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–179. Russell,_B. (1908)._Mathematical logic as based_on the_theory_of types._American_journal of mathematics_30, 222–262. Sainsbury,_R. M. (2009). Paradoxes_(3rd ed.). Cambridge_University_Press. Schroeder-Heister, P. (1984a). Generalized rules for quantifiers and the completeness of the intuitionistic_operators_∧, ∨,_→,_⊥,_∀, ∃. In E. Börger,_W. Oberschelp, M. M. Richter,_B. Schinzel,_and W. Thomas (Eds.), Computation and Proof Theory,_pp._399– 426. Springer_Press. Schroeder-Heister, P. (1984b). A natural extension of natural_deduction. The Journal of Symbolic Logic_49, 1284–1300. 171_Schroeder-Heister,_P._(2006). Validity concepts in_proof-theoretic semantics. Synthese 148, 525–571. Schroeder-Heister,_P. (2015). Proof-theoretic validity based on_elimination rules. In Haeusler, E. Hermann, W._de Campos Sanz, and B. Lopes_(Eds.), Why is this a_proof?, pp._159–176. London: College Publications. Schroeder-Heister,_P. and L._Tranchini (2017)._Ekman's paradox. Notre_Dame Journal of Formal Logic 58(4),_567–581. Schroeder-Heister, P._and L. Tranchini (2018). How to_ekman_a crabbé-tennant. Synthese._Stålmarck,_G._(1991). Normalization_theorems for full_first_order classical_natural_deduction. The Journal of Symbolic Logic_56,_129–149. Steinberger, F. (2009). Not so stable._Analysis 69(4), 655–661. Steinberger,_F._(2011). Harmony in a_sequent setting: a reply to_tennant. Analysis 71(2), 273–280. Tarski, A._(1936a). The_concept of_truth in formalized languages. In J. Corcoran (Ed.), Logic, Semantics, Metamathematics,_pp. 152–278. Oxford University Press. Tarski,_A. (1936b). The establishment_of scientific_semantics._In J. Cor