Is there a possible intersection of mathematical logic and machine learning? Why there is so comparatively little work in machine learning using  mathematical logic? Most of the research uses methods in (Bayesian) statistics, linear algebra, numerical analysis and so on. However, almost no use of, for instance, relevant logics is made.

My background is in philosophy, not computer science. But given that logic is in the business of analyzing  the notion of inference I was rather surprised to learn that machine learning and other branches of AI research do not rely on mathematical logic as much as I thought they would. What about non-monotonic logics in drawing inferences or modal logic? It seems to me that it is mostly philosophers and some, dare I say, older people in cs departments that work in logic?

Are the reasons practical (easier implementation of methods from other areas), sociological (machine learning/AI was started by non-logicians), or theoretical (the bulk of machine learning is mostly about categorising and analyzing data, because that is the hard part, and less drawing inferences).

As a closer: Is there any chance that mathematical logic will have greater influence on machine learning in the future?

All the best!
I think the reason why formal logic is not widely used is its strictness. Also, it has quite restricted area of application and requires the domain to be formalized first. In contrast to that, ML methods can be fuzzy, robust to noise and work well for real-life data.

Here is an interesting discussion on the topic:
https://www.quora.com/Whats-the-relationship-between-mathematical-Logic-and-Machine-Learning