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 thinking were we in situation to assert (9). But this may be more a matter of our likely knowledge (in such a situation). And this might make it misleading to assert (9) if Chris were female.* But would (9) be not only misleading but also false if Chris were female? Consider that in some cases one could roughly restate (9) as "All seniors except Chris (about whom I know nothing) are female".

Proof:

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))

Or, to put the point differently, (10) says that Sam is a female senior and that no one except Sam is a female senior.

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.

  1. All females (in class) except Sam are seniors. (Hint)
  2. All except Sam are seniors. (Hint)
  3. All but Chris and Sam are seniors. (Hint)
  4. Chris is the one female in class. (Hint)
  5. The female is a senior. (Hint)
  6. No female except Chris is a senior. (Hint)