Abstract

Abstract: In this document, we study a 3-polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3-category generated by this 3-polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3-dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4-dimensional rewriting. Outline In the first section of this paper, we give a 2-dimensional translation of the formulas of system SKS, a formal system for propositional classical logic [Brünnler 2004] expressed in the style of the calculus of structures [Guglielmi 2004]. The idea consists in the replacement of formulas by circuit-like objects organized in a 2-polygraph [Burroni 1993]. This construction is formalized in theorem 1.4.16. We proceed to section 2, whose purpose is to translate the proofs of SKS into 3-dimensional objects that form a 3-polygraph. There we note that every inference rule can be interpreted as a directed 3-cell between two circuits. We prove theorem 2.4.3 stating that the 3-polygraph we have built can be equipped with a proof theory which is the same as the SKS one. Section 3 is where the 3-dimensional nature of

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