5.  SYMBOLIZATION TO EXPRESS QUANTIFICATION

Back when Aristotle developed the rules for syllogisms, a key idea was that we were working with generalizations that did not allow for exceptions.  For instance, saying "all men are mortal" (which Aristotle used as an example to show that once we then agreed Socrates was a man it would follw that Socrates too was mortal) means that we are willing to accept that no human being lives forever.  One thing that Aristotle did not think about was how we might talk about an empty category--one that has no members.  Let's say that comic-book superheroes would be an example.  We know what we mean, but we also know that such characters are entirely imaginary.  In Aristotle's logic it is supposed to follow from the statement that "all superheroes are honest" that some superheroes--ones that we choose to name--are honest, and, more important, it cancels out the possibiity of saying that no superheroes are honest.

Modern logic has gone in a different direction.  We might make the generalization "all superheroes are honest" and also make the generalization "no superheroes are honest" without really contradicting ourselves if, in fact, there are no superheroes at at all and so no way of disproving either statement.  In the same way, modern logic does not allow a statement about some individuals being such-and-such to follow from a statement about all individuals being such-and-such.  Instead, we talk about generalizations in two ways: there are universal generalizations (statements about all the members of a category, even if none of them, such as our superheroes, really exist) and there are existential generalizations (statements about at least one individual in a category with the meaning that such an individual does exist). 

We symbolize universal generalizations by using a new "connective"--one of three letters (or variables) inside a parenthesis.  For instance, to express the idea that everyone is a student we would have (x)Sx or (y)Sy or (z)Sz--and the choice of letters at this point really does not matter.  For an existential generalization the usual way is to use the letter "E" printed  backwards, but to make use of a standard keyboard we will just use (Ex)Sx or (Ey)Sy or (Ez)Sz to say either "there are students" or "some individuals are students."   What matters in our understanding is that with an existential generalization we are definitely asserting that there is at least one individual in the category. 

When we want to say something about individuals in the category, it matters whether we are thinking in terms of everyone (or everything) or just some individuals.  For a universal statement such as "all students are ambitious" we are going to think in terms of conditionals: (x)(Sx->Ax).  This can be read as saying "for all x, if x is a student then x is ambitious."  For an existential statement such as "some students are ambitious" we are going to use a conjunction: (Ex)(Sx & Ax).  This can be read as saying "there is an x such that x is both a student and ambitious."  Note that the quantifiers in these last expressions are considered the main connectives, not the symbol inside the parentheses.  Also note that we are not allowing such a thing as an "unbound" variable: we do not use the variables apart from a quantifier.
please note the difference between saying everyone is an ambitious student -- (x)(Sx & Ax) -- and saying every student is ambitious  -- (x)(Sx -> Ax)

What this does is allow for opposite statements about non-existent individuals to be seen as equally "true."   For instance, we can have (x)(Sx->Hx) saying that all superheroes are honest and (x)(Sx->~Hx) saying that none of them are.  What we cannot have is (x)(Sx->Hx) and (Ex)(Sx & ~Hx) as both true.  If we mean to deny the statement that all superheroes are honest then we would just write ~(x)(Sx->Hx), and here the curl is the main connective.

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)
        red flag 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)

EXERCISES   (try these on your own, then see the answer key)         


red flagbe thinking about what is the main connective each time; in the examples below note how it changes:

Alice is a good student.  Sa & Ga
Someone is a good student.   (Ex)(Sx & Gx)
Not everyone is a good student.   ~(x)(Sx -> Gx)

Symbolize each of the following.

1.  Everyone is a good student.
2.  Everyone is ambitious and works hard.
3.  Everyone who is a good student is ambitious and works hard.
4.  There are good students.
5.  There are students who work hard.
6.  Only ambitious students do well.
7.  There are no lazy students.
8.  If there are good students then all the teachers are happy.
9.  Some students are lazy but everyone graduates.
10.  Someone who is lazy will not be a success.
11.  Not all students work hard although they are all ambitious.
12.  There is someone who is not ambitious but only those who are ambitious will graduate.
13.  If some do not graduate then there will be unhappy teachers.
14.  All freshmen and sophomores are ambitious.   Think what this really is saying; do we mean someone has to be both to meet the condition?
15.  Only seniors will graduate but not all seniors will graduate.

go to the answer key

Remember to keep thinking in terms of what is the main connective in the expression you are building.  Sometimes it is the quantifier itself, but other times you may be connecting two quantified expressions.  This is why it is crucial that you use your parentheses correctly.