[7.2.4.] Existential Instantiation (EI).

 

This implicational rule allows you to move from an existential quantification to an instance of that quantification (hence the name “existential instantiation”). For example:

 

1. ($x)Dx         Something is a duck.

 

We can use the rule EI to eliminate the quantifier from that premise:

 

2. Dy               y is a duck.

 

Here, “y” is a quasivariable. It refers to some individual, but it doesn’t say which individual (Daffy, Donald, the AFLEC duck, etc.)

 

There are two restrictions on the use of EI – you must learn these in order to use this rule correctly:

 

 

EI RESTRICTION #1

 

The variable in the existential quantification must be replaced with a quasivariable; it cannot be replaced with an individual constant.

 

So the following would be invalid:

 

1. ($x)Px                      p

2. Pd                            1 EI

 

The fact that that there is a philosopher doesn’t not imply that any particular thing is a philosopher (here “d” stands for Dumbo the elephant; so this is clearly invalid).

 

 

A proper application of EI is as follows:

 

1. ($x)Px                      p

2. Px                            1 EI

 

 

EI RESTRICTION #2

 

The quasivariable must not appear as such in any line already in the proof. It is fine if the letter you’re using already appears in the proof as a variable bound to a quantifier. But it cannot have already appeared as a quasivariable.

 

The following “proof” would violate this second restriction... (Fx = x is a frog)

 

1. ($x)Dx · ($x)Fx             p

2. ($x)Dx                           1 Simp

3. ($x)Fx                            1 Simp

4. Dy                                 2 EI

 

Up to here, each move has been valid. But if we continue as follows...

 

5. Fy                                  3 EI

 

we’ve done something illegitimate. This is because “y” on line 5 stands for the same (unknown) individual as “y” stands for on line 4. Line 4 asserts that y is a duck; line 5 asserts that y is a frog. Clearly, from the fact that something is a duck and something is a frog (which is asserted in line 1), it does not follow that there is one and the same thing, y, that is both a duck and a frog.

 

So when you apply EI, you must choose a letter for your quasivariable that has not appeared as a quasivariable in the proof up to that point. So we could use either “x” or “z”...

 

5. Fz                                  3 EI (this is OK because “z” hasn’t yet appeared in the proof)

 

or

 

5. Fx                                  3 EI      (this is OK because “x” is a bound variable in 1 and 2)

 

 

 

Exercise 9-2 (pp.212-213)

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

·         the textbook’s answer to #6 mistakenly omits one of the errors you need to identify

 

 

[7.3.] “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; it must be replaced with a quasivariable.

 

·         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. Ag                            1 EI      [So, Lady Gaga 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 quasivariable you are introducing cannot occur as a quasivariable on any other line (e.g., if you are introducing the quasivariable “x”, the letter “x” cannot be used as a quasivariable earlier in the proof—but it can have been used as a bound individual variable).

 

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

 

·         When using UG, the quasivariable you are binding cannot occur 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 p.217: 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.218-19.

 

Exercise 9-3 (pp.219-20):

·         [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 Tuesday March 13. For next time (two weeks from today—the first Tuesday after spring break):

·         complete exercises  9-2 and 9-3

·         read ch.9:6 (pp.220-224).