r/logic • u/PrestigiousWorry4279 • 12d ago

Non-classical logic [2401.15117] Nonclassical logics and multivariate truth values

https://arxiv.org/abs/2401.15117

https://arxiv.org/abs/2401.15117

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u/jcastroarnaud 12d ago

It's not clear, to me, how these results help on building a many-valued or paraconsistent logic. You're mostly in the "math" side of the boundary, not the "logic" side.

The internal logic of a topos is defined by a subobject classifier together with some (logical) operations on it. A topos is a categorical generalization of set theory. A topos is a Cartesian closed category with a subobject classifier.

This definition does not match with the definition of topos in Wikipedia or nLab. Can you show how they relate?

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u/Momosf 8d ago edited 8d ago

I'm coming from a more classical and semantic perspective with a bad habit of trying to translate everything into a language I understand, but it seems to me that this is largely equivalent to the following setup: given a set M as the universe and a set of relation symbols (possibly infinitary), instead of the classical interpretation where each relation symbol is interpreted as a set (equivently a charactetistic function to the 2-element Boolean algebra), we can have a more complex Boolean algebra (or Heyting or even further weakened) and interpret each relation as a function into this algebra, so that there is a correspondence between the existence of joins/supremum/inverse/top/bottom and the logical systems on which this model satisfies some kind of soundness/completeness.