A cautionary note about self-reference

TL;DR

The paper demonstrates that unconstrained self-reference in open languages leads to absurdities, but these arise from ill-defined functions, implying that restrictions on self-reference are justified to avoid such issues.

Contribution

It clarifies that prohibiting ill-defined functions naturally limits unconstrained self-reference in semantically open languages.

Findings

Unrestricted self-reference can cause logical absurdities.

Ill-defined functions are the root of self-reference issues.

Restrictions on ill-defined functions prevent paradoxes.

Abstract

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.