We propose a new notion of positivity for topological field theories (TFTs), based on S. Eilenberg’s concept of completeness for semirings. We show that a complete ground semiring, a system of fields on manifolds and a system of action functionals on these fields determine a positive TFT. The main feature of such a theory is a semiring-valued topologically invariant state sum that satisfies a gluing formula. The abstract framework has been carefully designed to cover a wide range of phenomena. We indicate how to employ the framework presented here in constructing a new differential topological invariant that detects exotic smooth structures on spheres.