[9.5.6.] Strategy. (Ch.10:8)

 

A brief review of the new restrictions:

 

RESTRICTION A: With regard to UI, you cannot use two different constants, or two different quasivariables, to replace multiple instances of the same bound variable:

 

1. (x)(y)(Fxy É Cyx)                 p

2. Fab                                       p

3. (y)(Fay É Cyb)                     1 UI     invalid

 

 

RESTRICTION B: When replacing a bound variable with a quasivariable by way of UI, the quasivariable you introduce must be a genuine quasivariable and therefore free, i.e., it cannot be bound to any remaining quantifiers.

 

1.       (x)(y)(Fxy É Cyx)               p

2.       (y)(Fyy É Cyy)                   1 UI     invalid

 

 

RESTRICTION C: You must bind every occurrence of a given quasivariable or constant when applying UG. In other words, when you apply UG to bind a constant or quasivariable to a quantifier, you must bind every occurrence of that constant or quasivariable in the sentence to which you are applying UG.

 

…

7. Fzw É Pzw                           4, 6 HS

8. (y)(Fzy É Pzw)                     7 UG    invalid

 

 

RESTRICTION D: As with UI, when you use EI to remove an existential quantifier, you must replace every occurrence of the variable you are freeing with the same quasivariable.

 

1. ($x)($y)(Fxy · Cyx)              p

2. ($y)(Fzy · Cyw)                    1 EI      invalid

 

 

RESTRICTION E: When you use EG to replace an individual constant or quasivariable with a bound variable, there must be no additional occurrences of the variable to be bound contained in the expression to which you are applying EG.

 

…

3. Lxy                           2 EI

4. ($x)Lxx                    3 EG     invalid — there is an “x” in 3, so you cannot replace the quasivariable “y” in 3 with a bound “x.”

 

 

**

 

 

Chapter 10:8 contains the following strategic tips.

 

1.       Some premises will contain both a universal quantifier and an existential quantifier, with the universal quantifier on the outside. In such cases, you are forced to remove the universal quantifier from that premise before removing the existential quantifier. Your strategy should still be to remove the existential quantifier at the earliest possible opportunity. For example,

 

1. (x)($y)~Gxy              p

2. ($y)~Gxy                  1 UI

3. ~Gxy                                    2 EI

 

Notice that you cannot move from (2) to the following...

 

3. ~Gxx                                    2 EI      invalid

 

...because “x” is already free in (2).

 

 

2.       Sometimes, in order to complete a proof, you must use new letters for quasivariables when applying EI and UI. For example (from p.244):

 

1. ($x)(y)Fxy                            p

2. (y)(x)(Fyx É Gxy)                 p

3. (y)Fxy                                   1 EI

4. Fxy                                       3 UI

5. (x)(Fxx É Gxx)                     2 UI     invalid

 

The move from 2 to 5 is invalid, since UI requires that you replace a bound variable with either an individual constant or a (genuine) quasivariable. In 5, the “x” that is introduced to replace the “y” in line 2 is bound to the universal quantifier, so it is not really a quasivariable. We can avoid this problem by using letters other than “x” and “y” for the quasivariables introduced at lines 3 and 4:

 

1. ($x)(y)Fxy                            p

2. (y)(x)(Fyx É Gxy)                 p

3. (y)Fwy                                  1 EI

4. Fwz                                      3 UI

5. (x)(Fwx É Gxw)                   2 UI

6. Fwz É Gzw                           5 UI

 

 

3.       Some proofs can be solved only with IP (Indirect Proof).

 

An example of such a proof, from p.245:

 

1. (x)($y)(Fx · Gy)        p  /\ (x)Fx

 

One obvious way to proceed is simply to remove the quantifiers with UI and EI, as we have done in the past:

 

2. ($y)(Fx · Gy)            1 UI

3. Fx · Gy                     2 EI  [notice that “x” is free in this line, which is justified by EI]

4. Fx                             3 Simp

5. (x)Fx                                    4 UG    invalid

 

The move from 4 to 5 is invalid, because of the restriction on UG that says that UG cannot be used to bind a quasivariable that appears free in a line justified by EI. Even though the quasivariable in question was not introduced into the proof by EI, it still appears in a line justified by EI, and so cannot be bound using UG.

 

The way around this impasse is to open an indirect proof at the very start:

 

1. (x)($y)(Fx · Gy)        p  /\ (x)Fx

2. ~(x)Fx                      AP

3. ($x)~Fx                    2 QN

4. ~Fx                           3 EI

5. ($y)(Fx · Gy)            1 UI

6. Fx · Gy                     5 EI

7. Fx                             6 Simp

8. Fx · ~Fx                   4, 7 Conj

9. (x)Fx                                    2-8 IP

 

 

 

Exercise 10-10 (pp.245-46)

·         complete all of these for next time

 

OPTIONAL: Exercise 10-11 (pp.246-47)

·         some of these are very difficult!

·         we won’t necessarily cover these in class, but I will email everyone the answers after class

 

 

Stopping point for Friday April 18. For next time, complete ex.10-10 [optional: 10-11] and then read ch.13:1 (pp.283-88).