6.1d: Symbolizations in PL with Quantifiers


Universe of Discourse: Cats (and only cats! for this symbolization felines are our only subject)

Mx: x is a mammal, Rx: x is a reptile; Wx: x is wild.

f: Felix the Cat (pretend he's real)
t: Tony the Tiger (of course, he's real, he was on TV)

Keep in mind that the subject under discussion, the "universe of discourse" is just cats. We are not concerned here with anything but cats. Forget about dogs and mice, etc.! This is very restrictive. We'll begin to remove this restriction in 6.2.

Don't forget how to get an answer. Also, there are good keyboard shortcuts: Instead of typing out '(^x)' for the universal quantifier, type 'Vx' (no parentheses needed). This will turn into '(^x)' after you press TAB. Similarly, for the existential quantifier, type '3x'.
  1. All are mammals (i.e., all cats are mammals) .
  2. Some (cats) are wild.
  3. Some (cats) are wild and some cats are not. (hint)
  4. All are wild.
  5. Not all are wild. (hint)
  6. Some are not wild. (hint)

    So, we notice the 5 and 6 say the same thing. Number 5 is most naturally symbolized with main connective '~': it's a "Not" sentence after all! While 6 is a "some" sentence, symbolized with main connective '%'. Still, they are logically equivalent. This will become very important.

  7. Some cats are reptiles.
  8. No cats are reptiles. (hint)
  9. None are reptiles.

    So, 7 is symbolized with a backwards-E. But 8 and 9 are logically equivalent and are just the negation of 7.

  10. Felix is not wild but some cats are wild. (hint)
  11. Neither Felix nor Tony is wild.
  12. Felix is not wild and Tony is not wild.

    Notice that 11 and 12 different grammar in their English, and you might well symbolize them in different ways, but they are logically equivalent. We even have a rule for this equivalence: DM.

  13. Felix and Tony are not both wild. (hint)
  14. Either Felix is not wild or Tony is not wild. (hint)

    Notice that 13 and 14 also are very different grammatically, and you might most naturally be symbolized in different ways. But they too are a logically equivalent pair. Our rule for this equivalence is also DM...but the second version.