Chapter Nine, Tutorial Two
Logic with Identity
In the last section, we saw that predicate logic with identity
was just predicate logic with the 2-place 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
- 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
- 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.
"Sam is not the same person as the Judge" may be symbolized as