Axiomatizing the Quote

Abstract

We study reflection in the Lambda Calculus from an axiomatic point of view. Specifically, we consider various properties that the quote operator must satisfy as a function on lambda terms.  The most important of these is the existence of a definable left inverse, a so-called evaluator for the quote operator.  Usually the quote operator encodes the syntax of a given term, and the evaluator proceeds by analyzing the syntax and reifying all constructors by their actual meaning in the calculus.  Working in Combinatory Logic, Raymond Smullyan (1994) investigated which elements of the syntax must be accessible via the quote in order for an evaluator to exist.  He asked three specific questions, to which we provide negative answers.  On the positive side, we give a characterization of quotes which possess all of the desired properties, equivalently defined as being equitranslatable with a standard quote.  As an application, we show that Scott's coding is not complete in this sense, but can be slightly modified to be such.  This results in a minimal definition of a complete quoting for Combinatory Logic.