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Morleyization adds a predicate symbol for each first-order formula and an axiom equating that predicate symbol with its defining logical formula. Every original first-order model has a unique expansion. Translating an expanded logical formula back to the original first-order language and then using its named predicate symbol gives quantifier elimination. Completeness of the original first-order theory implies completeness of the expanded first-order theory.

Add a predicate symbol $P_{\theta }$ for each $L$-formula $\theta (\vec{x})$, of arity equal to its displayed free-variable tuple. Let $T^{+}$ consist of $T$ together with all defining axiomsEvery first-order model of $T$ has a unique such expansion, interpreting the new predicate symbols by their defining sets. Conversely the reduct of every first-order model of $T^{+}$ satisfies $T$.

Given an arbitrary $L^{+}$-formula $\psi (\vec{x})$, replace each new atomic predicate symbol by its original $L$-definition, renaming bound variables to avoid capture. This produces an $L$-formula ${\psi }^{-}(\vec{x})$ withThe defining axiom for that particular logical formula now gives an atomic replacement:Thus the expanded first-order theory eliminates quantifiers. For a first-order sentence, the same translation is an $L$-sentence, decided by completeness of $T$; hence every expanded first-order sentence is decided and $T^{+}$ is complete. Empty free-variable tuples can be handled by zero-ary predicate symbols, or by the usual logical truth and falsity constants. This definitional expansion is called Morleyization.