Towards Depth-Bounded Natural Deduction for Classical First-order Logic

In this paper we lay the foundations of a new proof-theory for classical first-order logic that allows for a natural characterization of a notion of inferential depth. The approach we propose here aims towards extending the proof-theoretical framework presented in [6] by combining it with some ideas inspired by Hintikka’s work [18]. Unlike standard natural deduction, in this framework the inference rules that fix the meaning of the logical operators are symmetrical with respect to assent and dissent and do not involve the discharge of formulas. The only discharge rule is a classical dilemma rule whose nested applications provide a sensible measure of inferential depth. The result is a hierarchy of decidable depth-bounded approximations of classical first-order logic that expands the hierarchy of tractable approximations of Boolean logic investigated in [11, 10, 7].