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 $\theta\land\neg\psi$ consistent with $T$. This is the exact extension property needed in the <Henkin construction> proving the <omitting types theorem>.


Nonprincipal partial type

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.


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 Incoming links: Principal partial type