What would be involved in moving logic beyond FOL?

Carl Hewitt has opined that first-order logic (FOL) should not be regarded as logic of choice for various people such as computer scientists (e.g., he says "First-order theories are entirely inadequate for Computer Science" in a reply to me in Mathematics self-proves its own Consistency (contra Gödel et. al.). He recommends instead his Inconsistency-Robust Logic for mathematicians, computer scientists and others.

What would it take to really change our perspective to be grounded in a logic other than FOL? I think that looking at John Corcoran's First days of a logic course provides the beginnings of an answer (the article treats Aristotelian logic rather than FOL): an alternative logic should provide a natural basis grounding all of the concepts that we expect a new student of logic to grasp, and a rival to the kind of traditional logic course based on first-order logic should be comparably good or better for building on.

What would an introductory logic course look like for, e.g., Inconsistency-Robust Logic? Would it start with Aristotelian logic or would it start with something else? What would having taken such a course be good for, as a prerequisite?