Logical systems where both the order and multiplicity of premises matter In most logical systems, the order and multiplicity of premises do not matter. For example, the set $\{A,A\}$ has the same logical consequences as the set $\{A\}$, and the set $\{A,B\}$ has the same logical consequences as the set $\{B,A\}$. However, has anyone defined and studied a logical system where the order and multiplicity of premises matter? For example, consider a binary connective $f$. There is a rule that states that from the sequence $(A,B)$, we can infer $f(A,B)$, but we cannot always infer $f(A,B)$ from the sequence $(B,A)$. Also, we cannot always infer $f(A,A)$ from the sequence $(A)$, but we can infer it from the sequence $(A,A)$. That is just one example of a connective that depends on both order and multiplicity of premises. Anyway, has logical systems like that been studied in the mathematical literature?
 A: It seems like "noncommutative logic" really was the right keyword to google. These are substructural logics where we miss out the structural axioms

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*Weakening, which says we can add ~bonus assumptions~:

$$\frac{\Gamma \vdash \varphi}{\Gamma, A \vdash \varphi}$$

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*Contraction, which says we can ignore duplicate assumptions:

$$\frac{\Gamma, A, A \vdash \varphi}{\Gamma, A \vdash \varphi}$$

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*Exchange, which says we can permute our assumptions freely:

$$\frac{\Gamma_1, A, B, \Gamma_2 \vdash \varphi}{\Gamma_1, B, A, \Gamma_2 \vdash \varphi}$$
Following Andreas's comment, you can find "cyclic linear logic" discussed on the wikipedia page for noncommutative logic. You can also find a nice paper about cyclic linear logic here.
Additionally, here is a set of notes about linear logic, with applications to programming language theory. Chapter $6$ in particular is about noncommutative linear logic, and this set of notes also has a nice bibliography where you can find more.

I hope this helps ^_^
