Interesting, although you don't seem to explain what exactly symbolic learning/inference is. For those of us that don't know, it would help understand the title.
Symbolic propositional inference using a NN was demonstrated years ago following the work of Eugene Charniak.
That said, the understanding of artificial "intelligence" requires first the unpacking of "intelligence". Humans use both iconic and propositional means to represent and manipulate information. Just in the area of basic numeracy, i.e. arithmetic skills, there is evidence of at least two ways that humans encode quantities and count.
(I am coming from the assumption that biomimetics provides fruitful insights, of course.)
Start from what we mean by symbolic representation, keeping an eye out for over generalizations (mine included!).
Then find computational methods that approximate them. My dissertation lead me to the conclusion that AI will necessitate a hybrid of modules, assuming the biomimetic approach. But that is why I am a Cognitive Scientist.