# section_4.27_answers - PHIL12A Section questions 27 April...

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PHIL12A Section questions, 27 April 2011 Julian Jonker Revision problems 1. (Ex 11.39 and 11.40) Translate the following sentences into FOL. (a) Everything is either a cube or a tetrahedron. x(Cube(x) Tet(x)) (b) There are at least three tetrahedra. x y z(Tet(x) Tet(y) Tet(z) x 6 = y y 6 = z x 6 = z) (c) There is a dodecahedron unless there are at least two large objects. ¬ ( x y(x 6 = y Large(x) Large(y))) → ∃ xTet(x) or xDodec(x) ( x y(x 6 = y Large(x) Large(y)) (d) Not every cube is smaller than every tetrahedron. ¬∀ x(Cube(x) → ∀ y(Tet(y) Smaller(x,y))) or: x y(Cube(x) Tet(y) ∧ ¬ Smaller(x,y)) (e) Everything is the same size as something else. x y(x 6 = y SameSize(x,y)) 2. Show by a chain of equivalences that the following pairs of sentences are equivalent: (a) i. ¬∃ x(P(x) Q(x)) ii. x(P(x) → ¬ Q(x)) ¬∃ x(P(x) Q(x)) ⇔ ∀ x ¬ (P(x) Q(x)) De Morgan for quantiﬁers ⇔ ∀ x( ¬ P(x) ∨ ¬ Q(x)) De Morgan for propositional connectives ⇔ ∀ x(P(x) → ¬ Q(x)) wff equivalence for 1

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(b) I made a mistake on this one by leaving out a negation. It should have read: i. ¬¬ ( ¬ A ∨ ¬ B ) ∨ ¬¬ ( C ∧ ¬ D ) ii. ( A B ) → ¬ ( C D ) ¬¬ ( ¬ A ∨ ¬ B ) ∨ ¬¬ ( C ∧ ¬ D ) ⇔ ¬ ( ¬ A
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## This note was uploaded on 09/19/2011 for the course PHILOS 12A taught by Professor Fitelson during the Spring '08 term at Berkeley.

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section_4.27_answers - PHIL12A Section questions 27 April...

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