[7.] Predicate Logic Proofs.

 

[7.1.] “Proving Validity.” [Ch. 9:1]

 

Just as in sentential logic, in predicate logic it is possible to use the proof method to demonstrate that an argument is valid. For example:

 

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

2. (x)(~~Fx É Gx)                                p  /\ ($x)Hx

3. (x)(Fx É Gx)                                                2 DN

4. (x)(~Fx Ú Gx)                                  3 Impl

5. ($x)Hx                                             1, 4 MP

 

This example illustrates that in predicate logic (just as in sentential logic), the ten equivalence rules, including DN and Impl, can be applied to parts of lines.

 

And just as in sentential logic, the eight implicational rules, including MP, can be applied only to entire lines…

 

So, although the following argument is valid...

 

1. (x)(Fx É Gx)

2. (x)Fx                        /\ (x)Gx

 

the following application of the proof method does not show it to be valid:

 

1.      (x)(Fx É Gx)          p

2.      (x)Fx                      p          /\ (x)Gx

3.      (x)Gx                     1, 2 MP

 

This “proof” fails; premise one does not have the form p É  q , so MP cannot be applied to it. (From the point of view of sentential logic, the only sentence form that premise one instantiates is p. So far as MP is concerned, the first premise may as well be an atomic sentence.)

 

In order to prove this sort of argument valid, we need new rules: four implicational rules and one equivalence rule.

 

About the four implicational rules...

·         two of them are for removing quantifiers from lines:

·         Universal Instantiation (UI): removes a universal quantifier;

·         Existential Instantiation (EI): removes an existential quantifier;

·         two of them are for introducing quantifiers into lines:

·         Universal Generalization (UG): adds a universal quantifier;

·         Existential Generalization (EG): adds an existential quantifier.

 

The general strategy we will use in our predicate logic rules is as follows:

·         step one: remove quantifiers from premises;

·         step two: use the 20 original rules (eight implicational rules, ten equivalence rules, CP and IP) to derive new lines;

·         step three: add quantifiers to those derived lines in order to reach the conclusion of the argument.

 

 

[7.2.] “The Four Quantifier Rules.” [9:2]

 

See the inside cover of your textbook for summary statements of these new rules. They are not as easy to state as the original rules, and each of them takes some detailed explanation.

 

[7.2.1.] Universal Instantiation (UI).

 

This rule allows you to move from a universal quantification to an instance of that quantification (hence the name “universal instantiation”).

 

The rule may be stated:

·         “From any universal quantification we may validly infer any instance of it.”[1]

·         “[Universal i]nstantiation is an operation that consists in deleting a quantifier and replacing every variable bound by that quantifier with the same instantial letter.”[2]

 

The general idea behind this rule is: whatever is true of everything is true of any particular thing.

 

For example:                Ex = x is an elephant

Mx = x is a mammal

d = Dumbo

 

1. (x)(Ex É Mx)                                   p

2. Ed                                        p          /\ Md

 

(UI) allows you to derive line 3 (which is just about Dumbo) from line 1:

 

3. Ed É Md                              1 UI

4. Md                                       2, 3 MP

 

After all, line 1 asserts that for any member of the domain of discourse (and as before, we are assuming an unrestricted domain of discourse), if that thing is an elephant, then it is a mammal. It follows from this that if Dumbo is an elephant, Dumbo is a mammal.

 

From the point of view of simply applying UI correctly, it doesn't matter which individual you refer to in the instantiation, and thus it does not matter which individual constant you use. We could have instead inferred Ea É Ma, Eb É Mb or any other instantiation of (x)(Ex É Mx). If it is the case that, for anything, if it is an elephant then it is a mammal, then it is the case that, for any particular thing you choose, if it is an elephant then it is a mammal.

 

 

[7.2.1.1.] Deciding Which Instance to Infer.

 

But now we face an important question of strategy: how do you decide which instance of the universal quantifier to infer?

 

In deciding on an instance, you should keep in mind the conclusion you are trying to reach, as well as what constants are used in other premises. In the above argument about Dumbo (“d”), it would have done you no good to use a constant other than “d”. Deriving “Ea É Ma” from line one would have been a valid move, but it would have been strategically pointless, since it would not have helped you solve the proof.

 

Here is another example that illustrates this:

 

1. (x)(~Hx É Ix)                      p

2. (y)(~Ky Ú ~Iy)                     p

3. ~Hg                                      p          / \ ~Kg

 

The rule UI will allow you to use any constant you like in instantiating the universal quantifications on lines 1 and 2, but strategically it makes sense to use only the constant “g”:

 

4. ~Hg É Ig                              1 UI

5. ~Kg Ú ~Ig                            2 UI

6. Ig                                         3, 4 MP

7. ~~Ig                                     6 DN

8. ~Kg                                      5, 7 DS

 

 

[7.2.1.2.] Two More Things to Note About UI.

 

One: UI is an implicational rule, not an equivalence rule. So the quantification that is replaced must be an entire line in the proof. This means that you can apply this rule to a line only if the entire line is a universal quantification.

·         You cannot apply UI to (x)Fx É (x)Gx, which is a conditional of the form p É q, not a universal quantification with the form (x)(…x…) .

·         Nor can you apply it to (x)Fx Ú (x)Gx, which is disjunction of the form p Ú q.

·         You cannot even apply it to ~(x)(Fx É Gx), which is a negation of the form ~p.

 

Two: When using UI, you can replace the variable with either an individual constant, which stands for some definite individual (this is what we have done in all of the examples given above), or you can replace the variable with a quasivariable:

 

quasivariable (df.) (a.k.a. an unknown): a lower case letter (x, y, z…) that does not work as a variable but instead refers indefinitely to some or other individual, i.e. to some individual the identity of which is unknown. It is like a marker that refers to a specific individual that has not been definitely identified.

 

So in the proof about elephants and mammals, a valid move at line 3 would have been:

 

1. (x)(Ex É Mx)                                   p

2. Ed                                        p          /\ Md

3. Ex É Mx                              1 UI

 

The “x” in line 3 is a quasivariable. It looks like a free variable (an individual variable that is not bound to any quantifier), but it functions in a very different way, referring (indefinitely) to some specific individual in the domain of discourse. Were that “x” an unbound variable, line 3 wouldn’t be a sentence at all—it would be a mere sentence form.

 

Now, even though the move from 1 to 3 in the above proof is valid, it is bad from the view of strategy, for the same reason that using a constant other than “g” would have been bad: it would not have helped you solve the proof. So in this proof, there is no good reason to use UI to introduce a quasi-variable. But in some other proofs, you will have to use UI to introduce a quasi-variable…

 

 

[7.2.2.] Universal Generalization (UG).

 

This rule allows you to move from an instance of a universal quantification back to the quantification itself:

 

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 in a line that is justified by Existential Instantiation (EI).*

 

*We will look at this rule at greater length next time. For now, just know that it’s the rule that allows you to remove an Existential Quantifier, i.e., to move from an existential quantification to one of its substitution instances. For example:

 

1.      ($x)Fx                   p

2.      Fx                          1 EI

 

This constraint ensures that any quasivariable 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 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 that one specific individual is F does not imply that 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 stand for an arbitrarily chosen 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 in an undischarged assumed premise.

 

Your textbook explains this constraint as follows: “When we make 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.” (210)

 

So the following is invalid:

 

1. ~(x)Fx                      p

2. Fx                            AP

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

 

But you can bind that quasivariable with UG once the assumed premise has been discharged. This is because, once the assumed premise is discharged, “we are no longer depending on an assumption about the individual designated by the variable.” (210)

 

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 undischarged assumed premise (AP).

 

 

[7.2.3.] Existential Generalization (EG).

 

This rule may be the easiest of all of the new rules to understand. It is based on the simple idea that if some specific, known individual has a property, then it is true that there is something that has that property. For example,

 

1. Ed                p

2. ($x)Ex         1 EG

 

As with the other rules for adding and removing quantifiers, EG can only be applied to entire lines. So the following argument misapplies this rule:

 

            Mx = x is a mouse

Dx = x is a dog

m = Mickey

p = Pluto

 

1.      Mm · Dp                           p          /\ ($x)Mx · ($x)Dx

2.      ($x)Mx · ($x)Dx               1 EG    invalid!

 

To demonstrate the validity of this argument in a proof, you must to do the following:

 

1. Mm · Dp                             p          /\ ($x)Mx · ($x)Dx

2. Mm                                      1 Simp

3. Dp                                       1 Simp

4. ($x)Mx                                2 EG

5. ($x)Dx                                 3 EG

6. ($x)Mx · ($x)Dx                 4, 5 Conj

 

 

 

Exercise 9-1 (p.208)

·         Complete all problems for next time; we’ll check the odds at the start of class.

·         Remember that you can use all 18 implicational and equivalence rules, as well as CP or IP.

·         There is one more rule for quantifiers that we’ve not yet covered in detail: Existential Instantiation (EI)… you do not need it to solve any of the exercises in 9-1. We will cover it next time.

 

 

Stopping point for Thursday March 8. For next time:

·         complete exercise 9-1;

·         read Ch.9:3 (pp.209-212).