In the deterministic framework, it is not clear what this means.  Either a fact changes the answer, in which case it is relevant, or it doesn’t, in which case it isn’t.  There is no notion of “relevant, but only a little”.  In the probabilistic framework, we have a well-defined semantics.

Furthermore, if we assume that the world is really stochastic, then “weak influence” is a very common state of affairs.  One can show formally that, in many settings, things are only weakly relevant.  This type of analysis has led to the development of a wide range of approximate inference algorithms.