Gödel, Escher, Bell: Contextual Semantics of Logical Paradoxes

Quantum physics exhibits various non-classical and paradoxical features. Among them are non-locality and contextuality (e.g. Bell’s theorem or the Einstein-Podolsky-Rosen paradox). Since they are expected to constitute a key resource in quantum computation, several approaches have been proposed to provide high-level expressions for them. In one of these approaches, Abramsky and others use the mathematics of algebraic topology and characterize non-locality and contextuality as the same type of phenomena as M. C. Escher’s impossible figures. This article expands this topological insight and demonstrates that logical paradoxes arising from circular references (of the sort formalized by Gödel’s fixed-point or diagonalization lemma) share the same topological structure as the quantum paradoxes, by reformatting the topological model of contextuality into a semantics of logical paradoxes. This topological semantics indeed provides a unifying perspective from which previous approaches of philosophers and logicians to logical paradoxes can be understood as diverse ways of fine-tuning topologies to model paradoxes.