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I have this one question (Part 7) in exercise 11.20 that I can't seem to get the answer from.

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I tried ∀x∀y ((x ≠ y ∧ Larger(x,y)) → Dodec(x)) and ∀x∀y (Larger(x,y) → Dodec(x)), as well as many other things so far, and none of them have passed through GradeGrinder.

Thanks!

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  • 12 ? Too many. Try with the first couple of the,. What have you tried about 1 and 2 ? Commented Nov 21, 2019 at 16:10
  • And the two transaltions you have written above refer to what ? Commented Nov 21, 2019 at 16:12
  • Sorry I'm just referring to part 7 of this question :Only dodecahedra are larger than everything else. I'm fine with the other parts.
    – NiceOnions
    Commented Nov 21, 2019 at 16:15
  • "There is no x such that for every y, if not x=y, then Larger(x,y) and not Dodec(x)." Commented Nov 21, 2019 at 16:19
  • That didn't work either
    – NiceOnions
    Commented Nov 21, 2019 at 16:22

1 Answer 1

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I tried ∀x∀y ((x ≠ y ∧ Larger(x,y)) → Dodec(x)) and ∀x∀y (Larger(x,y) → Dodec(x)),

Translate it a bit at a time.

  • "Only dodecahdera are larger than everything else."

  • ∀x ("larger than everything else"(x) → Dodec(x))

  • ∀x (∀y (x ≠ y → Larger(x,y)) → Dodec(x))

If you need it in prenex form, use contraposition and duality, so you may apply null quantification.

  • ∀x (~Dodec(x)→~∀y (x ≠ y → Larger(x,y)))
  • ∀x (~Dodec(x)→Ǝy (x ≠ y & ~Larger(x,y)))
  • ∀x Ǝy (~Dodec(x)→(x ≠ y & ~Larger(x,y)))

Check the translation:

  • "If anything is not a dodecahedra, there is something else that it is not larger than."

  • ?= "Only dodecahedra are larger than everything else."

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