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PHIL12A Section answers, 20 April 2011 Julian Jonker 1 From last time... Compare especially (a), (d) and (e) below. 1. (Ex 11.19) (c) No cube with nothing to its left is between two cubes. x((Cube(x) ∧ ¬∃ yLeftOf(y,x)) → ¬∃ z wBetween(x,z,w)) (d) The only large cubes are b and c . Large(b) Cube(b) Large(c) Cube(c) ∧ ∀ x((Large(x) Cube(x)) (x = b x = c)) (e) At most b and c are large cubes. NB: last time I omitted the following important tip printed in the textbook. ‘There is a significant difference between this sentence and the previous one. This one does not imply that b and c are large cubes, while the previous one does. x((Large(x) Cube(x)) (x = b x = c)) 2 FOL translation exercises 1. The following sentence is ambiguous. I show you how to translate it twice, showing how the English supports two different FOL translations. Every medium dodecahedron is adjacent to a medium cube. There are two ways to translate this sentence. (Note how I use constructions like ‘for all’ and ’there is some’ in order to make explicit the role of quantifiers in the sentence.) (i) There is some medium cube such that all the medium dodecahedra are next to it. (ii) For every medium dodecahedron there is some medium cube that it is next to. 1

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Here are the FOL translations: (i) x[Cube(x) Medium(x) ∧ ∀ y((Dodec(y) Medium(y)) Adjoins(x,y))] (ii) x[(Dodec(x) Medium(x)) → ∃ y(Cube(y)
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