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From what I can tell, they are NOT universal or existential quantifiers, as those symbols are also used in the post. I think they are

https://en.wikipedia.org/wiki/Coproduct https://en.wikipedia.org/wiki/Product_(mathematics)

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This is correct. The idea is that you can conjunctively (using the product) or disjunctively (using the coproduct) quantify the variables over the formulas that use them, essentially getting a conjunction/disjuction of the formula, instantiated with each of the concrete values of the variable quantified over.

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