We are going to introduce a new sign that indicates that one thing is the same as another.


This is the equality sign, which tells us about identity.  By this we mean that a variable is given an explicit name (as in x = a) or that one variable would have the same reference as another ( as in x = y), or that two names refer to the same individual (as in a = b).  We also may negate this statement, but, although this is the convention in the Hurley text, we do not need to use a parenthesis (after all, ~a is not a wff so there will not be any confusion about the scope of the negation).

We can  use this sign in order to say the same individual might have two names, as in Norma Jean Baker was Marilyn Monroe (n=m) or deny it to make it clear that we have two distinct individuals, as in Batman was not Clark Kent  (~b=c).

Identity comes into play above all in expressing numerical quantity.  This is how.

When we say (Ex)Fx & (Ey)Fy we are not necessarily saying that there are two individuals.  We could get across the idea that there are at least two if we have the expression
(Ex)(Ey)(Fx & Fy & ~x = y)
We could get across the idea that there are no more than two if we write
(Ex)(Ey)(Fx & Fy & (z)(Fz -> (z = x v z = y)))
We can make sure we know that there are exactly two individuals if we write
(Ex)(Ey)(Fx & Fy & ~x=y & (z)(Fz -> (z = x v z = y)))
And supposing we want to stress uniqueness, as in saying that at most one individual will have a particular characteristic even though there may be no such individual at all, we could have
(x)(y)((Fx & Fy) -> y=x)

Obviously we do not want to do this when we are discussing more than a very few individuals.

A particular inference involving identity is this:  a = b |- Fa -> Fb   Id 
     careful though: if we start with ~a=b we cannot legitimately infer ~(Fa->Fb)

A replacement rule is this:  a=b :: b=a   Id

What it says is that when two individuals are actually the same, whatever is a characteristic of one is a characteristic of the other.  (How well this really works in expressing our thoughts might be questionable, however.  Clark Kent really is Superman, but we might hesitate to say that whatever is true of Clark--that he wears glasses, let's say--is true of his alter ego.)

One of the particular uses of the identity sign is in expressing what are called definite descriptions.  For instance, suppose we want to say that Alice is the student who won the Booster prize.  We would do it by saying that there is a student who won the Booster prize and there is just one such student and that individual is Alice:
(Ex)(Sx & Wxb & (y)(Wyb -> y=x) &  x=a))

WHAT WE MEAN BY "ONLY"

An important  thing to note is the difference between saying (1) "only chessplayers are athletes"  (which spells out that being a chessplayer is a necessary condition for being an athlete) and statements such as (2) "Alice is the only chessplayer" and (3) "Alice is the only chessplayer who is also an athlete" in which we limit the scope of a term to a specific individual.  (And, of course, do not confuse either usage with what is meant in a statement such as "if only I had studied for the test I would have passed it," in which the word "only" simply intensifies the thought that study was enough for passing.)

(1)  (x)(Ax -> Cx)      (2)  Ca -> (x)(Cx -> x=a)     (3)  Ca & Aa & (x)((Cx & Ax) -> x=a)