Truth preserving logic is part of a family of axiom systems called Modal logics. In my humble opinion I think that Truth preserving logic isn't the most interesting; I think the most interesting logic is the Proof or Justification preserving logic, and coming in at a close second, the break-even-betting-on-probabilistic-outcomes logic.

Classical logic is the common name for truth preserving logic, Intuitionist logic for justification preserving and Bayesian reasoning for the last one.