Chapter Nine, Tutorial Two
Predicate
Logic with Identity
In the last section, we saw that predicate logic with identity
was just predicate logic with the 2place predicate (or, relation symbol)
'I' required to correspond to identity ("=").
Because of this correspondence to identity we
require that interpretations satisfy four constraints:
 The relationship 'I' is reflexive: '(^x)Ixx'
true in any interpretation, i.e., it's a logical truth. The relationship
'I' is required to hold between any object in the universe of discourse and
itself. (That is to say: 'Iaa' is true for whatever object 'a' refers to.)
 The relationship 'I' is symmetric: '(^x)(^y)(Ixy>Iyx)'
is logially true. (So, if 'Iab' is true, then so is 'Iba'.)
 The relationship 'I' is transitive: '(^x)(^y)(^z)[(Ixy&Iyz)>Ixz)]'
is logically true.
 The substitution of one name for another name of the same object preserves
truth. (So, if P and 'Iab' are true then, P(a/b)
is also true.*
It's worth seeing that these constraints must surely hold if 'I' is to count
as identity. For examples, take our usual way of expressing simple truths
of arithmetic:
 1=1 and in general, n=n.
 If n=m, then m=n.
 If n=m and m=l, clearly all three names are for the same object and so
n=l.
 If n>(m+1) and m=l, we can substitute in (replace the 'm' by 'l' in
the first conjunct) to show that n>(l+1).
These four constraints allow us to symbolize in a number of equivalent ways.
Consider:
"Sam is not the same person as the Judge" may be symbolized as
 ~Isj
 ~Ijs
 ~Ijj
