Quantifiers are the way in which we indicate whether we are talking about everyone in a group or just about some in a group.

Earlier you learned how we formed expressions that would associate a predicate with an individual name.  For instance, to say Bob is a student we wrote Sb.  This meant that we worked with the units inside our sentences instead of just with single letters expressing the whole idea of the sentence.

Why do we use a predicate logic at all?  Think of the problem we would have with the oldest syllogism in the world:
                    Every human being is mortal.  Socrates is a human being.  Therefore, Socrates is mortal.
We can see easily enough it works because of how we are linking the individual ideas (or terms), but this could never be shown if we used just letters for the complete statements.  (For instance, H and S, therefore M.)
But we can see how it works if we have a way of expressing the relationship in the first sentence using a variable in place of an actual name.
                    Every human being is mortal.      (x)(Hx -> Mx)
Then we can see that if we talk about Socrates as a specific instance of the generalization we have this formula:
                     [If Socrates is a human being then he is mortal.]         Hs -> Ms
Once we assert that the antecedent is true  (Hs) we can see that the consequent (the conclusion here in the syllogism) has to be true as well (Ms).

For this course we are setting some definite restrictions on what counts as a WFF, and one reason is to get around the problem of working with an ordinary keyboard.  Traditionally the ideas of all and some have been represented with rotated versions of the letters "A" and "E."  We will just use just (x) for a universal quantifier and (Ex) for an existential quantifier.  Do study the following examples carefully.

Imagine we have a specific domain or universe of  discourse, such as all the creatures in a particular yard.  We might say

Everyone is a human being.    (x)Hx
No one is a human being.     (x)~Hx or ~(Ex)Hx
Some are human beings  (the same as saying that there are human beings).    (Ex)Hx
Some are not human beings.    (Ex)~Hx

We might go further and imagine that the yard is shared by quite mortal human beings and angels (which are immortal).

All human beings are mortal.   (x)(Hx -> Mx)
All angels are immortal.   (x)(Ax -> ~Mx) 
Some creatures in the yard are mortal.    (Ex)(Cx & Mx)
Some creatures in the yard are immortal.   (Ex)(Cx & ~Mx)
Only angels are immortal.   (x)(~Mx -> Ax)    (Do you see a parallel with how earlier we learned to work with necessary conditions?)

A key point in symbolization:  universals (talking about everyone) involve thinking in terms of conditionals while particular cases (or existential situations--the reason for using the letter "E" suggesting "there actually exists...") involve thinking in terms of conjunctions.   What is important here is that we can then assert both that all unicorns are white -- (x)(Ux ->Wx) -- and that no unicorns are white -- (x)(Ux -> ~Wx) -- without contradicting ourselves.  Sound strange?  Well,  if there are no unicorns at all, the conditionals are automatically true under truth table rules.    (This is a key difference between modern symbolic logic and the classical logic of syllogisms dating back to the Greeks.)   Another point following from it is that asserting that all unicorns are white does not imply that some unicorns are white (which we understand as saying that there are white unicorns so that I simultaneously claim that there are unicorns and that they are white).

Basic quantifier rules involve (1) the idea that if we make a generalization about everyone, then we can set up a specific instance for any individual we choose, and (2) if we make a statement about any individual we can make the more general statement that there is someone with that characteristic.  We refer to these steps as instantiation and generalization.

(x)Ax |-  Aa   UI   (universal instantiation)
Aa  |-  (Ex)Ax   EG   (existential generalization)

Practice exercises:

Write translations for the following sentences, using the letter "S" to represent students, "A" to represent those who are ambitious and "W" to represent those who work hard.  Also use the first letter of the names (lower case, remember).

Bob is a student but Carol is not.
Everyone is ambitious.
No one is working hard.
There are ambitious students and there are students who do not work hard.
Every student is ambitious.
Some students are not working hard.
If all students are ambitious then Bob is ambitious.
Everyone who works hard is ambitious.
Only those who are ambitious work hard.
Only those who work hard are ambitious.

When you are done click on to check your answers.