Suppose we want to say that every student reads books. This would be easy enough if we let the letter "R" represent the idea of reading books: (x)(Sx -> Rx)
But we would have a problem if we want to make a distinction between reading just some books and all the books there are. The way we get around it is by using more than one quantifier and also using two-place (dyadic) variables. Now we can use three predicates: S for "student" and R for the relationship of "someone reading something" and B for "book."
Let's look at how we can begin to build these more complicated stories:
Every student reads some books: (x)(Sx -> (Ey)(By & Rxy)) as though we had said "For everyone we could name (any "x"), if that person is a student then there are some objects (some "y's") such that they are books and person x is reading them."
What we are
doing always is thinking of
statements
about everyone as essentially hypothetical (using the symbol for
implication)
and statements about a limited number as existential (using the symbol
for conjunction). We do need two distinct variables (we cannot
risk
confusing ourselves). We could certainly have grouped the
quantifiers
together:
(x)(Ey)(Sx -> (By& Rxy))
We usually will not start this way so that
we minimize the confusion about the references for the variables x
and y.
In some cases we have no choice, as when we
might have the statement that only logic students understand every
proof.
(x)(y)((Py & Uxy) -> Lx)
The reason is that we cannot use a variable
(such as "y") without first seeing the quantifier that binds it.
Here are some useful models or templates for use in your symbolization.
Every
student reads some books. (x)(Sx
-> (Ey)(By & Rxy))
Every student reads every book. (x)(Sx -> (y)(By -> Rxy))
Some students read every book. (Ex)(Sx & (y)(By -> Rxy))
There are students who read a few
books. (Ex)(Sx &
(Ey)(By & Rxy))
No student reads every book. (x)(Sx
-> ~(y)(By -> Rxy))
Not all students read every book.
~(x)(Sx -> (y)(By -> Rxy))
Some students read no books. (Ex)(Sx
& (y)(By -> ~Rxy))
How do we use names in these expressions? Here are some examples.
Jack likes
every student. (x)(Sx
-> Ljx)
Every student likes Jack. (x)(Sx
-> Lxj)
Some students do not like Jack. (Ex)(Sx & ~Lxj)
There are students that Jack does not like. (Ex)(Sx & ~Ljx)