Chapter Eight, Tutorial Four

PD+ includes all the rules of SD+ and all the rules of PD. In addition, there is one new rule of replacement: QN. (In case you've forgotten, click here for a pop-up of the older rules of replacement introduced in chapter 5)

Quantifier Negation: the rule QN

You have already noticed that to say something like "not everyone is present" ('~(^x)Px') expresses the same idea as "someone is not present" ('(%x)~Px'). These two sentences of PL are logically equivalent: we have shown (8.2ex V) that from either one, the other can be derived.

In general, moving the tilde from one side of a quantifier to the other and changing the type of quantifier (from universal to existential or vice versa) always provides a logically equivalent sentence. The rule, "quantifier negation" or "QN", codifies this equivalence:

(Quantifier Negation)
~(^x)P (%x)~P
~(%x)P (^x)~P


Any accessible line which includes a tilde and a quantifier immediately following the other is a candidate for QN.

Now, consider the derivation beginning with one premise:

Premise 1 ~(^x)Ax&(%x)~Bx
1 QN 2

From the justification 1 QN, which of the following can be derived on line 2?

  1. (^x)~Ax
  2. (%x)~Ax&(%x)~Bx
  3. ~(^x)Ax&~(^x)Bx
  4. ~(^x)Ax