@maonu From the abstract. "The new approach encompasses the old one, typically if “I” win, “the world” loses, i.e., wins “NOT”. When logical artifacts are identified with their own rules of production, LOCATIVE phenomenons arise. In particular, one realises that usual logic (including linear logic) is SPIRITUAL, i.e., up to isomorphism. But there is a deeper locative level, with indeed a more regular structure. Typically the usual (additive) conjunction has the value of categorical product in usual logic, and enjoys commutativity, associativity, etc. up to isomorphism."

First, the "linear" in linear logic refers to causality. Not ax+b.

"Spiritual up to isomorphism" is a very winter solstice sort of thing to say. Happy holidays! And don't worry, the days will start getting longer soon, here in the north.

The rest of what he says is the distinction between a #cartesian and a #monoidal product. The latter lacks projections.

Cartesian products represent things you can take apart and put back together. Monoidal products (despite that "disassembly is sometimes the best" -- The Expanse) usually can't be put back together.

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Realising that without a recommendation engine posts can't get anywhere without hashtags, lemme add a bunch now: #categorytheory #Python #diagram #monoidal #GraphicalFormalism #PicturingQuantum #GraphicalLinearAlgebra #LogicalNAND
#CharlesSandersPeirce

@aadmaa It's true that in an inconsistent BINARY system you can prove anything, so it's trivial. Gödel himself developed a 3-valued logic after he wrote his other proofs, because he knew you could create an inconsistent system that is 1) complete, and 2) not trivial. #Binary #logic hates inconsistency because you can prove anything from \( A \wedge \lnot A \), this is known as "explosion". But in a 3-valued logic you can have Both, which is Valid. That is, "NOT FALSE" is different from "TRUE", and you can build a logic that is not only complete, it's a symmetric #monoidal closed #category, a mathematical structure of great generality.

In #enriched #category #theory, where one #considers #categories C enriched in a “#nice” #monoidal category V (generally one where V is #complete, #cocomplete, #closed #symmetric monoidal) there is in #general #no V-enriched #diagonal functor Δ:C→CJ to #speak of. For #example, when V is the category #Ab, there is no #preferred enriched functor J→I along which to #pull #back.

- The #axioms are partly chosen so that the category of H-modules is a rigid #monoidal category.

- A #braided #monoidal category is called symmetric if { \gamma } also satisfies {gamms _{B,A}\circ \gamma _{A,B}=Id} for all pairs of objects { A, B}. In this case action of { \gamma } on an { N}-fold tensor product factors through the symmetric group