This topic refers to the course Proof-theoretic harmony (Pistone-Tranchini). Here is a description and a bibliography for addressing it.
Paradoxical expressions are peculiar in that:
(i) their introduction and elimination rules display the kind of harmony characteristic of the logical connectives;
(ii) but the natural deduction systems containing them fail to normalize.
Whereas some authors argued that non-normalizability is the distinctive feature of paradoxical expression, others have questioned this idea on various grounds. Your task is to present the idea of paradox as non-normalizability and the main objections raised against it.
Bibliography
N. Tennant, "Proof and paradox", Dialectica 36(2-3):265-296, 1982
P. Schroeder-Heister and L. Tranchini, "Ekman Paradox", Notre Dame journal of formal logic 58(4):567-581, 2017
M. Petrolo and P. Pistone, "On paradoxes in normal for", Topoi 38(3):605-617, 2019