T9.1 2 of 4

Here's the idea from the last page's end:

"Skipper isn't the same person as William Trent" distinguishes two people.

Similarly, in PL, we may use a negated identity statement, like '~Iab', to distinguish two things. Here the referents of 'a' and 'b' are different if '~Iab' is true.

We say that two things are distinct if they are not identical.

This is actually quite useful, it allows us to use PL to quantify in much more sophisticated ways. We will see a number of examples in this next section:

Numerical Quantification in PL

To have a symbolization key in mind, suppose that our universe of discourse is some logic class including Sam (s) and Chris (c). We will want to symbolize sentences referring to the seniors in class (Sx: x is a senior) and to the females (Fx: is a female). As always in this chapter, we will interpret 'I' as identity:

      Ixy:  x=y

Now think about symbolizing the following.

(1) There are at least two seniors.

We could try to symbolize (1) as:

???       (%x)(%y)(Sx&Sy)

or as

???       (%x)Sx&(%y)Sy

But neither of these is correct! Why not? Because both say that there is something x which is a senior and something y which is a senior. But it's allowed that x and y be the same thing. (Recall our semantics from chapter seven.)

So, to say there are two different things x and y, we need to distinguish x from y and say they are not identical: '~Ixy'.

Hence, we may symbolize (1)

(1) There are at least two seniors.

as:

(%x)(%y)[~Ixy&(Sx&Sy)]

or, equivalently, as

(%x)(%y)[(Sx&Sy)&~Ixy]

How would we symbolize

(2) There is more than one senior.

Answer: the same way as for (1).

How about

(3) There are at least 3 seniors.

or

(4) There are more than 2 seniors.

(3) and (4) mean the same thing: there are three (or more) distinct things in the universe of discourse all of which are seniors:

(%x)(%y)(%z)[((~Ixy&~Iyz)&~Ixz)&((Sx&Sy)&Sz)]

Here we have to write that each of x, y, and z is distinct for the others: '(~Ixy&~Iyz)&~Ixz', then say that each is a senior: '(Sx&Sy)&Sz)'.


You can probably probably guess about symbolizing "There are at least four seniors". Right?


But here's a tricky one:

Which of the following symbolizes

(5) There is at least one senior.

Click on all correct symbolizations of 5:

  1. (%x)Sx
  2. (%x)(Sx&(%y)(~Ixy&~Sy))
  3. (%x)(Sx>Ixx)