"Symbolic learning" is simply "machine learning" (automatically learn function x -> y given examples of (x, y) pairs), but where the substrate is symbolic, i.e. the functions you learn are code-like, not curve-like.

The term is in opposition to "parametric learning" or "curve-fitting".

Symbolic learning does not mean that the system avoids numbers or probabilities. It means its learned representation are discrete, explicit, parsimonious, code-like.