REVIEWING QUANTIFIERS

In referring to a group we can either talk about everyone or just about some.  In the first case, as in the expression "everyone is happy," we are going to use a universal quantifier, which we will symbolize with a variable in parenthesis.   Also, we use the letters x and y and z as variables, which must always be "bound" to the quantifier.
        Everyone is happy.   (x)Hx
        Not everyone is happy.  ~(x)Hx
        All are unhappy.  (x)~Hx
        No one is happy.  (x)~Hx  note how we express the concept with a universal quantifier and a negated predicate  

For ideas such as "all students are ambitious" we think in terms of conditionals, as though we are saying "if person x is a student, then person x is ambitious."  What this allows is the idea that there may not in fact be any students at all, since the only way for the conditional to be false is to have the antecedent true (someone is a student) but the consequent false (that same someone is not ambitious).
        Everyone who is ambitious will be successful.   (x)(Ax -> Sx)
        Anyone who works hard is successful.   (x)(Wx -> Sx)
        No one who is lazy is rich.  (x)(Lx -> ~Rx)
        Anyone who is unhappy is not lucky.   (x)(~Hx -> ~Lx)
        Only someone who is lucky is rich.   (x)(Rx -> Lx)  note how we need to express the implicit necessary condition second in the expression
       

When we refer to just some in a group (and in symbolic logic this could be just one individual or any number), we use what is called an existential quantifier.  The original concept was that we were saying, for instance, "there exists a person x such that x is ..." and from the beginning the reversed letter "E" was used.  Again, we will use the normal letter and the parenthesis.  Ordinarily, whenever we symbolize with an existential quantifier we have the connective "&" just as ordinarily we use "->" with universal quantifiers.  Note we do not regard the terms "person" and "individual" as predicates that need to be symbolized.
        Some persons are ambitious.    (Ex)Ax
        There are ambitious students.    (Ex)(Ax & Sx)
        There are rich individuals who are not lucky.   (Ex)(Rx & ~Lx)
        It is wrong to say that some students are not ambitious.  ~(Ex)(Sx & ~Ax)

MORE COMPLEX SYMBOLIZATION

When we work with quantifiers the most basic thing to remember is that ordinarily a universal quantifier sets up a conditional relationship while an existential quantifier sets up a conjunction.  The English sentences may not sound that much different, but what we need to express can be very different.  The reason: we can make a generalization about everyone without it applying to any individual at all, as when I say that all unicorns are white, meaning only that if there is something that is a unicorn then it will be white.  Saying no unicorns are white does not actuallly contradict this, since in the absence of actual unicorns neither statement could be shown to be false.  However, once we say some unicorns are white we are committing ourselves (truthfully or not) to saying there are such things as unicorns.


 All ambitious students work hard.             No ambitious students work hard.
(x)[(Ax & Sx) -> Wx]                               (x)[(Ax & Sx) -> ~Wx]

Some ambitious students work hard.         Some ambitious students do not work hard.
(Ex)[(Ax & Sx) & Wx]                            (Ex)[(Ax & Sx) & ~Wx]

 In working with sentences that involve more than one quantifier the most important thing is to recognize how the ideas are meant to connect so that we know how to use our parentheses and brackets to identify the main connective (and connectives do include the curls and the quantifiers).  Sometimes we can work with equivalent expressions, as in the following, but once the arrow is not the main connective we also need to introduce another variable, regardless of the fact that it will apply to the same individuals.

 If every student is working hard then none of them will fail.         Every student working hard means that none of them will fail.
(x)(Sx -> Wx) -> (x)(Sx -> ~Fx)                                              (x)[(Sx -> Wx) -> (y)(Sy -> ~Fy)]

 Once we start working with expressions involving relationships (typically something with an active transitive verb) we need to be especially careful. 

Every student likes some classes.     Some students do not like any classes.     There are students who dislike some classes.
(x)[(Sx -> (Ey)(Cy & Lxy)]             (Ex)[(Sx & (y)(Cy -> ~Lxy)]                  (Ex)[(Sx & (Ey)(Cy & ~Lxy)]
(x)(Ey)[Sx ->(Cy & Lxy)]          (Ex)(y)[Sx & (Cy -> ~Lxy)]                    (Ex)(Ey)[Sx & (Cy & ~Lxy)]