I am struggling to figure out a couple questions from my book:
#7 – ∃x [Cube(x) → ∀x Small(x)] #8 – ∃x [Cube(x) → ∀x Cube(x)]
I never came accross this before, how can there be quantifiers for the same variable like that? I felt that was strange. But I was working through it this way:
1. ∃x [Cube(x) → ∀x Small(x)] 2. ∃x [~Cube(x) v ∀xSmall(x)] 3. ∃x ~~[~Cube(x) v ∀xSmall(x)] 4. ∃x ~[Cube(x) v ~∀xSmall(x)] 5. ∃x ~[Cube(x) v ∃x~Small(x)] 6. ∃x ~∃x[Cube(x) v ~Small(x)] 7. ∃x ∀x[Cube(x) v ~Small(x)]
Now that looks very strange to me and I'm thinking I messed up at part 5-6. I'm guessing that won't work, because there is another variable x before the ∃. But I don't know how else to go about this.
For the second part I got
~∀x Cube(x). I don't think that is right either. It would be true if there was anything not an cube in the world, but how can one conclude that there must be no cubes in the world? If they were all cubes, it would also be true.
Could someone help me understand how I should go about this?
Thanks for your help!