In Linear logic (LL), it is unprovable but when considering the resource interpretation it seems to me that from the resources $A, B$ we can produce the resources $A, B$.
By $A, B \vdash A, B$ I mean $A \otimes B \vdash A \wp B$
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Taroccoesbrocco
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Boris
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Do u mean disjunction by comma on RHS? – LoMaPh Nov 13 '17 at 08:59
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I edited my message – Boris Nov 13 '17 at 10:00
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I think by $\wp$ you're trying to produce ⅋. – Derek Elkins left SE Nov 13 '17 at 10:34
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Yes. But $\wp$ is also used for "par" – Boris Nov 13 '17 at 11:39
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The problem is that $\wp$'s meaning under "resource interpretation" is not clear. – LoMaPh Nov 13 '17 at 20:09
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@LoMahPh What do you think about that interpretation https://math.stackexchange.com/a/332654/344551 ? Is it correct or does it suffer from some problems ? – Boris Nov 14 '17 at 14:32
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According to What is the intuition behind the "par" operator in linear logic? which seems to be a quite coherent and nice interpretation.
Since $A_1, ..., A_n \vdash B_1, ..., B_n$ is $A_1 \otimes ... \otimes A_n \vdash B_1 \wp ... \wp B_n$
- The left part $\vdash$ should be seen as a space of resources we have and can use simultaneously.
- The right part of $\vdash$ should be seen as a space of resources we can't use simultaneously.
When using the monolateral formulation $\vdash A, A^\bot$ of $A \vdash A$ to get $\vdash A_1^\bot \wp ... \wp A_n^\bot, B_1 \wp ... \wp B_n$ we can clearly see that we can't use simultaneously the resources on the right part of $\vdash$. Let's consider that the axiom rule is a way to "trigger" the process of consumption.
Since the resources on the right can't be use simultaneously (they live in parallel), we can't connect the occurrences by consumption so the sequent become unprovable.
Boris
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