[8.5.] “Quantifier Negation (QN).” (Ch.9:6)

 

This is fifth and final rule for dealing with quantifiers. Because it is an equivalence rule it can be applied to parts of a line. It has four forms:

 

1.      (x)(. . . x . . .) :: ~($x)~(. . . x . . .)

 

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

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

 

In English: the fact that every x that is F is also G implies that it is false that there is an x such that it is false that if that x is an F, then it’s a G (and vice versa).

 

2.      ($x)(. . . x . . .) :: ~(x)~(. . . x . . .)

 

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                                  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 both F and G implies that it is false that, for all x, it is not the case that x is both F and G (and vice versa).

 

3.      (x)~(. . . x . . .) :: ~($x)(. . . x . . .)

 

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)                        p

 

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.      ($x)~(. . . x . . .) :: ~(x)(. . . x . . .)

 

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).

 

 

[8.5.1.] Identifying Incorrect Uses of QN.

 

We will go through exercise 9-4 (pp.215-216) 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) -- notice the typos in textbook; the argument should read:

 

(9) 1.    (y)~Fy É ($z)(Gz· Hz)

2.   ~($y)~Fy É ($z)(Gz · Hz)        

 

10. incorrect; ~($x)(Fx É Gx)

 

 

[8.5.2.] 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 (p.221)

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

 

 

Stopping point for Monday March 31. For next time, complete ex.9-5. No new reading. We will go through all of 9-5 on Wednesday, and Friday we will review for your third test, which is one week from today.