In a lot of introductions to Landau-Ginzburg theory, which gives the partition function in the form of a functional integral $$\mathcal{Z}[F]=\int \mathcal{D}\phi e^{-\beta F(\phi)}$$ it is said that a "locality" condition can be invoked to write $F$ as $$F(\phi)=\int d^nx f(\phi,\nabla \phi,...)$$
where $f$ can depend on the field and its derivatives. Why is this called locality? It is said that enforcing this guarantees that, for example, in the Ising model, only nearest neighbours affect each other directly, but I can't see why.
