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
welldefined 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. 