You know, discussion of relational logic
thus far, we have assumed that there is a
one to one relationship between ground
terms in our language and objects in the
application area we're trying to describe
For example, we have been assuming a
unique name for each person. In
Arithmetic, we've been assuming unique
term for each number. This is often called
Isomorphic Representation because if the
1-to-1 relationship between ground terms
and objects in the world. This approach
makes things conceptually simple and it's
a reasonably way to go in many
circumstances but not always. In natural
language for example, we often find it
convenient to use more than one term to
refer to the same real world object. For
example, we sometimes have multiple names
for the same person, Michael and Mike And
in elementary arithmetic, we frequently
use different terms to refer to the same
number, two + two, two  two, s of s of s
of s of zero and so forth. In Relational
Logic, we can axiomatize this
coreferentiality of terms in the form or
equations. An equation is where we are
stating that to possibly distinct terms,
referred to the same word object For
example, to express the coreferentiality
of f of (a) and f of (b), we write equal f
of (a), f of (b). We can also distinguish
terms; distinguish terms by writing
negated equations. To say that two terms
referred to different objects, we write
not equal f of (a), f of (b), for example.
So, since the quality of such common
relation and what follows a right
equations with the infix operator equal
sign. For example, writing f(a) = f(b) in
place of equal f(a), f(b). However, this
is just syntactic sugar. We must remember
that as far as our logic is concerned,
syntactically and semantically, an
equation is just a relational sentence
involving a relation constant like any
other. We start this lesson by
axiomatizing the equal relation as an
ordinary binary relation. We then discuss
the substitution of equals for equals in
other expressions. And finally, we talk
about how to expand our pre-syste m for
[inaudible] about equality.