Answer by Carlo Beenakker for Literature about formalization of "natural...
Quantum logic might be one example of "a non-standard way of doing logic motivated by the nature of reality," as described in Quantum Logic in Historical and Philosophical Perspective.
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In "Logic of sheaves of structures", X. Caicedo justifies the logic he introduces stating (more or less) that assertions about a point should really be understood as assertions about a neighbourhood of...
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