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Philosophy

Timeline for Relationship between Structural Contraction and the Absorption theorem?

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May 4 at 13:51 comment added lee pappas @concusedcius, I have added a proof of the theorem that G,A,A; B iff G, A and A; B. The proof uses the systin the link you posted. Here I have used a semicolon as the turn style.
May 4 at 13:39 history edited lee pappas CC BY-SA 4.0
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May 4 at 5:52 comment added confusedcius Sorry for being unclear, I’ve updated the question so hopefully that’s not a problem now. Essentially, it’s not surprising that absorption theorem holds of the material conditional. But I’m wondering about its relation to structural contraction, which has a specific form that is not used in your answer.
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May 4 at 0:09 comment added confusedcius When I said the structural rules, I mean the left and right contraction rule (see here). For example, the left rule is: Given Γ, A, A ⊢ Δ, derive Γ, A ⊢ Δ. This corresponds to something like the (sometimes implicit) Reiteration rule in natural deduction (and idk what the right rule corresponds to). The rules you stated need to be translated to sequent calculus, so for example, the conjunction rule would be the right conjunction rule. So you can't go from Γ, A, A ⊢ Δ to Γ, A∧A ⊢ Δ (not in the system I linked anyways).
May 4 at 0:09 comment added confusedcius What I mean by deny is that the rules are invalid. I had substructural logics in mind, where these logics simply dont allow some structural rules. But I was wondering if the contraction theorem is provable in any such logic that dont allow either the left or right contraction rule.
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S May 3 at 23:30 review First answers
May 6 at 20:21
S May 3 at 23:30 history answered lee pappas CC BY-SA 4.0
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