[7.5.] “Quantifier Negation (QN).” (Ch.9:6, pp.220ff.)

 

This is the fifth and final rule for dealing with quantifiers. Unlike the first four quantifier rules, this one is an equivalence rule, so it can be applied either to an entire line or just to a part of a line. It has four forms:

 

1.      (u)(. . . u . . .) :: ~($u)~(. . . u . . .)

 

This form allows you to replace a universal quantifier with an existential quantifier embedded between two tildes (and vice versa), e.g.,

 

1. (x)(Fx É Gx)                  p / \ ~($x)(Fx · ~Gx)

2. ~($x)~(Fx É Gx)                        1 QN                           

3. ~($x)~(~Fx Ú Gx)          2 Impl

4. ~($x)(~~Fx · ~Gx)         3 DeM

5. ~($x)(Fx · ~Gx)             4 DN

 

In English: the fact that every x that is F is also G implies that it is false that there is something that is F but not G. And because QN is an equivalence rule, it works in “the opposite direction,” as well:

 

1. ~($x)~(Fx É Gx)                        p

2. (x)(Fx É Gx)                  1 QN

 

 

2.      ($u)(. . . u . . .) :: ~(u)~(. . . u . . .)

 

This form allows you to replace an existential quantifier with a universal quantifier embedded between two tildes (and vice versa), e.g.,

 

1. ($x)(Fx · Gx)                 p /  ~(x)(Fx É ~Gx)

2. ~(x)~(Fx · Gx)               1 QN

3. ~(x)(~Fx Ú ~Gx)                        2 DeM

4. ~(x)(Fx É ~Gx)              3 Impl

 

In English: the fact that there is an x that is both F and G implies that it is false that everything that is F is not-G. And because QN is an equivalence rule, it works in “the opposite direction,” as well:

 

1. ~(x)~(Fx · Gx)               p

2. ($x)(Fx · Gx)                 1 QN

 

 

3.      (u)~(. . . u . . .) :: ~($u)(. . . u . . .)

 

This form allows you to move a tilde that immediately follows a universal quantifier to the front of that quantifier, thereby changing it from universal to existential (and vice versa); e.g.

 

 

1. (x)~(Fx · Gx)                 p                                  1. ~($x)(Fx · Gx)                     p

2. ~($x)(Fx · Gx)               1 QN                            2. (x)~(Fx · Gx)                       1 QN

 

In English: the fact that for all x, it is false that x is both F and G implies that there is no such thing as an x that is both F and G (and vice versa).

 

4.      ($u)~(. . . u . . .) :: ~(u)(. . . u . . .)

 

This form allows you to move a tilde that immediately follows an existential quantifier to the front of that quantifier, thereby changing it from existential to universal (and vice versa); e.g.

 

1. ($x)~(Fx · Gx)               p                                  1. ~(x)(Fx · Gx)                       p

2. ~(x)(Fx · Gx)                 1 QN                            2. ($x)~(Fx · Gx)                     1 QN   

 

In English: the fact that there is an x that is not both F and G implies that it is false that all x’s are F and G (and vice versa).

 

 

[7.5.1.] Identifying Incorrect Uses of QN.

 

We will go through exercise 9-4 (p.223-24) in class: this exercise asks you to identify incorrect uses of QN and to give a premise which could be validly inferred using that rule.

 

Answers to ex. 9-4:

 

1.      incorrect; ~($x)Fx

2.      correct

3.      incorrect; ~($x)(~Fx Ú Gx)

4.      incorrect; (y)(Ry · ~Ky)

5.      incorrect; (y)~(~Ry · Ky)

6.      correct

7.      correct

8.      correct -- notice the typos in textbook; the argument should read:

 

(8) 1.     (y)~[(Fy) É ($z)(Gz · Hz)]

2.        ~($y)[Fy É ($z)(Gz · Hz)]

 

9.      incorrect; ~($y)[Fy É ($z)(Gz · Hz)]

10.  incorrect; ~($x)(Fx É Gx)

 

 

[7.5.2.] Proofs Using QN.

 

Here’s a proof that contains multiple uses of QN:

 

1. ~(x)(Ax Ú Bx)                                  p

2. ~($x)~(Cx É Ax)                             p

3. ($x)~Cx É ($y)~Fy                         p          /\~(y)Fy

4. ($x)~(Ax Ú Bx)                                1 QN

5. (x)(Cx É Ax)                                               2 QN

6. ~(Ax Ú Bx)                                      4 EI

7. Cx É Ax                                          5 UI

8. ~Ax · ~Bx                                       6 DeM

9. ~Ax                                                 8 Simp

10. ~Cx                                                            7, 9 MT

11. ($x)~Cx                                         10 EG

12. ($y)~Fy                                          3, 11 MP

13. ~(y)Fy                                            12 QN

 

 

Exercise 9-5 (pp.224-25)

·         do #1-15 (omit 16-20, unless you’re up for a real challenge)

 

 

Stopping point for Tuesday March 27. For next time, complete ex.9-5. No new reading. We will go through all of 9-5 on Thursday and then review for your third test, which is one week from today. The study guide for that test is now on the class website.