T9.1 4 of 4

Further Uses of the Identity Relation

Here's another case requiring the identity relation for symbolization:

(9) All seniors except Chris are female.

Unfortunately, (9) is a bit ambiguous. It could mean

(9a) Chris is a senior non-female but every
*other* senior is female.

or it might be seen to merely say:

(9b) All seniors distinct from Chris are female.

Can you see the difference between (9a) and (9b)? The first entails that Chris isn't female. But (9b) is about the others and so tells us nothing about Chris (Chris might be male or female as far as (9b) is concerned).

Symbolizing may make the difference clearer. Both (9a) and
(9b) require universal statements about "every senior other than
Chris". To be a senior other than Chris is to be a senior distinct
from Chris: 'Sx&~Ixc'. But (9a) says *more*, that Chris is a
senior but not a female: 'Sc&~Fc'. So we get the symbolizations:

To symbolize (9a):

(Sc&~Fc)&(^x)[(Sx&~Ixc)>Fx]

and

To symbolize (9b):

(^x)[(Sx&~Ixc)>Fx]

Notice that only the first of these two says anything about Chris's gender.

* Which is the best way to symbolize (9)?* (9a)
seems a little closer to what most of us would be

Think about the *negation* of "All S except c are F." If we know that *not* all S except c are F, then don't we know that some S other than c is not-F?

If so, then 9a cannot be correct. (9a's negation does not have this consequence. Think it through...)

For these reasons, it would seem that (9) itself means only (9b) and should be symbolized without taking sides about Chris's status. Because this latter understanding of meaning is closer to the modern categorical logic described in the last chapter, the Café quizzes and exercises will assume that

All P except a are Q

may be seen as having hybrid form:

(^x)[(Px&~Ixa)>Qx]

Finally think about

(10) Sam is the only female senior.

We could express the idea of (10) in a number of different
ways. For instance, we could say "Sam is the one female senior in
class" or simply "Sam is *the* senior who is female".

The last of these uses what we have called a definite description,
"the senior who is female", which we can symbolize by a name
if we have one. But in our symbolization
key we do *not* have such a name. So, (10) and the like may be
symbolized as having hybrid form

Sam is a female senior & ~(%x)( x is distinct from Sam & x is a female senior )

which can be symbolized in pure PL as follows:

To symbolize (10):

(Fx&Ss) & ~(%x)(~Ixs&(Fx&Sx))

Whew! These get easier if you do a few on your own. So here's one last little quiz for this tutorial. If you need to use the answers or hints on the first pass through the quiz, you may want to see if you can later give the answers without any help.

Good!

If you needed to use the hints above, you may want to see if you can do these same problems on your own. Otherwise...

More symbolizations...

Next tutorial...

Back to chapter 9