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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 said point, since punctual properties only exist as limit idealizations of extended properties.

Are there any other pieces of literature in the same spirit that explore non-standard ways of doing logic in the name of philosophical stances about the nature of reasoning or of reality? In particular, are there any of them which turn out to have interesting (or "useful") developments?

(I understand that the question may appear somewhat vague and its interpretation vastly depends on the meaning one assigns to "non-standard" (for instance, fuzzy logic may or may not be subsumed under this category), but I would like to leave it as it is, so that the reader may answer as he pleases, not needing to worry about whether his answer is sufficiently relevant to the question.)

EDIT: for an English introduction to the ideas of the above article: The logic of sheaves, sheaf forcing and the independence of the Continuum Hypothesis by J. Benavides.

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  • $\begingroup$ Related: Logic in mathematics and philosophy. Arguably, pretty much every nonstandard logic has been motivated by some philosophical stance about the nature of reasoning. $\endgroup$
    – Timothy Chow
    Commented Mar 11 at 23:37

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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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    $\begingroup$ Thank you for your answer, this looks very interesting. Also, I have found an article that applies the logic in Caicedo's article to quantum machanics, I'll leave the link here if it can be of some interest: Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics. $\endgroup$
    – user524506
    Commented Mar 11 at 12:08
  • $\begingroup$ (sorry for the grammar, now I cannot edit anymore) $\endgroup$
    – user524506
    Commented Mar 11 at 12:14

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