= Solution
\b[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.
Back to article page
