 2011 Julian Jonker 1 How much do you know? 1. Translate the following sentences (these are the four ‘Aristotelian forms’). For each sentence, come up with a translation that uses the quantifier and one that uses the quantifier: (a) All cubes are small. x ( Cube ( x ) Small ( x )) ¬∃ x ( Cube ( x ) ∧ ¬ Small ( x )) (b) Some cubes are small. x ( Cube ( x ) Small ( x )) ¬∀ x ( Cube ( x ) → ¬ Small ( x )) (c) No cubes are small. ¬∃ x ( Cube ( x ) Small ( x )) x ( Cube ( x ) → ¬ Small ( x )) (d) Some cubes are not small. x ( Cube ( x ) ∧ ¬ Small ( x )) ¬∀ x ( Cube ( x ) Small ( x )) 2. Given any sentence starting with , how can you arrive at an equivalent sentence in which the symbol has been replaced with the symbol? And if you have a sentence starting with , how could you arrive at an equivalent sentence in which the symbol has been replaced with the symbol? Let P ( x ) be some well formed formula with x free. Then: xP ( x ) ¬∃ x ¬ P ( x ) and xP ( x ) ¬∀ x ¬ P ( x ) 1
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2 Slightly harder. .. 1. Translation with multiple quantifiers: (a) There are some dodecahedra that are small, and some that are large, but none are medium.
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Unformatted text preview: x Dodec ( x ) x ( Dodec ( x ) (( Small ( x ) Large ( x )) Medium ( x )) (b) Every large cube has a small tetrahedron on its left. x (( Cube ( x ) Large ( x )) y ( Tet ( y ) Small ( y ) LeftOf ( y,x )) (c) If there is a large tetrahedron, then there are no cubes behind it. x ( Tet ( x ) Large ( x )) x (( Tet ( x ) Large ( x )) y ( Cube ( y ) BackOf ( y,x )) (d) There is more than one large cube. x y ( Large ( x ) Cube ( x ) Large ( y ) Cube ( y ) x 6 = y ) (e) There is exactly one large cube. x y ( Large ( x ) Cube ( x ) (( Large ( y ) Cube ( y )) y = x )) (f) There are exactly two large cubes. x y z ( Large ( x ) Cube ( x ) Large ( y ) Cube ( y ) x 6 = y [( Large ( z ) Cube ( z )) ( z = x z = y )]) 2...
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section_3.30_answers - x Dodec ( x ) x ( Dodec ( x ) ((...

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