Quick sanity check: should this logic be powerful enough to prove every true fact about natural number arithmetic? ;)
I'm very pragmatic: If such a proof calculus is able to prove useful things, it serves its purpose. If someone found out that there is a deep connection to, say, natural number arithmetic, I would, of course, be very delighted. I personally think it's much more realistic to expect that connections to rather different mathematical topics will be found first: For example many people have contradictory opinions at the same time. If by such a calculus we had a model with strong predictive power how humans resolve such ambiguities by emotions, this could lead to a leap in developing artificial agents that work in an environment with contradictory data. Or to a development of a theory of error-correcting codes for data that is much more highly structured than words over some alphabet.
But one never knows...