We have already covered two restrictions of Existential Instantiation:

 

EI RESTRICTION #1

 

The variable in the existential quantification must be replaced with a quasivariable.

 

EI RESTRICTION #2

 

The quasivariable must not appear free (unbound) in any line already in the proof.

 

We now need to learn three restrictions on Universal Generalization (UG)...

 

 

[8.2.5.] Restrictions on Universal Generalization.

 

As we have seen, UG allows you to move from an instance of a universal quantification back to the quantification itself. Here is an example from last time:

 

Ex = x is an elephant

Mx = x is a mammal

Ax = x is an animal

 

1. (x)(Ex É Mx)                                    p

2. (x)(Mx É Ax)                       p          /\ (x)(Ex É Ax)

 

Our strategy will be:

·         use UI to remove the quantifiers from lines 1 and 2

·         apply the rule HS

·         use UG to introduce the quantifier back into the proof, thus arriving at the desired conclusion

 

This strategy may suggest to you that the following is an acceptable way to proceed:

 

3. Ed É Md                         1 UI

4. Md É Ad                        2 UI   [lines 3 & 4 use the individual constant “d” to instantiate lines 1 and 2]

5. Ed É Ad                         3, 4 HS

6. (x)(Ex É Ax)                  5 UG    invalid!!!

 

The move from 5 to 6 is invalid. We cannot use our new rule, UG, to move from a premise containing an individual constant, like “d”, to a quantification.

 

This is because of an important restriction on the use of UG:

 

 

UG RESTRICTION #1

 

1.      When using UG, the expression that is replaced by a bound variable must be a quasivariable

 

If we did not have this restriction, then the following argument would be valid: (s = Socrates; Px = x is a philosopher):

 

1.       Ps

2.       (x)Px

 

But the fact that Socrates is a philosopher does not imply that every individual thing that there is is a philosopher.

 

This means that in our initial use of UI to generate lines 3 and 4, we must replace the bound “x” in lines 1 and 2 with a quasivariable:

 

3. Ex É Mx                         1 UI

4. Mx É Ax                                    2 UI

 

We can then apply HS:

 

5. Ex É Ax                         3, 4 HS

 

Before we complete this proof, we need to look at the second restriction on UG:

 

 

UG RESTRICTION #2

 

2.      When using UG, the quasivariable that you are binding must not appear free in a line that is justified by EI.

 

This constraint ensures that any unknown appearing in a premise to which you apply UG will stand for a truly arbitrarily-chosen individual.

 

If a quasivariable was introduced into the proof with rule EI, then it was taken as having been entailed by an existential quantification. For example:

 

1. ($x)Fx                p

2. Fx                       1 EI

3. (x)Fx                  2 UG    invalid!!!

 

We can validly infer that x is F, based solely on the premise ($x)Fx, because x is a quasivariable we are allowing to stand for whatever individual is referred to by ($x)Fx.

 

But it would be illegitimate to move from the premise that x is F (where “x” is a quasivariable that was introduced by EI) to the premise that for all x, x is F. Because it was introduced by EI, “x” refers to one specific individual (although we do not know exactly which one), and the fact that x is F does not imply that for all members of the domain of discourse are F. That is, the fact that some indeterminately referenced individual is F does not imply that everything is F.

 

On the other hand, were “x” a quasivariable introduced by UI, then it would not stand for a specific individual, but for an arbitrary individual. That is, if it were introduced by UI, then any other individual could have been introduced instead; the choice of that specific (unknown) individual would have been arbitrary. So the use of UG in this proof is legitimate:

 

1. (x)Fx                  p

2. Fx                       1 UI

3. (x)Fx                  2 UG    valid

 

 

 

UG RESTRICTION #3

 

This restriction applies only to proofs with assumed premises (i.e., proofs using CP or IP):

 

3.      When using UG, the quasivariable that you are binding must not appear free in an assumed premise.

 

Your textbook explains this constraint as follows: “In making an assumption with a quasivariable, the variable does not name an arbitrary individual. Rather, it names an individual assumed to have a particular property. So we cannot bind that variable with UG so long as we are relying upon that assumption.” (206)

 

So the following is invalid (204):

 

1. ~(x)Fx                      p

2. Fx                             AP

3. (x)Fx                                    2 UG    invalid!!!

 

So, combining restrictions 2 and 3, we can say: the quasivariable that you bind with the use of UG cannot have been introduced into the proof by EI or in an assumed premise (AP).

 

 

 

Exercise 9-2 (pp.209-210)

·         do all six problems-- note which steps are invalid and explain why each is; we’ll go through ALL next time

 

Stopping point for Wednesday March 26. For next time, complete exercise 9-2 and read ch.9:4-5 (pp.210-15).