Past exam of the logic and set theory course of the University of Oxford / 2009 / A / 3 / d / ii

Neither constants alone nor functions alone gives the same uniform adequacy on all nonempty carriers. A one-element set has exactly one interpretation of each function symbol of any finite arity and of every constant symbol. In contrast it has two interpretations of a unary predicate: empty or full. Thus a fixed purely functional or constant-only language has only one structure on that singleton, and cannot encode both truth values of $\exists xP(x)$ by a same-carrier truth-preserving translation with reconstruction. This is the singleton truth-preservation obstruction to expressive completeness, even when logical equality is available.

Constants alone also cannot represent general variable-dependent relations on larger infinite carriers: a fixed formula mentions only finitely many named elements, and its remaining equality tests do not encode an arbitrary binary relation. Functions offer more flexibility. On carriers with at least two elements, predicates can be encoded by characteristic functions taking two distinguished values, and constants can be represented by constant-valued unary functions if explicit constant symbols are forbidden. Such a construction can work after adding axioms for distinct truth markers, but those axioms exclude the singleton case. Enlarging the carrier or adding a Boolean sort is another possible encoding; it changes the same-carrier sense established in the preceding part and must not be silently identified with it.