The most common problems with symbolization involve working with universal quantifiers, since anything that can confuse a student working with conditionals also comes in here along with some new issues.
Let's look again at a number of expressions that could be easily mixed up. Try to understand why the order and the grouping appear the way they do. Above all, note the difference between using the letter "A " as part of a quantifier and as a predicate.
Everyone is an ambitious student.
(or All are ambitious students.)
Ax(Ax
& Sx)
Every student is ambitious. Ax(Sx
-> Ax)
Anyone ambitious is a student. Ax(Ax
-> Sx)
All non-students are ambitious. Ax(~Sx
-> Ax)
Anyone who is a student is not ambitious.
(or No students
are ambitious.) Ax(Sx -> ~Ax)
Only students are ambitious. Ax(Ax
-> Sx)
Ambitious students work hard. Ax((Ax
& Sx) -> Wx)
Only ambitious students work hard.
Ax(Wx
-> (Ax & Sx))
Any student who does not work hard is not ambitious.
Ax((Sx & ~Wx) -> ~Ax)
Any non-student who does work hard is ambitious.
Ax((~Sx & Wx) -> Ax)
Working with existential quantifiers brings up a special problem. When we use Ex (or Ey or Ez) as a quantifier we have to remember that the "someone" in one part of an expression is not necessarily the same "someone" in another part of the expression.
There is someone who is ambitious and hardworking.
Ex(Ax & Wx)
There is someone ambitious and there is someone
hardworking. Ex(Ax) & Ex(Wx)
Some who are ambitious are hardworking.
Ex(Ax & Wx)
If some are ambitious then there will be those
who work hard. Ex(Ax) -> Ex(Wx)
(Why should we take for granted that there is a reference to the same persons?
Think about the sentence "If some are sick then there are those who must
be nurses.")
Is it okay to use two different variables in the last example, even though they might in fact refer to the same individuals? Yes, there is no problem with that at all. We could have Ex(Ax) -> Ey(Wy) but what we could not do is use a variable inside the parenthesis that does not match the variable used for the quantifier. Ex(Ay) is not a wff in our system. (Another way of putting this is that we are defining our game in such a way that we have no "free" variables; they are always "bound" by the variable in the quantifier.)