The semantics of relational logic is not
by itself to tell us which terms are equal
and which are not. In fact, it's possible
for every term to refer to a distinct
object in the real world and it's possible
for every term to refer to the same
object. Well, the semantics relational
logic does not constrain your quality
relation. The idea of coreferentiality
does. For example, it's not possible for
us to believe a=b and b=c and at the same
time, not believe a=c. We can capture
these constraints through axioms. First of
all the equality relation must be
reflexive. This means that the relation
holds of every term in the language for
itself for all x, x = x. Relation must
also be symmetric. If two terms refer to
the same thing, it does not matter which
one we write in the equation, for all x is
equals y implies y=x. Finally the relation
must be transitive. If we believe that a =
b refer to the same object and we believe
that b and c refer to the same object then
a and c must refer to the same object as
well. Let's see how we can use these
properties to solve some problems of
equality. Suppose we know that b = a and
we know that b = c. Let's prove that a = c
as well. As usual we start our proof with
our premises. B = a and b = c. We add our
axioms for equality First reflexivity then
symmetry and then transitivity. Now we go
to work on the proof. First, we use two
applications of universal elimination on
our symmetry axiom to derive the fact that
b = a implies a = b. I'm substituting b
for x and a for y. We then use implication
elimination on line six and line one to
produce a=b. We then use universal
elimination again to instantiate the
transitivity axiom this time with x
replaced by a, y replaced by b and z
replaced by c. We can join the result on
line seven with the premise on line two
to, to derive the conjunction on line
nine. And finally we use implication
elimination to derive our overall
conclusion so it works as expected though
it's a bit lengthy. We'll see a so much
faster way to solve problems like this in
j ust a short while. This exercise test,
of understanding of equality by asking you
to prove the results, using the basic
equality axioms.