Every presentation of linear logic I've seen seems to either omit or treat quantifiers as an after-thought. Even Girard says that there is "little to say" about them. However, if we view universal (existential) quantification as a generalization of conjunction (disjunction), then we would expect that we would have two of each. For instance, we might expect that there is a $\forall_\otimes$ corresponding to $\otimes$, where $\forall_\otimes x\in A. \varphi(x)$ could be interpreted as saying that you can simultaneously have $\varphi(a)$ for all $a\in A$; and a $\forall_\&$ corresponding to $\&$, where $\forall_\& x\in A. \varphi(x)$ could be interpreted as saying that you may pick any one $a\in A$ and obtain $\varphi(a)$. I'm wondering if there is something wrong with my thinking, or some good reason for why quantifiers are boring in linear logic.
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3I think the usual $\forall$ quantifier in linear logic is your $\forall_&$. It seems like it would be very hard to give rules for your $\forall_\otimes$, since $A$ might be infinite or of unknown cardinality. – Mike Shulman Aug 07 '17 at 17:40
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3A somewhat different approach to a "multiplicative quantifier" can be found in "bunched implication" (https://ncatlab.org/nlab/show/bunched+logic). – Mike Shulman Aug 07 '17 at 17:40