Summary: 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 by \textit{M. D'Agostino} et al. [Stud. Log. 108, No. 2, 291--357 (2020; Zbl 1455.03013)] by combining it with some ideas inspired by \textit{J. Hintikka}'s work [Logic, language-games and information. Kantian themes in the philosophy of logic. London: Clarendon Press; Oxford: University Press (1973; Zbl 0253.02005)]. 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 [\textit{M. D'Agostino} and \textit{L. Floridi}, Synthese 167, No. 2, 271--315 (2009; Zbl 1172.03003); \textit{M. D'Agostino} et al., Theor. Comput. Sci. 480, 43--68 (2013; Zbl 1315.03107); \textit{M. D'Agostino}, Theor. Comput. Sci. 606, 79--97 (2015; Zbl 1332.03002)].