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 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:

  1. 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.)
  2. The relationship 'I' is symmetric: '(^x)(^y)(Ixy>Iyx)' is logially true. (So, if 'Iab' is true, then so is 'Iba'.)
  3. The relationship 'I' is transitive: '(^x)(^y)(^z)[(Ixy&Iyz)>Ixz)]' is logically true.
  4. 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=1 and in general, n=n.
  2. If n=m, then m=n.
  3. If n=m and m=l, clearly all three names are for the same object and so n=l.
  4. 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

  1. ~Isj
  2. ~Ijs
  3. ~Ijj