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A full set of first-order sentences has existential witnesses among names in its own first-order language: if it proves an existential sentence, it proves some closed-term instance of the matrix. For a deductively closed set this may be stated by membership of the first-order sentences. In a function-free first-order language the witness is a constant symbol. A first-order consistent complete full first-order theory has a canonical term model.

A canonical first-order structure is generated by its names: every element is the value of a closed term. In the function-free setting permitted here, this means every element is the interpretation of a constant symbol. Equivalently, the natural map from closed terms modulo logical equality in the first-order structure onto its carrier is surjective. When canonical is defined specifically using constant names, the same definition and all the arguments below apply in this function-free setting.

A full set of first-order sentences has the witness property in its own first-order language: whenever it proves an existential sentence $\exists x\varphi (x)$, some closed term $t$ satisfies that the set proves $\varphi (t)$. In a deductively closed set, one can equivalently state this using membership of the first-order sentences. In the function-free setting the witness first-order term is a constant symbol. This is the Henkin witness property; fullness does not by itself replace the separate requirements of first-order consistency and completeness.