PHIL12A Section answers, 20 April 2011 - Philosophy

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(b) (Ex 12.5)1 ∀y[Cube(y) ∨ Dodec(y)]2 ∀x[Cube(x) → Large(x)]3 ∃x¬Large(x)4 ∃x[Dodec(x) ∧ Small(x)]This argument is not valid. Suppose that there is a medium dodecahedron only. Then the premises are true(check this!) but the conclusion is false.(c) (Ex 12.8)1 ∀x[Cube(x) ∨ (Tet(x) ∧ Small(x))]2 ∃x[Large(x) ∧ BackOf(x,c)]3 ∃x[FrontOf(c,x) ∧ Cube(x)]Proof: By the second premise there is something which is large and behind c. Let’s call this object e. ByExistential Instantiation we get that e is large and that it is behind c. But since BackOf and FrontOfare inverses, we know that c is in front of e.Now by the first premise everything is either a cube or a small tetrahedron, so by Universal Instantiation,e is either a cube or a small tetrahedron. But we know that e is large – so it must be cube. So c is in frontof e and e is a cube. But now we use Existential Generalization in order to obtain our conclusion: thatthere is something that c is in front of and that is a cube.(d) (Ex 12.9)1 ∀x[(Cube(x) ∧ Large(x)) ∨ (Tet(x) ∧ Small(x))]2 ∀x[Tet(x) → BackOf(x,c)]3 ∀x[Small(x) → BackOf(x,c)]Proof: We’re going to reason about an arbitrary object – it could be any one. For convenience, we’ll call itj. We’ll show by Conditional Proof that if j is small then it is behind c. But since j is arbitrary, this willbe true of every object.Suppose j is small. By Universal Instantiation on the first premise we get that j is either a large cubeor a small tetrahedron. Since it is small, we know it is a tetrahedron. By Universal Instantiation on thesecond premise we get that if j is a tetrahedron then it is behind c. So, by modus ponens, j is behind4
c. So this shows that if j is small, then it is behind c. (In other words, we have proved Small(j) →BackOf(j,c).But j was arbitrary, so by Universal Generalization, we get that any small object is behind c, which is thedesired conclusion.5
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PHIL12A Section answers, 20 April 2011 - Philosophy

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