Partial type ($p(\mathbf{x})$)

A partial type over a first-order theory $T$ is a set of first-order formulas in a fixed finite tuple of variables which is jointly consistent with $T$. A tuple realizes it when it satisfies every formula in the set. A complete type additionally decides every formula in those variables.

A partial type is nonprincipal when it is not a principal partial type. Equivalently, for every formula $\theta (\mathbf{x})$ consistent with $T$, some $\psi \in p$ makes $\theta \wedge \neg \psi$ consistent with $T$. This is the exact extension property needed in the Henkin construction proving the omitting types theorem.

A partial type $p(\mathbf{x})$ is principal over $T$ if some formula $\theta (\mathbf{x})$ with $T\cup \{\exists \mathbf{x}\theta \}$ consistent implies each formula of $p$ modulo $T$. For complete types this is the usual notion of an isolated type. For an incomplete type, the isolating formula need not belong to the type itself.