LESSON 6: Using Quantifiers

Assignment 6

In the last lesson we worked with predicate variables, but now our job is to work with four new signals that will let us talk about whether we have everyone in a group or just some (at least one).  First off, we are going to use V as a universal quantifier to talk about everyone, and we will use W as what we call an existential quantifier to say there is at least one person.  Let's use the same code as before:
       Fx: x is friendly
        Gx: x greets people
        Hx: x is happy
        a: Al
        b: Bob

FV: everyone is friendly
FNV:  no one is friendly or everyone is unfriendly
    Watch for the difference here:  FVN: not everyone is friendly (equivalent to FNW)
FW:  someone is friendly
FNW: someone is unfriendly

Now let's see what we can do to represent the idea that anyone with one characteristic also will have another.
FGD:  anyone who is friendly greets people (=only those who greet people are friendly)
GFD:  only individuals who are friendly greet people  (=everyone who greets people is friendly)
FGE:  there is someone friendly who is greeting people
GHND:  everyone greeting people is unhappy
GHNE: there is someone greeting people who is unhappy
And we can also talk about several characteristics at once:
FGAHD: Anyone who is both friendly and greets people is happy.
FGHAD:  Anyone who is friendly is both happy and greets people.
    Please note that D is also a universal quantifier and E an existential one.  An important point is that the relationship expressed in FHD can be true even through there is no one like this, but when we assert FGE there is at least one individual about whom it has to be true (why we call this an existential quantifier, since it implies existence in a way that the universal quantifier might not).

We are able to eliminate and introduce quantifiers just as we do with other signals, but we need to be careful not to say something beyound the evidence.  This is why there are some important restrictions.

FV |- Fa  (or any other individual)    V elim
FGD |- FaGaC   D elim
(again, any other name can be used, but it must be used for both parts of the string)
FW |- Fa   W elim
This is something  we can do in a derivation, as long as this is a "new name" that is not otherwise involved in the argument form.
FGE |-  FaGaA  E elim
Here too we need to observe this "new name rule."
Fa |- FV   V int
This is something we do in a derivation if we had the name come from V elim (as well as in one other situation to be discussed later)
FaGaC |-   FGD  D int
Once again, there is a restriction on this, just as there is with V int
Fa |-  FW   W int  (no restrictions)
FaGaA |- FGE    E int  (no restrictions)
And, finally, we have an important equivalence rule with several ways it can be used.  The rule itself is "quantifier negation": QN
FVN = FNW
FWN = FNV
FNVN = FW
FNWN = FV
applied to our signals D and E it gives us these basic instances
FGDN=FGNE
FGEN=FGND


In the first half of this course we are working with PLN, and in the second half, as we make the transition to a more standard notation, we will not have equivalents for D and E, but the rules and restrictions are otherwise the same.

Suggestion: before moving on, review the SLIPS material on variables and signals and then look at someting more about the new name rule.

Derivations use the same patterns we've already seen.  At this point we will have to make more frequent use of indirect derivations (later there is another technique that will prove very efficient).

Example 1.
Symbolize and provide a derivation for this argument.
Everyone who is friendly is happy and greets people.  Al does not greet people.  Therefore, Al is unfriendly.
FHGAD, GNa |- FNa
1.  FHGAD    Ass
2.  GNa          Ass
3.  FaHaGaAC            1  D elim
4.  HaGaANFNaC      3  contra
5.  HNaGNaOFNaC   4  deM
6.  HNaGnaO              2  O int  (the rule lets us introduce either front or back to what's already there)
7.  FNa                         5,6  C elim

Example 2.
Symbolize and provide a derivation for this argument.
Everyone who is either friendly or happy greets people.  Everyone is friendly.  Therefore, there is someone who greets people.
FHOGD, FV |- GW
note: to work with derivations such as these, we have to move from the quantifiers to some particular example (we say that we are instantiating), then go back to the quantifier (we say we are generalizing).
1.  FHOGD     Ass
2.  FV              Ass
3.  FaHaOGaC            1  D elim
4.  Fa                            2  V elim
5.  FaHaO                    4  O int
6.  Ga                           3,5  C elim
7.  GW                          6  W int

Example 3.
Symbolize and provide a derivation for this argument.
Nobody who is unhappy is friendly.  Some who are unhappy do greet people.  Therefore, there are those who are unfriendly who do greet people.
HNFND, HNGE |- FNGE
1.  HNFND    Ass
2.  HNGE      Ass
3.  HNaGaA                 2  E elim
So that we are not trapped by the new name rule, we eliminate the existential quantifiers first.
4.  HNaFNaC               1  D elim
5.  HNa                         3   A elim
6.  FNa                          4,5  C elim
7.  Ga                            3  A elim
8.  FNaGaA                  6,7  A int
9.  FNGE                      8  E int

Example 4.
Symbolize and provide a derivation for this argument.
Everyone who is unfriendly is unhappy.  All who are friendly greet people.  Therefore, everyone who is happy greets people.
FNHND, FGD |- HGD
1.  FNHND     Ass
2.  FGD           Ass
3.  FNaHNaC                1   D elim
4.  FaGaC                      2   D elim
5.  HaFaC                      3   contra
6.          HGDN             Hyp
7.          HGNE               6  QN
8.          HaGNaA           7  E elim
9.          Ha                      8  A elim
10.        Fa                       5,9  C elim
11.        Ga                      4,10  C elim
12.        GNa                   8  A elim
13.        GaGNaA           11,12  A int
This is the critical move in an indirect proof.  It shows that the hypothesis HGDN has to be false, since this is the only way that there could be an obviously false statement in a legitimate proof.Again this plays off our understanding of a valid argument: when all the premises are true, the conclusion cannot be false.  Thus, we can be sure in a valid argument that there has to be at least one false premise when there is a false conclusion.
14.  HGD                        6-13  hyp elim
 

Go on to assignment 6.
Please note that there will be a review quiz before the assignment that you submit.