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Springer book http://www.springer.com/la/book/9783642128202 "New Structures for Physics" (which contains lot of metaphysics despite the physics in its title) elaborates categorical and monoidal categorical logics that are based on the category theory. Category, as applied to logic, consists from objects (propositions) and arrows from object to object (proofs; given consequence relation among objects). So, the concrete category reflects some world (real or imagined) and the category draws the consequence relation from this world.

The question is - what consequences exists in this world? Can we classify all the possible consequences (e.g. consequences can arise from the inheritace relation among the objects, from the aggregation relation among the objects)? Maybe such classification can uncover the deeper, finer structure of the category for given logics?

Is there branch of philosophy that studies the types of consequences (or the semantical structures in the real world and of the commonsense knowledge more generally)? Maybe phenomenology, which reflects about real world, is the appropriate direction for this?

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    I am familiar with the mathematical category theory and its use in modern physics, but what you are asking is unclear to me. If by "semantical structures in the real world" and "commonsense knowledge" you mean something outside the mathematized hard science those are too vague, informal and diverse to fall under something so schematic. Semantics of natural languages is notoriously difficult and many believe unformalizable at all. I am not sure phenomenology means what you think, maybe try semiotics en.wikipedia.org/wiki/Semiotics or pragmatics en.wikipedia.org/wiki/Pragmatics
    – Conifold
    Oct 23, 2016 at 23:37

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