There is a path to a kind of logic that seems to be well-defined but chaotic. The connection seems to be thru Homotopy Type Theory (HOTT) If we look at HoTT we

There is a path to a kind of logic that seems to be well-defined but chaotic. The connection seems to be thru Homotopy Type Theory (HOTT) If we look at HoTT we find a mapping between logic to topology. There is work at CMU by Kristina Sojakova http://reports-archive.adm.cs.cmu.edu/anon/2015/CMU-CS-15-105.pdf http://reports-archive.adm.cs.cmu.edu/anon/2015/CMU-CS-15-10... using logic to prove the equivalence S^1 x S^2 = T^2 (S^1 is a circle, T^2 is a torus) There is work on the relationship between the torus and Stephen Smale's work on strange attractors https://link.springer.com/content/pdf/10.1007/s00209-014-1388-1.pdf https://link.springer.com/content/pdf/10.1007/s00209-014-138... Strange attractors show a natural fractal character for some initial values, leading to chaotic behavior on the torus. https://en.wikipedia.org/wiki/Attractor https://en.wikipedia.org/wiki/Attractor It should be possible to develop a logic based on a torus that was sound but leads to chaotic behavior. That would certainly form a basis for "nonlinear logic". My claim would be that a "nonlinear logic", aka chaotic logic, would be a "higher order logic". In that higher order logic we would find "emergent behavior" which is the kind of behavior that is unpredictable but sound. Mapping from chaotic functions on a torus to a supporting logic would escape deductive logic, and likely provide a logic that is not covered by Godel. The chaotic behavior is sensitive to starting values where values differing by epsilon could be stable or chaotic, I'm not sure how to capture this notion of "nearby values" in a logic. Note that chaotic values don't diverge to infinity but remain "within the logic" but never achieve what might likely be called a "fixed point".