  If a semantically open language has no constraints on self-reference, one can
prove an absurdity. The argument utilizes co-referring names 'a0' and 'a1', and
the definition of a functional expression 'The reflection of x = y'. The
definition enables a type of self-reference without deploying any semantic
terminology--yet given that a0= a1, the definition implies the that 'a0' =
'a1', which is absurd. In truth, however, 'the reflection of x = y' expresses
an ill-defined function. And since there is a general ban on ill-defined
functions, there is no real cause for concern. Still, the moral would be that
the prohibition on ill-defined functions entails that self-reference cannot be
unconstrained, even in a semantically open language.
