13. QUANTIFIER  NEGATION

A final equivalence (replacement) rule involves what happens when we have the curl as the main connective in a quantified expression, as in ~(x)Fx or ~(Ex)Fx.   ~(x)Fx expresses the idea that not everyone is "F" (we have at least one individual who is not) and ~(Ex)Fx expresses the idea that there is not anyone who is "F"  (we have no individuals who are).  In the same way, ~(x)(Fx->Gx) tells us that there has to be at least one individual who is not both "F'" and "G" while ~(Ex)(Fx->Gx) tells us that there is no one who is both.

Accordingly, we have the following examples of an equivalence rule we will call quantifier negation (QN).
~(x)Fx ::  (Ex)~Fx     QN
~(x)(Fx->Gx)  ::  (Ex)(Fx & ~Gx)    QN
~(Ex)Fx  ::  (x)~Fx     QN
~(Ex)(Fx & Gx)   ::  (x)(Fx -> ~Gx)    QN

What we are doing is exchanging quantifiers.  In more complicated expressions, such as those that involve relationships, we can talk about driving the curl through.
          ~(x)(Ey)(Fxy -> ~Gyx)  can then become (Ex)(y)(Fxy & Gyx).

red flag To save steps you will not need to go through each move, but you will need to be careful to see that you have changed each quantifier as well as the main connective inside the parenthesis (note too how the example also builds in DN).
EXERCISES (ON YOUR OWN)

Symbolize and provide derivations to show the equivalence for the following pairs of sentences.

1.  Not everyone who studies will pass.  Someone who did not pass had studied.
2.  Unless everyone is ambitious there will be individuals who do not succeed.  If everyone succeeds then everyone was ambitious.

Go to the answer key.