[8.3.] “Mastering the Four Quantifier Rules.” (Ch.9:5)

 

This section of the textbook revisits the restrictions on the quantifier rules.

 

Restrictions involving individual constants:

 

·         When using EI, the bound variable you are replacing cannot be replaced with an individual constant.

 

·         When using UG, you cannot replace an individual constant with a bound variable.

 

As your textbook points out, even beginners are unlikely to violate these restrictions, since doing so would yield inferences that are pretty obviously invalid…

 

A violation of the constant-constraint on EI:

 

1. ($x)Ax                      p          [Something is an axe murderer.]

2. Aa                            1 EI      [So, Kofi Annan is an axe murderer.]

 

A violation on the constant-constraint on UG:

 

            1. Vh                            p          [The Hope Diamond is valuable.]

            2. (x)Vx                        1 UG    [So, everything is valuable.]

 

 

Restrictions that require checking previous lines:

 

·         When using EI, the variable you are introducing cannot occur free on any other line.

 

·         When using UG, the variable you are binding cannot occur free in a line justified by EI.

 

·         When using UG, the variable you are binding cannot occur free in an undischarged assumed premise.

 

Mistakes with the second restriction on EI are especially common among beginners. To help avoid them, follow this strategy: always apply EI as early in your proof as possible; when you have a choice among which rules to apply first in your proof, apply EI first.

 

An example from pp.213-14: if you were to use UI before EI, you would get stuck:

 

1. (x)(Ax É Bx)                        p

2. ($x)Ax                      p          / \ ($x)Bx

3. Ax É Bx                   1 UI                             [here “x” is a quasivariable]

 

At this point, you cannot apply EI to get “Ax” because “x” appears free in line 3. This will not happen if you apply EI before UI:

 

1. (x)(Ax É Bx)                        p

2. ($x)Ax                      p          / \ ($x)Bx

3. Ax                            2 EI

4. Ax É Bx                   1 UI

5. Bx                            3, 4 MP

6. ($x)Bx                      5 EG

 

Examine the Walk-Through and Examples on pp.214-15.

 

Exercise 9-3 (p.216):

·         [we will do a few evens in class]

·         do all of these; we will go through at least the odds in class next time

 

 

Stopping point for Friday March 28. For next time, complete ex. 9-3, and read ch.9:6